Math - Maths - MCQ Test

Question 3 : Euclids division lemma, the general equation can be represented as .......

Question 5 : State whether the following statement is true or false.The following number is irrational $7\sqrt {5}$

Question 6 : What is the HCF of $4x^{3} + 3x^{2}y - 9xy^{2} + 2y^{3}$ and $x^{2} + xy - 2y^{2}$?

Question 7 : State the following statement is True or False 35.251252253...is an irrational number

Question 8 : $2\times 2\times 2\times 3\times 3\times 13 = 2^{3} \times 3^{2} \times 13$ is equal to

Question 9 : Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion$\displaystyle \frac{7}{210}$

Question 11 : Fundamental theorem of arithmetic is also called as ______ Factorization Theorem.

Question 12 : ................. states the possibility of the prime factorization of any natural number is unique. The numbers can be multiplied in any order.

Question 14 : The statement dividend $=$ divisor $\times$ quotient $+$ remainder is called 

Question 15 : Euclid's division lemma states that for two positive integers a and b, there exist unique integers q and r such that $a = bq + r$, where r must satisfy

Question 16 : To get the terminating decimal expansion of a rational number $\dfrac{p}{q}$. if $q = 2^m 5^n$ then m and n must belong to .................

Question 18 : In a division sum the divisor is $12$  times the quotient and  $5$  times the remainder. If the remainder is  $48$  then what is the dividend?

Question 20 : If $a=\sqrt{11}+\sqrt{3}, b =\sqrt{12}+\sqrt{2}, c=\sqrt{6}+\sqrt{4}$, then which of the following holds true ?

Question 21 : For three irrational numbers $p,q$ and $r$ then $p.(q+r)$ can be

Question 22 : Assertion: The denominator of $34.12345$ is of the form $2^n \times 5^m$, where $m, n$ are non-negative integers. Reason: $34.12345$ is a terminating decimal fraction.

Question 23 : According to Euclid's division algorithm, HCF of any two positive integers a and b with a > b is obtained by applying Euclid's division lemma to a and b to find q and r such that $a = bq + r$, where r must satisfy

Question 25 : For finding the greatest common divisor of two given integers. A method based on the division algorithm is used called ............

Question 27 : ............. states that for any two positive integers $a$ and $b$ we can find two whole numbers $q$ and $r$ such that $a = b \times q + r$ where $0 \leq r < b .$

Question 28 : Which of the following irrational number lies between $\dfrac{3}{5}$ and $\dfrac{9}{10}$

Question 30 : Let $x=\dfrac { p }{ q } $ be a rational number, such that the prime factorization of $q$ is of the form $2^n 5^m$, where $n, m$ are non-negative integers. Then $x$ has a decimal expansion which terminates.

Question 32 : State True or False:$4\, - \,5\sqrt 2 $ is irrational if $\sqrt 2 $ is irrational.

Question 33 : Assertion: $\displaystyle \frac{13}{3125}$ is a terminating decimal fraction. Reason: If $q=2^n \cdot 5^m$ where $n, m$ are non-negative integers, then $\displaystyle \frac{p}{q}$ is a terminating decimal fraction.

Question 34 : Using fundamental theorem of Arithmetic find L.C.M. and H.C.F of $816$ and $170$.

Question 36 : H.C.F. of $x^3 -1$ and $x^4 + x^2 + 1$ is

Question 37 : State whether the following statement is True or False. 3.54672 is an irrational number.

Question 39 : The number of possible pairs of number, whose product is 5400 and the HCF is 30 is

Question 40 : The greatest number that will divided $398, 436$ and $542$ leaving $7,11$ and $14$ remainders, respectively, is

Question 43 : We need blocks to build a building. In the same way _______ are basic blocks to form all natural numbers .

Question 45 : The LCM of 54 90 and a third number is 1890 and their HCF is 18 The third number is

Question 47 : State whether the given statement is True or False : $2\sqrt { 3 }-1 $ is an irrational number.

Question 51 : State whether the given statement is True or False : $5-2\sqrt { 3 } $ is an irrational number.

Question 52 : Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion. $\dfrac {29}{343}$

Question 53 : Which of the following irrational number lies between 20 and 21

Question 55 : The H. C. F. of $252$, $324$ and $594$ is ____________.

Question 56 : State whether the given statement is True or False : $4-5\sqrt { 2 } $ is an irrational number.

Question 57 : State whether the given statement is True or False : $\sqrt { 3 } +\sqrt { 4 } $ is an irrational number.

Question 58 : Without actually performing the long division, state whether the following rational number will have terminating decimal expansion or a non-terminating repeating decimal expansion. Also, find the numbers of places of decimals after which the decimal expansion terminates. $\dfrac { 13 }{ 3125 } $

Question 59 : Assuming  that x,y,z  are positive real numbers,simplify the following : $ (\sqrt{x})^{-2/3}\sqrt{y^{4}}\div \sqrt{xy^{-1/2}} $

Question 60 : If the H.C.F. of $A$ and $B$ is $24$ and that of $C$ and $D$ is $56,$ then the H.C.F. of $A, B, C$ and $D$ is

Question 61 : State whether the given statement is True or False : $3+\sqrt { 2 } $ is an irrational number.

Question 62 : Say true or false:A positive integer is of the form $3q + 1,$ $q$  being a natural number, then you write its square in any form other than  $3m + 1$, i.e.,$ 3m $ or $3m + 2$  for some integer $m$.

Question 63 : State whether the following statement is true or not:$7-\sqrt { 2 } $ is irrational.

Question 64 : In algebra $a \times b$ means $ab$, but in arithmetic $3 \times 5$ is

Question 66 : State whether the given statement is True or False : The number $6+\sqrt { 2 } $ is irrational.

Question 67 : In a question on division if four times the divisor is added to the dividend then how will the new remainder change in comparison with the original remainder?

Question 68 :  The square of any positive odd integer for some integer $ m$ is of the form

Question 71 : If we apply Euclid"s division algorithm for $20, 8,$ then the correct answer will be

Question 72 : A number when divided by $114$ leaves the remainder $21.$ If the same number is divided by $19$ the remainder will be

Question 74 : $n$  is a whole number which when divided by  $4$  gives  $3 $ as remainder. What will be the remainder when  $2n$  is divided by $4$ ?

Question 75 : State whether the given statement is True or False :If $p,  q $ are prime positive integers, then $\sqrt { p } +\sqrt { q } $ is an irrational number.

Question 76 : State true or false of the following. The predecessor of a two digit number cannot be a single digit number.

Question 77 : If the square of an odd positive integer can be of the form $6q + 1 $ or  $6q + 3$ for some $ q$ then q belongs to:

Question 78 : Mark the correct alternative of the following. The HCF of $100$ and $101$ is _________.

Question 81 : Use Euclid's division lemma to find the HCF of the following 27727 and 53124

Question 82 : State whether True or False : All the following numbers are irrationals. (i) $\dfrac { 2 }{ \sqrt { 7 }  } $ (ii) $\dfrac { 3 }{ 2\sqrt { 5 }  }$ (iii) $4+\sqrt { 2 } $ (iv) $5\sqrt { 2 } $

Question 85 : The H.C.F. of two expressions is x and their L.C.M is $ \displaystyle x^{3}-9x  $  IF one of the expression is $ \displaystyle x^{2}+3x  $  then,the other expression is 

Question 86 : In a division sum, the divisor is $10$ times the quotient and five times the remainder. What is the dividend, if the remainder is $46?$

Question 87 : If a = 0.1039, then the value of $\sqrt{4a^2-4a+1}+3a$ is :

Question 88 : If HCF of $210$ and $55$ is of the form $(210) (5) + 55 y$, then the value of $y$ is :

Question 89 : The H.C.F of $ 144 $ and $ 198 $ is

Question 90 : According to Euclid's division algorithm, using Euclid's division lemma for any two positive integers $a$ and $b$ with $a > b$ enables us to find the

Question 92 : State true or false of the following. If a and b are natural numbers and $a < b$, than there is a natural number c such that $a < c < b$.

Question 93 : State whether the given statement is true/false:$\sqrt{p} + \sqrt{q}$, is irrational, where p,q are primes.

Question 94 : State whether the given statement is True or False : $2-3\sqrt { 5 }$ is an irrational number.

Question 95 : If these numbers form positive odd integer 6q+1, or 6q+3 or 6q+5 for some q then q belongs to:

Question 96 : The divisor when the quotient, dividend and the remainder are respectively $547, 171282$ and $71$ is equal to 

Question 97 : Use Euclid's division lemma to find the HCF of the following 8068 and 12464

Question 99 :  One and only one out of  $n, n + 4, n + 8, n + 12\  and \ n + 16 $ is ......(where n is any positive integer)

Question 100 : In a division sum a student took  $63$  as divisor instead of  $36$. His answer was  $24$. What is the correct answer? 

Question 101 : When a natural number x is divided by 5, the remainder is 2. When a natural number y is divided by 5, the remainder is 4. The remainder is z when x+y is divided by 5. The value of $\dfrac { 2z-5 }{ 3 } $ is

Question 102 : The sum of two numbers is $2$ and their difference is $1$. Find the numbers.

Question 103 : A choir is singing at a festival. On the first night $12$ choir members were absent so the choir stood in $5$ equal rows. On the second night only $1$ member was absent so the choir stood in $6$ equal rows. The same member of people stood in each row each night. How many members are in the choir?

Question 104 : The unit digit of a number is $x$ and its tenth digit is $y$ then the number will be 

Question 105 : Examine whether the point $(2, 5)$ lies on the graph of the equation $3x\, -\, y\, =\, 1$.

Question 106 : In a zoo there are some pigeons and some rabbits. If their heads are counted these are $300$ and if their legs are counted these are $750$ How many pigeons are there?

Question 108 : The graph of the linear equation $2x -y = 4$ cuts x-axis at

Question 110 : Choose the correct answer which satisfies the linear equation: $2a + 5b = 13$ and $a + 6b = 10$

Question 111 : Let PS be the median of the triangle with vertices $P\left( 2,2 \right), Q\left( 6,-1 \right), R\left( 7,3 \right).$The equation of the line passing through $\left( 1,-1 \right)$and parallel to PS is

Question 112 : The survey of a manufacturing company producing a beverage and snacks was done. It was found that it sells orange drinks at $ $1.07$ and choco chip cookies at $ $0.78$ the maximum. Now, it was found that it had sold $57$ food items in total and earned about $ $45.87 $ of revenue. Find out the equations representing these two. 

Question 113 : A member of these family with positive gradient making an angle of$\frac{\pi }{4}$ with the line3x-4y=2, is

Question 114 : The solution of the equation $2x - 3y = 7$ and $4x - 6y = 20$ is

Question 115 : If $(a, 3)$ is the point lying on the graph of the equation $5x\, +\, 2y\, =\, -4$, then find $a$.

Question 116 : If the system of equation, ${a}^{2}x-ay=1-a$ & $bx+(3-2b)y=3+a$ possesses a unique solution $x=1$, $y=1$ then:

Question 117 : What is the equation of straight line passing through the point (4, 3) and making equal intercepts on the coordinate axes ?

Question 118 : Solve the following equations: $x + \dfrac {4}{y} = 1$, $y + \dfrac {4}{x} = 25$.Then $(x,y)=$

Question 119 : The linear equation $y = 2x + 3$ cuts the $y$-axis at 

Question 120 : State whether the given statement is true or false:The graph of a linear equation in two variables need not be a line.

Question 121 : Assem went to a stationary shop and purchased $3$ pens and $5$ pencils for $Rs.40$. His cousin Manik bought $4$ pencils and $5$ pens for $Rs. 58$. If cost of $1$ pen is $Rs.x$, then which of the following represents the situation algebraically?

Question 124 : If $x + y = 25$ and $\dfrac{100}{x + y} + \dfrac{30}{x - y} = 6$, then the value of $x - y$ is

Question 126 : The value of $k$ for which the system of equations $3x + 5y= 0$ and $kx + 10y = 0$ has a non-zero solution, is ________.

Question 127 : Equation of a straight line passing through the point $(2,3)$ and inclined at an angle of $\tan^{-1}\dfrac{1}{2}$ with the line $y+2x=5$, is:

Question 129 : $\dfrac{1}{3}x - \dfrac{1}{6}y = 4$ $6x - ay = 8$ In the system of equations above, $a$ is a constant. If the system has no solution, what is the value of $a$

Question 132 : The solution of the simultaneous equations $\displaystyle \frac{x}{2}+\frac{y}{3}=4\: \: and\: \: x+y=10 $ is given by

Question 133 : The graph of the lines $x + y = 7$ and $x - y = 3$ meet at the point

Question 134 : State whether the given statement is true or false:Every point on the graph of a linear equation in two variables does not represent a solution of the linear equation.

Question 135 : The values of x and y satisfying the two equation 32x+33y=31, 33x+32y=34 respectively will be

Question 136 : The  linear equation, such that each point on its graph has an ordinate $3$ times its abscissa is $y=mx$. Then the value of $m$ is

Question 138 : Five tables and eight chairs cost Rs. $7350$; three tables and five chairs cost Rs. $4475$. The price of a table is

Question 139 : A line which passes through (5, 6) and (-3. -4) has an equation of

Question 140 : The graph of the line $5x + 3y = 4$ cuts the $y$-axis at the point

Question 141 : For what value of k does the system of equations$\displaystyle 2x+ky=11\:and\:5x-7y=5$ has no solution?

Question 142 : If $2x + y = 5$, then $4x + 2y$ is equal to _________.

Question 143 : Equation of a straight line passing through the origin and making an acute angle with $x-$axis twice the size of the angle made by the line $y=(0.2)\ x$ with the $x-$axis, is:

Question 144 : What is the equationof Y-axis? Hence, find the point of intersection of Y-axis and the line $y\,=\, 3x\, +\, 2$.

Question 145 : Two perpendicular lines are intersecting at $(4,3)$. One meeting coordinate axis at $(4,0)$, find the coordinates of the intersection of other line with the cordinate axes.

Question 146 : The number of pairs of reals (x, y) such that $x =x^2+y^2$ and $y =2xy$ is

Question 147 : If x and y are positive with $x-y=2$ and $xy=24$ , then $ \displaystyle \frac{1}{x}+\frac{1}{y}$   is equal to

Question 148 : What is the nature of the graphs of a system of linear equations with exactly one solution?

Question 149 : What is the equation of the line through (1, 2) so that the segment of the line intercepted between the axes is bisected at this point ?

Question 150 : If (a, 4) lies on the graph of $3x + y = 10$, then the value of a is

Question 151 : Some students are divided into two groups A & B. If $10$ students are sent from A to B, the number in each is the same. But if $20$ students are sent from B to A, the number in A is double the number in B. Find the number of students in each group A & B.

Question 152 : Solve the following pairs of linear (simultaneous) equation by the method of elimination by substitution:$y =\, 4x\, -\, 7$, $16x\, -\, 5y\, =\, 25$

Question 153 : If $\displaystyle \frac{x+y-8}{2} = \frac{x+2y-14}{3}=\frac{3x-y}{4}$, then the values of $x$ and $y$ is

Question 154 : Solve the following pair of linear (simultaneous) equations by the method of elimination: $0.2x+0.1y= 25$ $2\left ( x-2 \right )-1.6y= 116$

Question 155 : Solve the following pair of simultaneous equations:$\displaystyle \frac{8}{x}\, -\, \frac{9}{y}\, =\, 1;\,\frac{10}{x}\, +\, \frac{6}{y}\, =\, 7$

Question 156 : Solve the following pair of simultaneous equations:$\displaystyle\, y\, -\, \frac{3}{x}\, =\, 8\, ;\, 2y\, +\, \frac{7}{x}\, =\, 3$

Question 157 : Solve the following pair of equations by reducing them to a pair of linear equations: $\displaystyle \frac {2}{\sqrt x}+\frac {3}{\sqrt y}=2, \frac {4}{\sqrt x}-\frac {9}{\sqrt y}=-1$

Question 158 : Solve : $\displaystyle \frac{2}{x}+\displaystyle \frac{2}{3y}= \displaystyle \frac{1}{6}$ and $\displaystyle \frac{3}{x}+\displaystyle \frac{2}{y}= 0$.  Hence, find $'m'$ for which $y= mx-4$.

Question 159 : If $2x=t+\sqrt{t^2+4}$ and $3y=t-\sqrt{t^2+4}$ then the value of  $y$ when $x=\dfrac {2}{3}$, is ____.

Question 160 : Find the value of x and y using cross multiplication method: $x-  2y = 1$ and $x + 4y = 6$

Question 161 : Solve the following pair of equations by reducing them to a pair of linear equations: $\dfrac {10}{(x+y)}+\dfrac {2}{(x-y)}=4, \dfrac {15}{(x+y)}-\dfrac {5}{(x-y)}=-2$

Question 163 : The ratio of the present ages of mother and son is $ 12: 5$. The mother's age at the time of the birth of the son was $21$ years. Find their present ages.

Question 164 : If $(3)^{x + y} = 81$ and $(81)^{x - y} = 3$, then the values of $x$ and $y$ are

Question 165 : If$\displaystyle \frac{x+y}{x-y}=\frac{5}{3}\: and\: \frac{x}{\left ( y+2 \right )}=2$ the value of (x , y) is

Question 166 : Solve the following pair of equations by cross multiplication rule. $ax + by + a = 0, bx + ay + b = 0$

Question 167 : For what value of $\alpha$, the system of equations $\alpha x+3y=\alpha-3$ $12x+\alpha y=\alpha$ will have no solution

Question 168 : The simultaneous equations, $\displaystyle y = x + 2|x| $ & $y = 4 + x - |x|$ have the solution set 

Question 169 : Solve the following pair of equations by reducing them to a pair of linear equations: $\dfrac{2x-3y}{xy}=4$ and $\dfrac{15x+3y}{xy}=30$

Question 170 : Solve the following pair of equations :$x\, -\, y\, =\, 0.9$ $\displaystyle \frac{11}{2\, (x\, +\, y)}\, =\, 1$

Question 171 : Solve the following pair of equations: $\displaystyle \frac{a}{x}\, -\, \displaystyle \frac{b}{y}\, =\, 0$ $\displaystyle \frac{ab^{2}}{x}\, +\, \displaystyle \frac{a^{2}b}{y}\, =\, a^{2} \, +\, b^{2}$

Question 172 : Solve: $4x+\displaystyle \frac{6}{y}= 15$ and $6x-\displaystyle \frac{8}{y}= 14$. Hence find the value of $k$, if $y= kx-2$.

Question 173 : Solve the equations using cross multiplication method: $3x + 2y = 10$ and $4x - 2y = 4$

Question 174 : Solve the equations using elimination method: $x - y = 2$ and $-x y = -10$

Question 176 : Find the value of $x$ and $y$ using cross multiplication method: $6x + y = 18$ and $5x + 2y = 22$

Question 177 : Solve the following pairs of equations by reducing them to a pair of linear equations. $\displaystyle \frac{3}{x+1}-\frac{1}{y+1}=2$ and $\dfrac{6}{x+1}-\dfrac{1}{y+1}=5$

Question 178 : Solve the following pair of simultaneous equations:$\displaystyle \frac{1}{x}\, +\, \frac{1}{y}\, =\, 5\,;\, \frac{1}{x}\, -\, \frac{1}{y}\, =\, 1$

Question 179 : If $y=a+\cfrac { b }{ x } $, where $a$ and $b$ are constants, and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, then $a+b$ equals.

Question 180 : Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is $20$ years.Four years ago, the product of their ages in years was $48$.

Question 181 : Find the value of x and y using cross multiplication method: $x + y = 15$ and $x - y = 3$

Question 182 : If $2p + 3q = 18$ and $4p^{2} + 4pq - 3q^{2} - 36 = 0$ then what is $(2p + q)$ equal to?

Question 183 : What is the value of $a$ for the following equation: $3a + 4b = 13$ and $a + 3b = 1$? (Use cross multiplication method).

Question 184 : If $bx+ay=a^2+b^2$ and $ax-by=0$, then the value of $(x-y) $ is

Question 185 : Based on equations reducible to linear equations Solve for x and y $\dfrac {2}{x}+\dfrac {3}{y}=2; \dfrac {1}{x}-\dfrac {1}{2y}=\dfrac {1}{3}$

Question 186 : If the product of two numbers is $10$ and their sum is $7$, which is the greatest of the two numbers?

Question 187 : Solve the following pair of simultaneous equations:$\displaystyle \frac{6}{x}\, -\, \frac{2}{y}\, =\, 1\,;\, \frac{9}{x}\, -\, \frac{6}{y}\,=\, 0$

Question 188 : With Rs. $1000$ a rancher is to buy steers at Rs. $25$ each and cows at Rs. $26$ each. If the number of steers $s$ and the number of cows $c$ are both positive integers, then:

Question 189 : Find the solution of $x$ and $y$ using cross multiplication method: $3x - y = 1$ and $x + 2y = 5$

Question 190 : If $y=a+\dfrac { b }{ x } $, where $a$ and $b$ are constants and if $y=1$ when $x=-1$, and $y=5$ when $x=-5$, what is the value of $a+b$?

Question 191 : If $1$ is added to each of the two certainnumbers, their ratio is $1:2$; and if $5$issubtracted from each of the two numbers, theirratio becomes $5:11$. Find the numbers.

Question 192 : Find the fraction such that it becomes $\displaystyle \frac{1}{2}$ if 1 is added to the numerator, and $\displaystyle \frac{1}{3}$ if 1 is added to the denominator.

Question 193 : Solve the equations using elimination method: $x - 6y = 9$ and $2x - y = 7$

Question 194 : Solve the following pairs of linear (simultaneous) equation by the method of elimination by substitution:$6x = 7y +7$, $7y - x = 8$

Question 195 : The sum of the digits of a two-digit number is 5. The digit obtained by increasing the digit in tens' place by unity is one-eighth of the number. Then the number is

Question 197 : Based on equations reducible to linear equations Solve for x and y: $\dfrac {11}{2x}-\dfrac {9}{2y}=-\dfrac {23}{2}; \dfrac {3}{4x}+\dfrac {7}{15y}=\dfrac {23}{6}$

Question 198 : Solve the equations using elimination method: $x - 4y = -20$ and $4x + 4y = 20$

Question 199 : What is the value of $x$ for the following equations: $x - 5y = 10$ and $x + y =4$? (Use cross multiplication method).

Question 200 : Given that $3p + 2q = 13$ and $3p - 2q = 5$, find the value of $p + q$

Question 201 : Find the value of x and y using cross multiplication method: $3x + 4y = 43$ and $-2x + 3y = 11$

Question 202 : Based on equations reducible to linear equations Solve for x and y: $\dfrac {16}{x+3}+\dfrac {3}{y-2}=5; \dfrac {8}{x+3}-\dfrac {1}{y-2}=0$

Question 203 : A straight line $L$ through the point $(3,-2)$ is inclined at an angle $60^{o}$ to the line $\sqrt{3}x+y=1$. lf $L$ also intersects the $x-$axis, then the equation of $L$ is

Question 204 : Equation of a straight line passing through the point $(2, 3)$ and inclined at an angle of $\tan^{-1} \left(\dfrac{1}{2}\right)$ with the line $y + 2x = 5$ is

Question 205 : Based on equations reducible to linear equations Solve for x and y: $\dfrac {24}{2x+y}-\dfrac {13}{3x+2y}=2; \dfrac {26}{3x+2y}+\dfrac {8}{2x+y}=3$

Question 206 : Equations $\displaystyle \left ( b-c \right )x+\left ( c-a \right )y+\left ( a-b \right )=0$ and $\displaystyle \left ( b^{3}-c^{3} \right )x+\left ( c^{3}-a^{3} \right )y+a^{3}-b^{3}=0$ will represent the same line if

Question 208 : The axes being inclined at an angle of $30^o$, the equation of straight line which makes an angle of $60^o$ with the positive direction of x-axis and x-intercept 2 is

Question 209 : If the equations $y = mx + c$ and $x  \cos  \alpha + y \sin  \alpha = p$ represent the same straight line, then

Question 210 : Based on equations reducible to linear equations, solve for $x$ and $y$: $\dfrac {x-y}{xy}=9; \dfrac {x+y}{xy}=5$

Question 211 : Equation of straight line $ax + by + c= 0$, where $3a + 4b + c = 0$, which is at maximum distance from $(1, -2)$,is

Question 212 : Equations of the two straight lines passing through the point $(3, 2)$ and making an angle of $45 ^ { \circ }$ with the line $x - 2 y = 3$, are

Question 213 : A straight line L through the point $(3, - 2)$ is inclined at an angle of 60$^o$ to the line $\sqrt 3 x + y = 1$. If $L$ also intersects the $x-$axis, then the equation of $L$ is

Question 214 : A line has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through an angle $\alpha $, keeping the origin fixed, the line makes equal intercepts on the coordinate axes, then $\tan$ $\alpha $=

Question 215 : The ratio between the number of passangers travelling by $1^{st}$ and $2^{nd}$ class between the two railway stations is 1 : 50, whereas the ratio of$1^{st}$ and $2^{nd}$ class fares between the same stations is 3 : 1. If on a particular day, Rs. 1325 were collected from the passangers travelling between these stations by these classes, then what was the amount collected from the $2^{nd}$ class passangers ?

Question 216 : The equation of the straight line which passes through $(1, 1)$ and making angle $60^o$ with the line $x+ \sqrt 3y +2 \sqrt 3=0$ is/are.

Question 217 : The equation of the line passing through the point $P(1, 2)$ and cutting the lines $x + y - 5 = 0$ and $2x - y = 7$ at $A$ and $B$ respectively such that the harmonic mean of $PA$ and $PB$ is $10$, is

Question 218 : The sum of three numbers is $92$. The second number is three times the first and the third exceeds the second by $8$. The three numbers are: 

Question 219 : A line perpendicular to the line $\displaystyle 3x-2y=5$ cuts off an intercept $3$ on the positive side of the $x$-axis. Then 

Question 220 : The cost of an article $A$ is $15$% less than that of article $B.$ If their total cost is $2,775\:Rs\:;$ find the cost of each article$.$

Question 221 : The equations of two equal sides of an isosceles triangle are $ 3x + 4y = 5 $and $4x - 3y = 15$. If the third side passes through $(1, 2)$, its equation is

Question 222 : The ratio of the areas of two similar triangles is $25:16$. The ratio of their perimeters is ..............

Question 223 : Triangle A has a base of x and a height of 2x. Triangle B is similar to triangle A, and has a base of 2x. What is the ratio of the area of triangle A to triangle B?

Question 224 : State true or false: The ratio of the areas of two triangles of the same height is equal to the ratio of their bases.

Question 225 : State true or false: The ratio of the areas of two triangles on the same base is equal to the ratio of their heights.

Question 227 :  If the two legs of a right angled $\Delta$ are equal and the square of the hypotenuse is $100,$ then the length of each leg is:

Question 228 : In a right triangle the square of the hypotenuse is equal to twice the product of the legs. One of the acute angles of the triangle is:

Question 229 : If three sides of a right-angled triangle are integers in their lowest form, then one of its sides is always divisible by

Question 230 : Which of the following can be the sides of a right angled triangle ?

Question 231 : In $\triangle{ABC}$, $\angle{B}=90$, $AB=8\:cm$ and $BC=6\:cm$.The length of the median BM is

Question 232 : In $\Delta$ ABC, $\angle B = 90$, AB = 8 cm and BC = 6 cm. The length of the median BM is

Question 233 : We use ........... formula to find the lengths of the right angled triangles.

Question 234 : In the $\triangle LMN$ $\displaystyle $, angle L is $\displaystyle { 65 }^{ o }$ $\displaystyle $, angle M is a right angle, what would be angle N?

Question 235 : A............can never be made up of all odd numbers or two even numbers and one odd number.

Question 236 : Find hypotenuse of right angled triangle if the sides are $12,4\sqrt 3$

Question 237 : A right angled triangle has $24,7cm $ as its sides . What will be its hypotenuse?

Question 238 : Can we construct sets of Pythagorean Triples with all even numbers?

Question 239 :  A Pythagorean Triplet always...............of all even numbers, or two odd numbers and an even number.

Question 240 : It is easy to construct sets of Pythagorean Triples, When m and n are any two ............... integers.

Question 241 : Is it true that a Pythagorean Triple can never be made up of all oddnumbers?

Question 242 : If the measures of sides of a triangle are $(x^2-1) cm, (x^2 +1) cm$, and $2x cm$, then the triangle will be: 

Question 243 : In a $\Delta$ABC, if $AB^2\, =\, BC^2\, +\, AC^2$, then the right angle is at:

Question 244 : The length of the hypotenuse of a right angled $\Delta$ le whose two legs measure 12 cm and 0.35 m is:

Question 246 : Select the correct alternative and write the alphabet of that following : Out of the following which is the Pythagorean triplet ?

Question 247 : If the two legs of a right angled triangle are equal and the square of the hypotenuse is $100cm^2$, then the length of each leg is _________.

Question 248 : A right-angles triangle has hypotenuse $13$ cm, one side is $12$ cm, then the third side is _________.

Question 249 : If the lengths of the sides of a triangle does not satisfy the rule of $\displaystyle { a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }$, then that triangle does not contain a

Question 250 : If the hypotenuse of a right angled triangle is 15 cm and one side of it 6cm less than the hypotenuse, the other side b is equal to.

Question 251 : Which of the following cannot be the sides a right angle triangle?

Question 252 : Given the measures of the sides of the triangle , identify which measures are in the ratio 3 : 4 : 5

Question 253 : In $\Delta ABC,$ if $AB =6\sqrt{3}$ cm, $AC=12$ cm and $BC=6$ cm, then angle B is equal to:

Question 254 :  In a right angle triangle, the hypotenuse is the greatest side. State whether the above statement is true or false.

Question 255 : A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.

Question 256 : A ladder $13m$ long rests against a vertical wall. If the foot of the ladder is $5m$ from the foot of the wall, find the distance of the other end of the ladder from the ground.

Question 258 : The sides of a triangle are given below. Check whether or not the sides form a right-angled triangle.$3cm, 8cm, 6cm$

Question 259 : The hypotenuse of a grassy land in the shape of a right triangle is $1$ meter more than twice the shortest side. If the third side is $7$ meters more than the shortest side, find the sides of the grassy land.

Question 260 : In $\Delta$ ABC, angle C is a right angle, then the value of tan $A + tan B $is

Question 261 : Which of the following numbers form pythagorean triplet?  i) $2, 3, 4$ ii) $6, 8, 10$ iii) $9, 10, 11$ iv) $8, 15, 17$

Question 262 : Which of the following could be the side lengths of a right triangle?

Question 263 : Triangle ABC is right -angled at C. Find BC, If AB = 9 cm and AC = 1 cm. In each case, answer correct to two place of decimal.

Question 264 : The hypotenuse 'c' and one arm 'a' of a right triangle are consecutive integers. The square of the second arm is:

Question 265 : There is a Pythagorean triplet whose one member is $6$ and other member is $10$

Question 266 : In$ \displaystyle \bigtriangleup $ ABC , angle C is a right angle, then the value of$ \displaystyle \tan A+ \tan B is $

Question 267 : The sides of a triangle are given below. Check whether or not the sides form a right-angled triangle.$13cm, 12cm, 5cm$

Question 268 : In $\triangle ABC$, $\angle C={90}^{o}$. If $BC=a, AC=b$ and $AB=c$, find $b$ when $c=13 \ cm$ and $a=5 \ cm$.

Question 269 : In $\triangle ABC$, $\angle C={90}^{o}$. If $BC=a, AC=b$ and $AB=c$, find $a$ when $c=25 \ cm$ and $b=7 \ cm$.

Question 270 : The sides of a triangle are given below. Check whether or not the sides form a right angled triangle.$50cm, 80cm, 100cm$

Question 271 : $4\, RN^{2}\, =\, PQ^{2}\, +\, 4\, PR^{2}$ State whether the above statement is true or false.

Question 272 : The areas of two similar triangles are $121$ cm$^{2}$ and $64$ cm$^{2}$, respectively. If the median of the first triangle is $12.1$ cm, then the corresponding median of the other is:

Question 273 : If in $\triangle $s $ABC$ and $DEF,$ $\angle A=\angle E=37^{\circ}, AB:ED=AC:EF$ and $\angle F=69^{\circ},$ then what is the value of $\angle B\: ?$

Question 274 : $\triangle ABD \sim \triangle DEF$ and the perimeters of $\triangle ABC$ and $\triangle DEF$ are $30 cm$ and $18 cm$ respectively. If $BC = 9 cm$, calculate measure of $EF$.

Question 275 : If in$\displaystyle \triangle ABC$ and$\displaystyle\triangle DEF$,$\displaystyle \frac{AB}{DE}=\frac{BC}{FD}$ then they will be similar if

Question 276 : Two isosceles triangles have equal vertical angles and their areas are in the ratio $16:25$. Find the ratio of their corresponding heights.

Question 277 : Which among the following is/are correct? (I) If the altitudes of two similar triangles are in the ratio $2:1$, then the ratio of their areas is $4 : 1$. (II) $PQ \parallel BC$ and $AP : PB=1:2$. Then, $\dfrac{A(\triangle APQ)}{A(\triangle ABC)}=\dfrac{1}{4}$

Question 278 : The ratio of areas of two similar triangles is $81 : 49$. If the median of the smaller triangle is $4.9\ cm$, what is the median of the other?

Question 279 : D and F are mid-points of sides AB and AC of a triangle ABC. A line through F and parallel to AB meets BC at point E. Find AB (in cm), if EF = 4.8 cm.

Question 280 : The sides of a triangle are in the ratio 4 : 6 : 7. Then

Question 281 : The perimeter of two similar triangles $\triangle ABC$ and $\triangle DEF$ are $36$ cm and $24$ cm respectively. If $DE=10 $ cm, then $AB$ is :

Question 282 : The areas of two similar triangles are $49 \ {cm}^{2}$ and $64 \ {cm}^{2}$ respectively. The ratio of their corresponding sides is:

Question 283 : Two equilateral triangles with side $4 \ cm$ and $6 \ cm$ are _____ triangles.

Question 284 : If $\triangle ABC\sim \triangle QRP,\dfrac{Ar(ABC)}{Ar(QRP)}=\dfrac{9}{4}$,$AB=18\ cm$ and $BC=15\ cm$; then $PR$ is equal to:

Question 285 : ABC is an isosceles triangle right angled at B. Similar triangles ACD and ABE are constructed in sides AC and AB. Find the ratio between the areas of $\triangle ABE$ and $\triangle ACD$.

Question 286 : If $\triangle ABC\sim \triangle  PQR,$  $ \cfrac{ar(ABC)}{ar(PQR)}=\cfrac{9}{4}$,  $AB=18$ $cm$ and $BC=15$ $cm$, then $QR$ is equal to:

Question 287 : Assertion: $\triangle ABC$ and $\triangle DEF$ are two similar triangles such that $BC= 4$ cm, $EF = 5$ cm and $A(\triangle ABC) = 64 \:cm^2$, then $A(\triangle DEF) = 100\:cm^2$. Reason: The areas of two similar triangles are in the ratio of the squares of the corresponding altitudes.

Question 288 : Aline segment DE is drawn parallel to base BC of $\Delta\,ABC$ which cuts ABat point D and AC at point E. If AB = 5 BD and EC = 3.2 cm. Find the length ofAE.

Question 290 : The perimeter of rectangle is $140$ cm. If the sides are in the ratio $3 : 4$, find the lengths of the four sides and the two diagonals

Question 291 : For $\Delta ABC$ & $\Delta PQR$, if $m \angle A = m \angle R$ and $m \angle C = m \angle Q$, then $ABC \leftrightarrow$ ............. is a similarity.

Question 292 : In $\Delta ABC$, a line is drawn parallel to $BC$ to meet sides $AB$ and $AC$ in $D$ and $E$ respectively. If the area of the $\Delta ADE$ is $\dfrac 19$ times area of the $\Delta ABC$, then the value of $\dfrac {AD}{AB}$ is equal to:

Question 293 : In triangle ABC, AD is perpendicular to BC and $AD^{2}\, =\, BD\, \times\, DC.$ Find $\angle BAC$

Question 294 : In $\Delta ABC$, DE is || to BC, meeting AB and AC at D and E. If AD = 3 cm, DB = 2 cm and AE = 2.7 cm, then AC is equal to:

Question 295 : The perimeters of two similar triangles are $24$ cm. and $18$ cm. respectively. If one side of first triangle is $8$ cm., what is the corresponding side of the other triangle.

Question 296 : Two angles of triangle ABC are $\displaystyle 85^{\circ}$ and $\displaystyle 65^{\circ}$ whilethe two angles of another triangle DEF are $\displaystyle 30^{\circ}$ and $\displaystyle 65^{\circ}$.Which of the statements is correct?

Question 297 : If $\Delta ABC \sim \Delta QRP, \displaystyle \frac{ar (ABC)}{ar (PQR)} = \frac{9}{4}, AB = 18 cm$ and $BC=15 cm$; then PR is equal to

Question 298 : Given $\Delta ABC-\Delta PQR$. If $\dfrac{AB}{PQ}=\dfrac{1}{3}$, then find $\dfrac{ar\Delta ABC}{ar\Delta PQR'}$.

Question 299 : In triangle ABC, AB = AC = 8 cm, BC = 4 cm and P is a point in side AC such that AP = 6 cm. Prove that $\Delta\,BPC$ is similar to $\Delta\,ABC$. Also, find the length of BP.

Question 300 : Area of similar triangles are in the ratio $25:36$ then ratio of their similar sides is _________?

Question 301 : In a right triangle, the square of the hypotenuse is $x$ times the sum of the squares of the other two sides. The value of $x$ is:

Question 302 : $ABC$ and $BDE$ are two equilateral triangles such that $D$ is the mid point of $BC$. Ratio of the areas of triangle $ABC$ and $BDE$ is

Question 303 : Through a point $P$ inside the triangle $ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal area. If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is

Question 304 : $\displaystyle \Delta ABC$ and $\displaystyle \Delta DEF$ are two similar triangles such that $\displaystyle \angle A={ 45 }^{ \circ  },\angle E={ 56 }^{ \circ  }$, then $\displaystyle \angle C$ =___.

Question 305 : The sides of a triangle are $3x+4y,\,4x+3y$ and $5x+5y$ units, where $x,y>0$.The triangle is ______________.

Question 306 : The lengths of the sides of a right triangle are $5x + 2$, $5x$ and $3x - 1$. If $x > 0$ then the length of each side is?

Question 307 : In Pythagoras theorem triplets the lengths of the sides of the right angled triangle are in the ratio.

Question 308 : State true or false: In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P, then $\displaystyle \Delta APB$ is similar to $\displaystyle \Delta CPD.$

Question 309 : In a triangle, sum of squares of two sides is equal to the square of the third side.

Question 310 : State true or false: In parallelogram $ ABCD $. $ E $ is the mid-point of $ AB $ and $ AP $ is parallel to $ EC $ which meets $ DC $ at point $ O $ and $ BC $ produced at $ P $. Hence$ BP= 2AD $

Question 311 : Triangles ABC and DEF are similar. If their areas are 64 $cm^2$ and 49 $cm^2$ and if AB is 7 cm, then find the value of DE.

Question 312 : The hypotenuse of a right triangle is $6$ m more than twice the shortest side. If the third side is $2$ m less than the hypotenuse, find the hypotenuse of the triangle.

Question 313 : In $\Delta PQR, \angle Q = 90^{\circ}$ and $QS$ is the median. If $QS = 7.4$ cm, then $PR$ is:

Question 314 : State true or false: In $\triangle ABC$, $\angle A$ is obtuse and $AB= AC$. $P$ is any point in side $BC$. $\displaystyle PM \perp AB$ and $\displaystyle PN \perp AC.$ Then, $\displaystyle PM \times PC= PN \times PB$

Question 315 : D and E are respectively the points on the sides AB and AC of a $\displaystyle \Delta ABC$ such that $AB = 12 cm$, $AD = 8 cm$, $AE = 12 cm$ and $AC = 18 cm$, then

Question 316 : The angles of the $\Delta ABC$ and $\Delta DEF$ are given as follows;$A = 90\displaystyle ^{\circ},\ \ B = 30\displaystyle ^{\circ}, \ \ D = 90\displaystyle ^{\circ},$ and $E = 30\displaystyle ^{\circ}$If the side $BC$ is twice the side $EF$, which of the following statement is true?

Question 317 : The triangle formed by the vertices $A(1,0,1) \quad B(2,-1,4) $ and $C(3,-4,-1)$ is

Question 318 : In two similar triangles ABC and PQR, if their corresponding altitudes AD and Ps are in the ratio 4:9, find the ratio of the areas of $\triangle ABC$ and $\triangle PQR$.

Question 319 : The areas of two similar triangles are $81\ cm^{2}$ and $49\ cm^{2}$. If the altitude of the bigger triangle is $4.5\ cm$, find the corresponding altitude of the smaller triangle.

Question 320 : If two triangles are similar then, ratio of corresponding sides are:

Question 321 : Instead of walking along two adjacent sides of a rectangular field,a boy book a short - cut along the diagonal of the field and saved a distance equal to 1/2 the longer side. The ratio of the shorter side of the rectangle to the longer side was:.

Question 322 : Let $\displaystyle \Delta XYZ$ be right angle triangle with right angle at Z. Let $\displaystyle A_{X}$ denotes the area of the circle with diameter YZ. Let $\displaystyle A_{Y}$ denote the area of the circle with diameter XZ and let $\displaystyle A_{Z}$ denotes the area of the circle diameter XY. Which of the following relations is true?

Question 323 : $\frac{a}{r}$, a, ar are the sides of a triangle. If the triangle is a right angled triangle, then $r^2$ is given by

Question 324 : Match the column. <table class="wysiwyg-table"><tbody><tr><td>1. In $\displaystyle \Delta ABC$ and $\displaystyle \Delta PQR$, $\displaystyle \frac{AB}{PQ}=\frac{AC}{PR},\angle A=\angle P$ </td><td>(a) AA similarity criterion </td></tr><tr><td>2. In $\displaystyle \Delta ABC$ and $\displaystyle \Delta PQR$, $\displaystyle \angle A=\angle P,\angle B=\angle Q$ </td><td>(b) SAS similarity criterion </td></tr><tr><td>3. In $\displaystyle \Delta ABC$ and $\displaystyle \Delta PQR$, $\displaystyle \frac{AB}{PQ}=\frac{AC}{PR}=\frac{BC}{QR}$ $\angle A=\angle P$ </td><td>(c) SSS similarity criterion </td></tr><tr><td>4. In $\displaystyle \Delta ACB,DE||BC$ $\displaystyle \Rightarrow \frac{AD}{BD}=\frac{AE}{CE}$ </td><td>(d) BPT</td></tr></tbody></table>

Question 326 : Simplify:$20(y + 4) (y^2 + 5y + 3) \div 5(y + 4)$

Question 327 : What must be added to $x^3-3x^2-12x + 19$, so that the result is exactly divisible by $x^2 + x-6$?

Question 328 : $\alpha $ and $\beta $ are zeroes of polynomial $x^{2}-2x+1,$ then product of zeroes of a polynomial having zeroes $\dfrac{1}{\alpha }$  and    $\dfrac{1}{\beta }$ is

Question 329 : Factorise the expressions and divide them as directed.$4yz(z^2 + 6z-  16)\div  2y(z + 8)$

Question 330 : State whether true or false:Divide: $4a^2 + 12ab + 91b^2 -25c^2 $ by $ 2a + 3b + 5c $, then the answer is $2a+3b+5c$.

Question 331 : State whether True or False.Divide : $a^2 +7a + 12 $ by $  a + 4 $, then the answer is $a+3$.

Question 332 : If $\alpha$ and $\beta$ are the zeroes of the polynomial $4x^{2} + 3x + 7$, then $\dfrac{1}{\alpha }+\dfrac{1}{\beta }$ is equal to:

Question 333 : Simplify:Find$\ x(x + 1) (x + 2) (x + 3) \div  x(x + 1)$

Question 336 : If $\alpha , \beta$ are the roots of equation $x^2 \, - \, px \, + \, q \, = \, 0,$ then find the equation the roots of which are $\left ( \alpha ^2  \, \beta ^2 \right )  \,  and  \,  \,  \alpha \, + \,\beta $.

Question 338 : If $a\ne 2$, which of the following is equal to $\cfrac { b\left( { a }^{ 2 }-4 \right) }{ ab-2b } $?

Question 341 : If $\alpha , \beta$ are the zeros of the polynomials $f(x) = x^2+x+1 $ then $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=$________.

Question 342 : What is $\dfrac {x^{2} - 3x + 2}{x^{2} - 5x + 6} \div \dfrac {x^{2} - 5x + 4}{x^{2} - 7x + 12}$ equal to

Question 343 : Work out the following divisions.$10y(6y + 21) \div 5(2y + 7)$

Question 344 : Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and their coefficients.$2s^2-(1+2\sqrt 2)s+\sqrt 2$

Question 345 : Find the Quotient and the Remainder when the first polynomial is divided by the second.$-6x^4 + 5x^2 + 111$ by $2x^2+1$

Question 348 : State whether True or False.Divide: $12x^2 + 7xy -12y^2 $ by $ 3x + 4y $, then the answer is $x^4+2x^2+4$.

Question 349 : State whether the following statement is true or false.After dividing $ (9x^{4}+3x^{3}y + 16x^{2}y^{2}) + 24xy^{3} + 32y^{4}$ by $ (3x^{2}+5xy + 4y^{2})$ we get $3x^{2}-4xy + 8y^{2}$

Question 350 : Divide the first expression by the second. Write the quotient and the remainder. $\displaystyle x^2-\frac{1}{4x^2}; x-\frac{1}{2x}$

Question 351 : Find the expression which is equivalent to : $\displaystyle \frac { { x }^{ 3 }+{ x }^{ 2 } }{ { x }^{ 4 }+{ x }^{ 3 } } $?

Question 352 : Factorise the expressions and divide them as directed.$12xy(9x^2-  16y^2)\div  4xy(3x + 4y)$

Question 354 : The product of the roots of the quadratic equation $2x^{2}-8x+3=0$ is

Question 356 : The degree of the remainder is always less than the degree of the divisor.

Question 358 : What must be subtracted from $4x^4 - 2x^3 - 6x^2 + x - 5$, so that the result is exactly divisible by $2x^2 + x - 1$?

Question 360 : Work out the following divisions. $96abc(3a -12) (5b +30)\div  144(a-  4) (b+  6)$

Question 361 : Is $(3x^{2} + 5xy + 4y^{2})$ a factor of $ 9x^{4} + 3x^{3}y + 16x^{2} y^{2} + 24xy^{3}  + 32y^{4}$?

Question 362 : The common quantity that must be added to each term of $a^{2}:b^{2}$ to make itequal to $a:b$ is:

Question 363 : If $P=\dfrac {{x}^{2}-36}{{x}^{2}-49}$ and $Q=\dfrac {x+6}{x+7}$ then the value of $\dfrac {P}{Q}$ is:

Question 365 : Choose the correct answer from the alternatives given. If the expression $2x^2$ + 14x - 15 is divided by (x - 4). then the remainder is

Question 366 : Apply the division algorithm to find the remainder on dividing $p(x) = x^4 -3x^2 + 4x + 5$ by $g(x)= x^2 +1 -x.$

Question 367 : The remainder when$4{a^3} - 12{a^2} + 14a - 3$ is divided by $2a-1$, is

Question 369 : Divide the first expression by the second. Write the quotient and the remainder. $a^2-b^2 ; a-b$

Question 374 : Find the value of a & b, if  $8{x^4} + 14{x^3} - 2{x^2} + ax + b$ is divisible by $4{x^2} + 3x - 2$

Question 375 : When ${ x }^{ 2 }-2x+k$ divided the polynomial ${ x }^{ 2 }-{ 6x }^{ 3 }+16{ x }^{ 2 }-25x+10$ the reminder is (x+a), the value of is

Question 376 : If a and b are the rootsof the quadratic equation $\displaystyle { 6x }^{ 2 }-x-2=0$from an equation whose roots are$\displaystyle { a }^{ 2 }$ and$\displaystyle { b }^{ 2 }$?

Question 377 : A quadratic equation with rational coefficients has both roots real and irrational, ifthe discriminant is

Question 382 : If the equation$\displaystyle{ px }^{ 2 }+2x+p=0$ hastwo distinct roots if.

Question 383 : The condition that one root is twice the other root of the quadratic equation$\displaystyle x^{2}+px+q=0$ is

Question 384 : When $x^3 -2x^2+ ax -b$ is divided by $x^2-2x -3$, the remainder is $x -6$. The values of $a$ and $b$ are respectively :

Question 385 : Let $ p $and $q $be roots of the equation $x^{2}-2 x+A=0 $and let $r $and $s $be the roots of the equation $x^{2}-18 x+B=0 . $If $p<q< $ $r<s $are in arithmetic progression, then the values of $A $and $B $are

Question 386 : The sum of all real roots of the equation ${|x-2|}^2+|x-2|-2=0$ is

Question 387 : $\displaystyle \propto \& \beta $ are the roots of equation $\displaystyle 3{ x }^{ 2 }+3x+3=0$, then $\displaystyle \frac { 1 }{ \propto  } +\frac { 1 }{ \beta  } $

Question 388 : The equation$ \displaystyle \frac{\left ( x+2 \right )\left ( x-5 \right )}{\left ( x-3 \right )\left ( x+6 \right )}= \frac{x-2}{x+4} $ has

Question 389 : $\left[2x\right]-2\left[x\right]=\lambda$ where $\left[.\right]$ represents greatest integer function and $\left\{.\right\}$ represents fractional part of a real number then 

Question 390 : The equation $(\cos {p}- 1) {x}^{2}+\cos {p}x+\sin {p} =0$, in the variable ${x}$ has real roots. Then ${p}$ can take any value in the interval

Question 391 : If $3{p}^{2}=5p+2$ and $3{q}^{2}=5q+2$, where $p\ne q$, then $pq$ is equal to

Question 393 : If $\alpha, \beta$ are the root of quadratic equation $ax^2+bx+c=0$,then $\displaystyle \left ( a\alpha +b \right )^{-3}+\left ( a\beta +b \right )^{-3}=$

Question 394 : Workout the following divisions $a(a + 1) (a + 2) (a + 3) \div a(a + 3)$

Question 395 : If $\displaystyle \left ( 14x^{2}+13x-15 \right )$ is divided by $\displaystyle \left ( 7x-4 \right )$, the degree of the remainder is

Question 396 : On dividing $x^3-3x^2+x+2$ by polynomial $g(x)$, the quotient and remainder were $x -2$ and $4 - 2x$ respectively, then $g(x)$ is

Question 397 : The condition that the equation $x^2 + px + q = 0$, whose one root is the cube of the other root is :

Question 398 : Quadratic polynomial having sum of it's zeros is 5 and product of it's zeros is - 14 is

Question 399 : Simplify: $\displaystyle \frac { 36ab\left( a+2 \right) \left( a+3 \right)  }{ 12a\left( a+3 \right)  } $

Question 400 : Using long division method, divide the polynomial$4p^3-4p^2+6p -\displaystyle \frac{5}{2}$ by $2p-1$

Question 401 : If the roots of the equation $x^2 + 2bx + c = 0$ are $\alpha$ and $\beta$, then $b^2 - c$ is equal to

Question 402 : The simplified form of the expression given below is :-$\eqalign{& \underline {{y^4} - {x^4}} - \underline {{y^3}} \cr & \dfrac{{x\left( {x + y} \right)\;x}}{{{y^2} - xy + {x^2}}} \cr} $

Question 403 : If $\displaystyle \alpha$ and$\displaystyle \beta$ are roots of$\displaystyle { x }^{ 2 }-2x-1=0$, find the value of$\displaystyle { a }^{ 2 }\beta +{ \beta }^{ 2 }\alpha$.

Question 404 : Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing $f(x) =10x^4 +17x^3-62x^2+30x -3$ by $g(x) =2x^2-x+1$

Question 405 : What must be added to $f(x)=4x^4+2x^3+2x^2+x-1$ so that the resulting polynomial is divisible by $g(x)=x^2+2x-3$

Question 406 : When $(x^{3} - x^{2} - 5x - 3)$ is divided by $(x - 3)$, the remainder is

Question 407 : Workout the following divisions $11a^3b^3(7c - 35) \div 3a^2b^2 (c - 5)$

Question 408 : If $\alpha, \beta$ be two zeroes of the quadratic polynomial $ax^2 + bx - c = 0$, then $\displaystyle \frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} =$_______.

Question 409 : If $p$ and $q$ are the roots of the equation $ax^2 +bx +c =0$, then the value of $\dfrac {p}{q}+\dfrac {q}{p}$ is

Question 410 : If ${(5{x}^{2}+14x+2)}^{2}-{(4{x}^{2}-5x+7)}^{2}$ is divided by ${x}^{2}+x+1$, then the quotient $q$ and the remainder $r$ are given by:

Question 412 : If $p, q$ are the distinct roots of the equation $x^2 + px + q = 0$, then

Question 413 : Divide $\displaystyle 4{ x }^{ 2 }{ y }^{ 2 }\left( 6x-24 \right) \div 4xy\left( x-4 \right) $

Question 415 : Assertion: If $ p x^{2}+q x+r=0 $ is a quadratic equation $ (p, q, r \in R $ ) such that its roots are $ \alpha, \beta $ and $ p+q+r<0, p-q +r<0 $ and $ r>0, $ then $ [\alpha]+[\beta]=-1, $ where [.] denotes greatest integer function. Reason: If for any two real numbers $ a $ and $ b $, function $ f(x) $ is such that $ f(a) f(b)<0 \Rightarrow f(x) $ has at least one real root lying in $ (a, b) $

Question 416 : If the roots of the equation, $ax^2+bx+c=0$, are of the form $\alpha / (\alpha -1)$ and $(\alpha +1)/\alpha$, then the value of $(a+b+c)^2$ is

Question 417 : If equation $\displaystyle p{ x }^{ 2 }+9x+3=0$ has real roots, then find value of $p$.

Question 418 : If the polynomial $3x^2-x^3-3x+5$ is divided by another polynomial $x-1-x^2$, the remainder comes out to be $3$, then quotient polynomial is

Question 419 : The roots of the equation $x^2 + kx -12=0$ will differ by unity only, when

Question 420 : $\alpha $ and $\beta $ are the roots of ${ x }^{ 2 }+2x+C=0$. If ${ \alpha  }^{ 3 }+{ \beta  }^{ 3 }=4$, then the value of $C$ is

Question 421 : Write whether the following statement is true or false. Justify your answer.A quadratic equation with integral coefficients has integral roots.

Question 422 : If $\displaystyle \alpha ,\beta$are the roots of the quadratic equation$\displaystyle { x }^{ 2 }-8x+p=0$, find the value of p if$\displaystyle { \alpha }^{ 2 }+{ \beta }^{ 2 }=40$.

Question 423 : If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $ax^2+bx+c$, then $\alpha + \beta=\dfrac {-b}{a}$ & $\alpha \beta=\dfrac {c}{a}$.

Question 425 : If $\cos{\cfrac{\pi}{7}},\cos{\cfrac{3\pi}{7}},\cos{\cfrac{5\pi}{7}}$ are the roots of the equation $8{x}^{3}-4{x}^{2}-4x+1=0$ The value of $\sec{\cfrac{\pi}{7}}+\sec{\cfrac{3\pi}{7}}+\sec{\cfrac{5\pi}{7}}=$

Question 426 : Divide $\displaystyle 10{ a }^{ 2 }{ b }^{ 2 }\left( 5x-25 \right)$ by $15ab\left( x-5 \right) $

Question 427 : Evaluate: $\displaystyle \frac { 35\left( x-3 \right) \left( { x }^{ 2 }+2x+4 \right)  }{ 7\left( x-3 \right)  } $

Question 428 : If $\alpha,\beta$ are the roots of $ { x }^{ 2 }+px+q=0$, and $\gamma,\delta$ are the roots of  $ { x }^{ 2 }+rx+s=0$, evaluate $ \left( \alpha -\gamma  \right) \left( \alpha -\delta  \right) \left( \beta -\gamma  \right) \left( \beta -\delta  \right) $ in terms of $p,q,r$ and $s$.

Question 429 : The number of integers $n$ for which $3x^3-25x+n=0$ has three real roots is$?$

Question 430 : Find the value of p for which the given equation has real roots. $\displaystyle8p{ x }^{ 2 }-9x+3=0$

Question 431 : $\displaystyle \frac{x^{-1}}{x^{-1} + y^{-1}} + \frac{x^{-1}}{x^{-1} - y^{-1}}$ is equal to

Question 432 : Total number of polynomials of the form ${ x }^{ 3 }+a{ x }^{ 2 }+bx+c$ that are divisible by ${ x }^{ 2 }+1$, where $a,b,c\in \left\{ 1,2,3,......10 \right\} $ is equal to

Question 433 : State the following statement is True or False The zeros of the polynomial $(x - 2) (x^{2} + 4x + 3)$ are $2,-1 and -3$

Question 434 : Let $f(x)=2{ x }^{ 2 }+5x+1$. If we write $f(x)$ as $f(x)=a(x+1)(x-2)+b(x-2)(x-1)+c(x-1)(x+1)$ for real numbers $a,b,c$ then

Question 436 : If $\alpha$ and $\beta$ are the roots of the equation $ \displaystyle 5x^{2}-x-2=0, $  then the equation for which roots are $ \displaystyle \dfrac{2}{\alpha }$ and $\dfrac{2}{\beta } $ is

Question 437 : The difference of the roots of$\displaystyle 2y^{2}-ky+16=0$ is 1/3 Find k

Question 438 : Divide $\displaystyle x\left( x+1 \right) \left( x+2 \right) \left( x+3 \right)$ by $\left( x+3 \right) \left( x+2 \right) $

Question 439 : Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $

Question 440 : $x_1$ and $x_2$ are the real roots of $ax^2+bx+c=0$ and $x_1x_2 < 0$. The roots of $x_1(x-x_2)^2+x_2(x-x_1)^2=0$ are

Question 441 : If$\alpha ,\beta $ are roots of the equation $2x^{2}+6x+b=0$ where $b<0$, then find least integral value of$\displaystyle \left ( \dfrac{\alpha ^{2}}{\beta }+\dfrac{\beta ^{2}}{\alpha } \right )$.

Question 442 : Simplify: $\displaystyle \frac { 45\left( { a }^{ 4 }-3{ a }^{ 3 }-28{ a }^{ 2 } \right)  }{ 9a\left( a+4 \right)  } $

Question 443 : The equation $\displaystyle x^{2}+Bx+C=0$ has 5 as the sum of its roots and 15 as the sum of the square of its roots. The value of C is

Question 444 : If the equation $\displaystyle\left( { p }^{ 2 }+{ q }^{ 2 } \right) { x }^{ 2 }-2\left( pr+qs \right) x+{ r }^{ 2 }+{ s }^{ 2 }=0$ has equal rootsthen

Question 445 : If $\alpha, \beta$ be the roots $x^2+px-q=0$ and $\gamma, \delta$ be the roots of $x^2+px+r=0$, then $\dfrac{(\alpha -\gamma)(\alpha -\delta)}{(\beta -\gamma )(\beta -\delta)}=$

Question 447 : Let $\alpha$ and $\beta$ be the roots of equation $x^2-6x-2=0$. If $a_n=\alpha^n-\beta^n$, for $n\geq 1$, then the value of $\dfrac{a_{10}-2a_8}{2a_9}$ is equal to?

Question 448 : If the roots of $ax^2+bx+c=0, \neq 0,$ are p,q ($p \neq q $), then the roots of $cx^2-bx+a=0$ are.

Question 450 : Suppose $\alpha ,\beta .\gamma $ are roots of ${ x }^{ 3 }+{ x }^{ 2 }+2x+3=0$. If $f(x)=0$ is a cubic polynomial equation whose roots are $\alpha +\beta ,\beta +\gamma ,\gamma +\alpha $ then $f(x)=$

Question 451 : Harmonic conjugate of the point $C(5, 1)$ with respect to the point $A(2, 10)$ and $B(6, -2)$ is?

Question 454 : $P$ is the point $(-5,3)$ and $Q$ is the point $(-5,m)$. If the length of the straight line $PQ$ is $8$ units, then the possible value of $m$ is:

Question 455 : The point at which the two coordinate axes meet is called the

Question 456 : An isosceles triangle has vertices at (4,0), (-4,0), and (0,8) The length of the equal sides is

Question 457 : The coordinates of $A$ and $B$ are $(1, 2) $ and $(2, 3)$. Find the coordinates of $R $, so that $A-R-B$  and   $\displaystyle \frac{AR}{RB} = \frac{4}{3}$.

Question 459 : The points which trisect the line segment joining the points $(0,0)$ and $(9,12)$ are

Question 460 : A line is of length $10$ m and one end is $(2,-3)$, the $x$ - co-ordinate of the other is $8$, then its $y$- coordinate is:

Question 461 : Which of the following are the co-ordinates of the centre of the circle that passes through $P(6, 6), Q(3, 7)$ and $R(3, 3)$?

Question 462 : The ratio, in which the line segment joining (3, -4) and (-5, 6) is divided by the x-axis is

Question 463 : Which of the following points is not 10 units from the origin ?

Question 464 : The coordinates of $A, B$ and $C$ are $(5, 5), (2, 1)$ and $(0, k)$ respectively. The value of $k$ that makes $\overline {AB} + \overline {BC}$ as small as possible is

Question 465 : How far is the line 3x - 4y + 15 = 0 from the origin?

Question 466 : The vertices P, Q, R, and S of a parallelogram are at (3,-5), (-5,-4), (7,10) and (15,9) respectively The length of the diagonal PR is

Question 467 : If the distance between the points $(4, p)$ and $(1, 0)$ is $5$, then the value of $p$ is:

Question 468 : If a point $P\left(\displaystyle\frac{23}{5}, \frac{33}{5}\right)$ divides line AB joining two points $A(3, 5)$ and $B(x, y)$ internally in ratio of $2:3$, then the values of x and y will be.

Question 470 : Find the value of $x$ if the distance between the points $(2, -11)$ and $(x, -3)$ is $10$ units.

Question 471 : If Q$\displaystyle \left ( \frac{a}{3},4 \right )$ is the mid-point of the line segment joining the points A(-6,5) and B(-2,3), then the value of 'a' is

Question 472 : The coordinates of the midpointof a line segment joining$P ( 5,7 )$ and Q $( - 3,3 )$ are

Question 473 : Find the distance from the point (5, -3) to the line 7x - 4y - 28 = 0

Question 474 : If $A$ and $B$ are the points $(-3,4)$ and $(2,1)$, then the co-ordinates of the point $C$ on $AB$ produced such that $AC=2BC$ are 

Question 475 : A(2,6) and B(1,7) are two vertices of a triangle ABC and the centroid is (5,7) The coordinates of C are

Question 476 : If a point $C$ be the mid-point of a line segment $AB$, then $AC = BC = (...) AB$.

Question 477 : Slope of the line $AB$ is $-\dfrac {4}{3}$. Co-ordinates of points $A$ and $B$ are $(x, -5)$ and $(-5, 3)$ respectively. What is the value of $x$

Question 478 : The coordinates of the point of intersection of X-axis and Y-axis is( 0,0) State true or false.

Question 479 : A Cartesian plane consists of two mutually _____ lines intersecting at their zeros.  

Question 480 : A rectangular hyperbola whose cente is C is cut by any circle of radius r in four point P, Q, R, S. The value of$CP^{2}+CQ^{2}+CR^{2}+CS^{2}$ is equal to :

Question 481 : Given the points $A(-1,3)$ and $B(4,9)$.Find the co-ordinates of the mid-point of $AB$

Question 482 : The distance between the points (sin x, cos x) and (cos x -sin x) is

Question 483 : In what ratio, does $P(4, 6)$ divide the join of $A(-2, 3)$ and $B(6, 7)$

Question 485 : The vertices of a triangle are $(-2,0) ,(2,3)$ and  $(1, -3)$ , then the type of the triangle is 

Question 487 : Let $\mathrm{P}(\mathrm{x}_{1},\mathrm{y}_{1})\mathrm{b}\mathrm{e}$ any point on the cartesian plane then match the following lists:   <table class="table table-bordered"><tbody><tr><td> LIST - I    </td><td> LIST - II</td></tr><tr><td> $\mathrm{A})$ The distance from $\mathrm{P}$ to X-axis</td><td>1) $0$</td></tr><tr><td> $\mathrm{B})$ The distance from $\mathrm{P}$ to Y-axis</td><td>2) $|\mathrm{y}_{1}|$</td></tr><tr><td> $\mathrm{C})$ The distance from $\mathrm{P}$ to origin is </td><td> 3) $\sqrt{x_{1}^{2}+y_{1}^{2}}$ </td></tr><tr><td> </td><td>4)$ |x_{1}|$                                   </td></tr></tbody></table>

Question 488 : The ratio in which the line joining the points $(3, 4)$ and $(5, 6)$ is divided by $x-$axis :

Question 489 : The centroid of the triangle with vertices (2,6), (-5,6) and (9,3) is

Question 490 : The distance between the points $(3,5)$ and $(x,8)$ is $5$ units. Then the value of $x$ 

Question 491 : A pair of numerical coordinates is required to specify each point in a ......... plane.

Question 492 : If the points (1,1) (2,3) and (5,-1) form a right triangle, then the hypotenuse is of length

Question 493 : A point R (2,-5) divides the line segment joining the point A (-3,5) and B (4,-9) , then the ratio is

Question 494 : Find the distance from the point (2, 3) to the line 3x + 4y + 9 = 0

Question 495 : Given the points $A(-3, 7)$ and $B(5, -9)$, determine the coordinates of point P on directed line segment that partitions in the ratio $\dfrac{1}{4}$.

Question 496 : The ratio in which the line segment joining (3,4) and (-2,1) is divided by the y-axis is

Question 497 : The point which lies in the perpendicular bisector of the line segment joining the points A (-2, -5) and B (2,5) is

Question 498 : Find the co-ordinates of the mid point of a point that divides AB in the ratio 3 : 2.

Question 499 : The points $(-2, -1), (1, 0),(4, 3),$ and $(1, 2)$ are the vertices

Question 500 : If A(x,0), B(-4,6), and C(14, -2) form an isosceles triangle with AB=AC, then x=

Question 501 : The point which lies on the perpendicular bisector of the line segment joining the points $P(-2,0)$ and $Q(2,5)$ is:

Question 502 : The coordinate of point which divides the line segment joining points $A(0,0)$ and $B(9,12)$ in the ratio $1:2$, are

Question 503 : The line joining $(5, 0)$ to $(10\cos\theta, 10\sin\theta)$ is divided internally in the ratio $2:3$ at $P$, then the locus of $P$ is

Question 504 : The point which divides the line segmentjoining the points (3, 5) and (8, 10) internallyin the ratio 2 :3 is:

Question 505 : The ratio in which X-axis divides the line segment joining $(3,6)$ and $(12,-3)$ is

Question 506 : The ratio in which the joint of (-3, 10), (6, -8)is divided by (-1, 6),

Question 507 : In what ratio does the point P(-2, 3) divide theline segment joining the points A(-3, 5) andB(4, -9) internally?

Question 508 : There are two point $P(1,-4)$ and $Q(4,2)$. Find a point X dividing the line PQ in the ratio $1:2$

Question 509 : The mid-point of line segment joining thepoints (3, 0) and (-1, 4) is :

Question 510 : The point P divides the line segment joining the points $\displaystyle A\left ( 2,1 \right )$ and $\displaystyle B\left ( 5,-8\right )$ such that $ \frac{AP}{AB}=\frac{1}{3}$ If P lies on the line $\displaystyle 2x+y+k=0$ then the value of k is-

Question 511 : The points $A$ $(x_1, y_1), B (x_2, y_2)$ and $C (x_3, y_3)$ are the vertices of $\Delta $ ABC. The median $AD$ meets $BC$ at $D$. Find the coordinates of points Q and R on medians BE and CF, respectively such that $BQ : QE = 2 : 1$ and $CR : RF = 2 : 1$.

Question 512 : The ratio in which the line $3x+y-9=0$ divides the line segment joining points (1, 3) and (2, 7) is:

Question 513 : In what ratio does the point $\begin{pmatrix} \dfrac { 1 }{ 2 },\dfrac { -3 }{ 2 } \end{pmatrix}$ divide the line segment joining the points $(3,5)$ and $(-7,9)$?

Question 514 : Find the coordinates of the point $P$ which divides line segment $QR$ internally in the ratio $m:n$ in the following example: $Q \equiv (6, -5), R \equiv (-10, 2)$ and $m:n = 3:4$

Question 515 : If X-axis divides the line joining $(3,-4)$ and $(5,6)$ in the ratio $a:b $, then what is the value of $\dfrac{a}{b}$?

Question 516 : Find the ratio in which the line segment joining the points $(3,5)$ and $(-4,2)$ is divided by y-axis.

Question 517 : What is the ratio in which $P(2, 5)$ divides the line joining the points $(8, 2)$ and $(-6, 9)$?

Question 518 : What is the approximate slope of a line perpendicular to the line $\sqrt{11}x+\sqrt{5}y=2$?

Question 519 : x-axis divides line segment joining points (2, -3) and (5,6) in the ratio

Question 520 : The ratio by which the line $2x + 5y - 7 = 0$ divides the straight line joining the points $(-4, 7) $ and $(6, -5)$ is

Question 521 : The straight line $3x+y=9$ divides the line segment joining the points $(1,\,3)$ and $(2,\,7)$ in the ratio

Question 522 : If the line joining A(2, 3) and B(-5, 7) is cut by X - axis at P, then find AP : PB.

Question 523 : If the line $2x+y=k$ passes through the point which divides the line segment joining the point $(1,1)$ & $(2,4)$ in the ratio $3:2$ then $k$ equal

Question 524 : If the point P (2, 1) lies on the segment joining Points A (4, 2) and B (8, 4) then

Question 525 : Length of the median from B on AC where A (-1, 3), B (1, -1), C (5, 1) is

Question 526 : The coordinates of one end of a diameter of a circle are $(5, -7)$. If the coordinates of the centre be $(7, 3)$, the co ordinates of the other end of the diameter are

Question 527 : What will be the value of $y$ if the point $\begin{pmatrix} \dfrac { 23 }{ 5 },y \end{pmatrix}$, divides the line segment joining the points $(5,7)$ and $(4,5)$ in the ratio $2:3$ internally?

Question 528 : The coordinates of the point which divides the line segment joining the points $(-7, 4)$ and $(-6, -5)$ internally in the ratio $7 : 2$ is:

Question 529 : Find the coordinates of the point which divides the line segment joining $(-3,5)$ and $(4,-9)$ in the ratio $1:6$ internally.

Question 530 : Let $A(-6,-5)$ and $B(-6,4)$ be two points such that a point $P$ on the line $AB$ satisfies $AP=\cfrac{2}{9}AB$. Find the point $P$.

Question 531 : Find the midpoint of the segment joining the points $(4, -2)$ and $(-8,6)$.

Question 532 : If $P(2, 2), Q(-2, 4)$ and $R(3, 4)$ are the vertices of $\Delta PQR$ then the equation of the median through vertex R is _______.

Question 533 : In what ratio is the line segment joining the points $(4, 6)$ and $(-7, -1)$ Is divided by $X$-axis ?

Question 534 : If the coordinates of opposite vertices of a square are $(1,3)$ and $(6,0)$, the length if a side od a square is

Question 535 : Find the distance between the points $(-1,-3)$ and the midpoint of the line segment joining $(2,4)$ and $(4,6)$.

Question 536 : $A(5,1)$, $B(1,5)$ and $C(-3, -1)$ are the vertices of $\Delta ABC$. The length of its median AD is:

Question 537 : The ratio in which the line $3x+y=9$ divides the line sequent joining the points $(1,3)$ and $(2,7)$ is given by

Question 538 : If the point $(x_1 + t (x_2 -x_1), y_1+t (y_2-y_1))$ divides the join of $(x_1, y_1)$ and $(x_2, y_2)$ internally, then

Question 539 : The point which is equi-distant from the points $(0,0),(0,8) and (4,6)$ is 

Question 540 : If P(x, y) is any point on the line joining thepoints (a, 0) and (0, b) then the value of$\displaystyle \frac{x}{a} + \frac{y}{b}$

Question 541 : The line segment joining the points $(3, -4)$ and $(1, 2) $ is trisected at the points P and Q. If the and co-ordinates of P and Q are $(p, -2)$ and $(\frac{5}{3}, q)$ respectively, find the value of p and q.

Question 542 : In how many maximum equal parts, a rectangular cake can be divided using three straight cuts?

Question 543 : Point $P$ divide a line segment $AB$ in the ratio $5:6$ where $A(0,0)$ and $B(11,0)$. Find the coordinate of the point $P$:

Question 544 : If we take $11$ points on a ray which is drawn at acute angle to a line segment, then the line segment can be divided into maximum _____ equal points.

Question 545 : The coordinates of the third vertex of an equilateral triangle whose two vertices are at $(3, 4), (-2 3)$ are ______.

Question 547 : State whether the following statements are true or false . Justify your answer. Point $ A(-6 , 10) , B(-4 , 6) $ and $ C(3 , -8) $ are collinear such that $ AB = \dfrac{2}{9} AC $ .

Question 548 : State whether the following statements are true or false . Justify your answer. The points $ (0 , 5) , (0 , -9) $ and $ (3 , 6) $ are collinear .

Question 549 : Select the correct option. The value of $p$, for which the points $A(3,1) , B (5, p)$ and $C (7, -5)$ are collinear, is

Question 550 : Number of points with integral co-ordinates that lie inside a triangle whose co-ordinate are (0,0), (0, 21) and (21, 0)

Question 551 : The vertices of a triangle are $A(3,4)$, $B(7,2)$ and $C(-2, -5)$. Find the length of the median through the vertex A.

Question 552 : If two vertices of a parellelogram are $(3,2)$ and $(-1,0)$ and the diagonals intersect at $(2, -5)$, then the other two vertices are:

Question 553 : The points $(-2,2)$, $(8, -2)$ and $(-4, -3)$ are the vertices of a:

Question 554 : Three points $\left( {0,0} \right),\left( {3,\sqrt 3 } \right),\left( {3,\lambda } \right)$ from an equilateral triangle, then $\lambda $ is equal to

Question 555 : If $\displaystyle(-1,2),(2,-1)$ and $\displaystyle(3,1)$ are any three vertices of a parallelogram then the fourth vertex $\displaystyle(a,b)$ will be such that

Question 556 : The point whose abscissa is equal to its ordinate and which is equidistant from $A(5,0)$ and $B(0,3)$ is

Question 558 : $\mathrm{P}_{1},\ \mathrm{P}_{2},\ldots\ldots.,\ \mathrm{P}_{\mathrm{n}}$ are points on the line $y=x$ lying in the positive quadrant such that $\mathrm{O}\mathrm{P}_{\mathrm{n}}=n\cdot\mathrm{O}\mathrm{P}_{\mathrm{n}-1}$, where $\mathrm{O}$ is the origin. If $\mathrm{O}\mathrm{P}_1=1$ and the coordinates of $\mathrm{P}_{\mathrm{n}}$ are $(2520\sqrt{2},2520\sqrt{2})$, then $n$ is equal to

Question 559 : If the line $2x+y=k$ passes through the point which divides the line segment joining the points $(1, 1)$ and $(2, 4)$ in the ratio $3 : 2$ ,then $k$ equals:

Question 560 : If $(-6, -4), (3, 5), (-2, 1)$ are the vertices of a parallelogram, then remaining vertex can be

Question 561 : $ABC$ is an isosceles triangle. If the coordinates of the base are $B(1,3)$ and $C(-2,7)$. The vertex $A$ can be

Question 562 : If $Q(0, 1)$ is equidistant from $P(5, -3)$ and $R(x, 6)$, find the values of x. Also find the distances QR and PQ.

Question 563 : The mid point of the segment joining $(2a, 4)$ and $(-2, 2b)$ is $(1, 2a+1)$, then value of b is

Question 564 : The points given are $(1, 1)$, $(-2, 7)$ and $(3, 3)$.Find distance between the points.

Question 565 : If a line intercepted between the coordinate axes is trisected at a point $A(4, 3),$ which is nearer to $x-$axis, then its equation is

Question 566 : Find the point on the x-axis which is equidistant from the points $(-2,5)$ and $(2, -3)$. Hence find the area of the triangle formed by these points

Question 567 : Determine the ratio in which the line $3x+y-9=0$ divides the line segment joining the points $(1,3)$ and $(2,7)$

Question 568 : Find the ratio in  which the point $P(2,y)$ divides the line segment joining the point $A(-2,2)$ and $B(3,7)$. Also find the value of $y$.

Question 569 : If $\displaystyle A \left(\frac{2c}{a},\frac{c}{b}\right),B\left(\frac{c}{a},0\right)$ and $\displaystyle C\left(\frac{1+c}{a},\frac{1}{b}\right) $ are three points, then

Question 570 : $ABC$ is an equilateral triangle. If the coordinates of two of its vertices are ($1, 3)$ and $(-2, 7)$ the coordinates of the third vertex can be

Question 571 : If the coordinates of the extermities of diagonal of a square are $(2,-1)$ and $(6,2)$, then the coordinates of extremities of other diagonal are

Question 572 : If the three distance points $\left( { t }_{ i\quad }2{ at }_{ i }+{ { at }^{ 3 }_{ i } } \right) \quad for\quad i=1,2,3$ are collinear then the sum of the abscissae of the points is

Question 573 : If $a> 0$ and $P(-a, 0), Q(a, 0)$ and $R(1,1) $ are three points such that $\displaystyle \left|(PR)^{2}-(QR)^{2} \right| = 12,$ then

Question 574 : If $P \left( \dfrac{a}{3},\dfrac{b}{2} \right)$ is the mid-point of the line segment joining $A(-4,3)$ and $B(-2,4)$ then $(a,b)$ is 

Question 575 : If $P\left( x,y,z \right) $ is a point on the line segment joining $Q\left( 2,2,4 \right) $ and $R\left( 3,5,6 \right) $ such that the projections of $OP$ on the axis are $\cfrac { 13 }{ 5 } ,\cfrac { 19 }{ 5 } ,\cfrac { 26 }{ 5 } $ respectively, then $P$ divides $QR$ in the ratio

Question 577 : The probability of an event happening and the probability of the same event not happening (or the complement) must be a

Question 578 : A coin is tossed $400$ times and the data of outcomes is below:  <table class="wysiwyg-table"><tbody><tr><td> Outcomes </td><td> $H$</td><td> $T$</td></tr><tr><td> Frequency</td><td> $280$</td><td> $120$</td></tr></tbody></table> Find:(i) $P(H)$, i.e., probability of getting head(ii) $P (T)$, i.e., probability of getting tail. (iii) The value of $P (H) + P (T)$.

Question 579 : A pair of dice is thrown. Find the probability of getting a sum of $8$ or getting an even number on both the dices.

Question 580 : If the probability of the occurrence of an event is P then what is the probability that the event doesn't occur.

Question 581 : Two dice are thrown. Find the odds in favour of getting the sum $4$.

Question 582 : The probability of an event $A$ lies between $0$ and $1$, both inclusive. Which mathematical expression best describes this statement?

Question 583 : If $P(A) = \dfrac{5}{9}$, then the odds against the event $A$ is

Question 584 : Ticket numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5 ?

Question 586 : A pair of dice is thrown once The probability that the sum of the outcomes is less than 11 is

Question 587 : A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the number between 1 to 15. What is the probability that it will point to an odd number.

Question 588 : A biased coin with probability $p , 0 < p < 1 ,$ of heads is tossed until a head appears for thefirst time. If the probability that the number of tosses required is even, is $2 / 5 ,$ then $p$ equal to

Question 589 : Out of the digits $1$ to $9$, two are selected at random and one is found to be $2$, the probability that their sum is odd is

Question 590 : According to the property of probability, $P(\phi) = 0$ is used for

Question 591 : Vineeta said that probability of impossible events is $1$. Dhanalakshmi said that probability of sure events is $0$ and Sireesha said that the probability of any event lies between $0$ and $1$. in the above, with whom will you agree?

Question 592 : If the events $A$ and $B$ mutually exclusive events such that $P(A)=\dfrac {1}{3}(3x+1)$ and $P(B)=\dfrac {1}{4}(1-x)$, then the aet of possible values of $x$ lies in the interval:

Question 593 : A bag contains 5 blue and 4 black balls. Three balls are drawn at random. What is the probability that 2 are blueand 1 is black?

Question 594 : A bulb is taken out at random from a box of 600 electricbulbs that contains 12 defective bulbs. Then theprobability of a non-defective bulb is

Question 595 : What is the maximum value of the probability of an event?

Question 596 : A die is thrown .The probability that the number comes up even is ______ .

Question 597 : If the odd in favour of an event are $4$ to $7$, find the probability of its no occurence.

Question 599 : The probability of guessing the correct answer to a certain test is $\displaystyle\frac{x}{2}$. If the probability of not guessing the correct answer to this questions is $\displaystyle\frac{2}{3}$, then $x$ is equal to ______________.

Question 600 : One hundred identical coins each with probability p as showing up heads are tossed. If $0 < p < 1$ and the probability of heads showing on 50 coins is equal to that of heads on 51 coins, then the value of p is

Question 602 : The probability expressed as a percentage of a particular occurrence can never be

Question 603 : If I calculate the probability of an event and it turns out to be $7$, then I surely know that

Question 604 : A fair dice has faces numbered $0, 1, 7, 3, 5$ and $9$. If it is thrown, the probability of getting an odd number is

Question 605 : A missile target may be at a point P with probability$\displaystyle \frac{9}{10}$ or at a point Q with probability$\displaystyle \frac{1}{10}$ we have 20 shells each of which can be fired either at point P or Q Each shell may hit the target independently of the other shoot with probability$\displaystyle \frac{2}{3}$ Then number of shells must be fired at point P to hit any target with maximum probability is

Question 606 : An investigator interviewed 100 students to determine the performance of three drinks: milk, coffee, and tea. The investigator report that 10 students take all three drinks, 20 had milk and coffee, 30 had coffee and tea;  25 students take milk and tea; 12 students take milk only; 5 students take coffee only and 8 students take tea only. Then the number of students who did not take any of three drink is 

Question 607 : If a leap year is selected at random what is the probability that it will contain $53$ Tuesdays?

Question 608 : The $2013$ U.S. Census recorded the highest educational attainment of all adults aged $25$ years or older in country $T$, one of the most educated parts of the country. The results are given in the two-way table below. <table class="wysiwyg-table"><tbody><tr><td></td><td>Men</td><td>Women</td><td>Total</td></tr><tr><td>High School Diploma</td><td>7535</td><td>7234</td><td>14769</td></tr><tr><td>Bachelor's Degree</td><td>17170</td><td>23455</td><td>40625</td></tr><tr><td>Master's Degree</td><td>45105</td><td>41078</td><td>86183</td></tr><tr><td>Professional Degree</td><td>23501</td><td>23405</td><td>46906</td></tr><tr><td>Doctoral Degree</td><td>16232</td><td>15817</td><td>32049</td></tr><tr><td>Total</td><td>10953</td><td>110989</td><td>220532</td></tr></tbody></table>According to the data presented in the table above, if one was told to choose a person at random out of the entire population aged $25$ years or older in country $T$, find the percentage probability that the person he/she chooses turns out to be a man with a doctoral degree.

Question 609 : In a race, the odds in favour of horses $A, B, C, D$ are $1:3, 1:4, 1:5$ and $1:6$ respectively. Find probability that one of them wins the race.

Question 610 : $n$ is an integer chosen at random from the set $\{2,5,6 \}$ and $p$ another integer chosen at random from the set $\{6,9,10 \}$. What is the probability that the two numbers $n$ and $p$ are even?

Question 611 : The odds in the favour of an event are $3 : 5$.The probability of occurrence of the event is?

Question 612 : The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chance of each event is

Question 613 : A bag contains $7$ black socks, $12$ white socks, and $17$ red socks. If you pick one sock at random from the bag, what is the probability that it will NOT be white?

Question 614 : From a batch of $100$ items of which $20$ are defective, exactly two items are chosen, one at a time, without replacement. Calculate the probability that the first item chosen is defective.

Question 615 : A pair of dice is thrown seven times . Getting a total of numbers on the two dice to be seven is considered as a success . Find the probability of getting $7$ in exactly $2$ trials out of $7$.

Question 616 : One of the two events must occur. If the chance of one is$\displaystyle \frac{2}{3}$ of the other, then odds in favour of the other are

Question 617 : If a person throw $3$ dice the probability of getting sum of digit exactly $15$ is

Question 618 : A fair die is rolled once. STATEMENT - 1 : The probability of getting a composite number is 1/3 STATEMENT - 2 : There are three possibilities for the obtained number (i) the number is a prime number (ii) the number is a composite number (iii) the number is 1, and hence probability of getting a prime number $=$1/3.

Question 619 : A family is going to choose two pets at random from among a group of four animals: a cat, a dog, a bird, and a lizard. Find the probability that the pets that the family chooses will include the lizard.

Question 620 : Cards are drawn one-by-one without replacement from a well shuffled pack of 52 cards. Then the probability that aface card (Jack, Queen or King) will appear for the first time on the third turn is equal to

Question 621 : If 10 persons are to sit around a round table, the odds against two specified persons sitting together is

Question 622 : The odds against a certain events are $5:2$ and the odds in favour of another events are $6:5$. The probability that at least one of the events will happens is:

Question 623 : A die is rolled. If the outcome is an odd number, what is the probability that it is prime?

Question 624 : $A$ and $B$ are two events. Odds against $A$ and $2:1$. Odds in favor of $A\cup B$ are $3:1$. If $x\le P\left( B \right) \le y$, then the ordered pair $(x,y)$ is

Question 625 : In a box, there are $8$ red, $7$ blue and $6$ green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

Question 626 : Odds $8$ to $5$ against a person who is $40$yr old living till he is $70$ and $4$ to $3$ against another person now $50$ till he will be living $80$. Probability that one of them will be alive next $30$yr.

Question 627 : In a single cast with two dice, the odds against drawing $7$ is

Question 628 : The odds is favour of winning a race for three horses $A, B$ and $C$ respectively $1:2, 1:3$ and $1:4$. Find the probability for winning of any one of them.

Question 629 : The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second.The chances of the events are

Question 630 : In a given race, the odds in favour of four horses $A, B, C$ & $D$ are $1 : 3, 1 : 4, 1 : 5$ and $1 : 6$ respectively. Assuming that a dead heat is impossible, find the chance that one of them wins the race

Question 631 : A die is thrown once.find the probability of getting a prime number less than $5.$

Question 632 : Two cards are drawn at random from a pack of $52$ cards. The probability of these two being "Aces" is

Question 633 : A coin tossed $100$ times. The no. of times head comes up is $54$.What is the probability of head coming up?

Question 634 : The table below shows the relative investment in alternative energy sources by type. One column shows the relative investment in $2007$ of $\$75$ million total invested in alternative energy. The other column shows the projected relative investment in $2017$ given current trends. The total projected investment in alternative energy in $2017$ is $\$254$ million. <table class="wysiwyg-table"><tbody><tr><td></td><td>Actual $2007$ Investment</td><td>Projected 2017 Investment</td></tr><tr><td>Biofuels</td><td>0.310</td><td>0.34</td></tr><tr><td>Wind </td><td>0.40</td><td>0.32</td></tr><tr><td>Solar </td><td>0.27</td><td>0.30</td></tr><tr><td>Fuel Cells </td><td>0.02</td><td>0.04</td></tr><tr><td>Toatl</td><td>1.00</td><td>1.00</td></tr></tbody></table>Based on the information in the table, if an investment was made in alternative energy in $2007$, what is the probability that the money was invested in wind resources?

Question 635 : There are three events $A$, $B$ and $C$ out of which one and only one can happen. The odds are $7$ to $3$ against $A$ and $6$ to $4$ against $B$. The odds against C are

Question 636 : A gumball machine contains $40$ blue gum balls, $20$ red gumballs, $15$ gumballs, and $25$ purple gumballs. What is the probability that a person gets a red gumball?

Question 637 : The odds in favor of standing first of three students appearing in an examination are $1:2,2:5$ and $1:7$ respectively. The probability that either of them will stand first, is

Question 638 : $A$ and $B$ each throw a dice. The probability that "$B$" throw is not smaller than "$A$" throw, is

Question 639 : If $\dfrac {1 + 3p}{3}, \dfrac {1 - 2p}{2}$ are probabilities of two mutually exclusive events, then p lies the interval

Question 640 : If x is chosen at random from the set $\left \{2, 3, 4, 5, 6\right \}$, and y is chosen at random from the set $\left \{11, 13, 15\right \}$, find the probability that $xy$ is even.

Question 641 : A woman has 10 keys out of which only one opens a lock She tries the keys one after the another(keeping aside the failed ones) till she suceeds in opening the lock. What is the chance that it is the seventh key that works?

Question 642 : In a ODI cricket match, probability of loosing the game is $\dfrac{1}{4}$. What is the probability of winning the game ?

Question 643 : A problem in statistics is given to three students whose chance of solving it are $ \dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}$ respectively. The probability that the question is solved is

Question 644 : $A, B$ are two events of a simple space.Assertion (A):- $A, B$ are mutually exclusive $\Rightarrow P\left ( A \right )\leq P\left ( \bar{B} \right )$Reason (R):- $A, B$ are mutually exclusive  $\Rightarrow P\left ( A \right )+ P\left ( B \right )\leq 1$

Question 645 : There are 5 letters and 5 different envelopes. The number of ways in which all the letters can be put in wrong envelope, is.

Question 646 : The chance of one event happening is the square of the chance of a $2^{nd}$ event, but odds against the first are the cubes of the odds against the 2nd. Find the chances of first event. (Assume that both events are neither sure nor impossible)

Question 647 : $(a)$ The probability that it will rain tomorrow is $0.85$. What is the probability that it will not rain tomorrow? $(b)$ If the probability of winning a game is $0.6$, what is the probability of losing it?

Question 648 : Let $A$ and $B$ be two events with $P(A) = \dfrac {1}{3}, P(B) = \dfrac {1}{6}$ and $P(A\cap B) = \dfrac {1}{12}$. What is $P(B|\overline {A})$ equal to?

Question 649 : The probability that atleast one of the events A and B occurs, is $0.6$. If A and B occur simultaneously with probability $0.2$, then $P(\bar{A})+P(\bar{B})$ is equal to?

Question 650 : If a positive integer $n$ is picked at random from the positive integers less than or equal to $10$, what is the probability that $5n + 3 \leq 14$  ?

Question 651 : Results on the bar exam of Law School Graduates <table class="wysiwyg-table"><tbody><tr><td></td><td>Passed bar exam</td><td>Did not pass bar exam</td></tr><tr><td>Took review course</td><td>18</td><td>82</td></tr><tr><td>Did not take review course</td><td>7</td><td>93</td></tr></tbody></table>The table above summarizes the results of $200$ law school graduates who took the bar exam. If one of the surveyed graduates who passed the bar exam is chosen at random for an interview, what is the probability that the person chosen did not take the review course?

Question 652 : Two dice are tossed. What is the probability that neither die is a $4$?

Question 653 : Two persons $A$ and $B$ have respectively $n + 1$ and $n$ coins, which they toss simultaneously. Then probability $P$ that $A$ will have more heads than $B$ is:

Question 654 : A box contains $9$ tickets numbered $1$ to $9$ inclusive. If $3$ tickets are drawn from the box without replacement. The probability that they are alternatively either {odd, even, odd} of {even, odd, even} is

Question 655 : Each of a and b can take values 1 or 2 with equal probability. The probability that the equation $ax^2 + bx + 1 = 0$ hasreal roots, is equal to

Question 656 : There are two events $A$ and $B$. If odds against $A$ are as $2:1$ and those in favour of $ A \cup B$ are $3:1$ , then

Question 657 : In throwing $3$ dice, the probability that atleast $2$ of the three numbers obtained are same is

Question 658 : A fair coin is tossed five times. Calculate the probability that it lands head-up at least twice.

Question 659 : A party of $23$ persons take their seats at a round table. The odds against two specified persons sitting together is

Question 660 : There are four letters and four addressed envelopes. The probability that all letters are not dispatched in the right envelope is:

Question 661 : The chance of an event happening is the square of the chance, of a second event but the odds against the first are the cubes of the odds against thefirst are the cubes of the odds against the second. Find the chance of each.

Question 662 : A fair coin is flipped $5$ times.  The probability of getting more heads than tails is $\dfrac{1}{2}$

Question 663 : In a set of games it is $3$ to $5$ in favour of the winner of the previous game.. Then the probability that a person who has won the first game shall win at least $2$ out of the next $5$ games is ?

Question 664 : In a group of $13$ cricket players, four are bowlers. Find out in how many ways can they form a cricket team of $11$ players in which atleast $2$ bowlers are included.

Question 665 : There are two bags $A$ and $B$. Bag A contains $3$ white and $4$ red balls whereas bag $B$ contains $4$ white and $3$ red balls. Three balls are drawn at random (without replacement) from one of the bags and are found to be two white and one red. Find the probability that these were drawn from bag $B$.

Question 666 : There are only three events $A,B,C$ one of which must and only one can happen; the odds are $8$ to $3$ against $A,5$ to $2$ against $B$; find the odds against $C$

Question 667 : The probability that an electronic device produced by a company does not function properly is equal to $0.1$. If $10$ devices are bought, then the probability, to the nearest thousandth, than $7$ devices function properly is

Question 668 : A number is randomly selected from the set $\left \{6, 7, 8, 8, 8, 10, 10, 11\right \}$. Find the probability the number will be less than the mean.

Question 669 : The odds that a book will be favorably reviewed by three independent critics are $5$ to $2,$ $4$ to $3$ and $3$ to $4$ respectively. What is the probability that of the three reviews a majority will be favorable?

Question 670 : If $2$ cards are drawn from a pack of $52$, then the probability that they are from the same suit is___

Question 671 : A man and his wife appear for an interview for two posts. The probability of the man's selection is $\dfrac{1}{5}$ and that of his wife selection is $\dfrac{1}{7}$. The probability that at least one of them is selected, is:

Question 672 : A coin whose faces are marked 3 and 5 is tossed 4 times; what are the odds against the sum of the numbers thrown being less than 15?

Question 673 : If odds against solving a question by three students are $2:1, 5:2$ and $5:3$ respectively, then probability that the question is solved only by one students is

Question 674 : For two events $A$ and $B , P ( B ) = P ( B / A ) = 1 / 3$ and $P ( A / B ) = 4 / 7 ,$ then Option a : $P \left( B ^ { \prime } / A \right) = 2 / 3$ Option b : $P \left( A / B ^ { \prime } \right) = 3 / 7$ Option c : $A$ and $B$ are mutually exclusive Option d: $A$ and $B$ are independent

Question 675 : A cow is tied to a pole, fixed to the midpoint of a side of a square field of dimensions $40\ m\times 40\ m$, by means of $14\ m$ long rope. Find the area that the cow can graze.

Question 677 : If the radius of a circle is tripled, the ares becomes.

Question 678 : Area of a sector having radius 12 cm and arc length 21 cm is

Question 680 : The length of minor arc $\widehat {AB}$ of a circle with radius $7$ units  is $14$. Find the length of major arc $\widehat {AB}$.

Question 681 : Circular dome is a 3D example of which kind of sector of the circle?

Question 683 : The region between an arc and two radii joining the centre to the end points of the arc is called

Question 684 : If 'c' be the circumference and 'd' be the diameter then the value of$ \displaystyle \pi $ is equal to-

Question 685 : If one side of a square is 2.4 m. Then what will be the area of the circle inscribed in the square?

Question 686 : State true or false: Sector is the region between the chord and its corresponding arc.

Question 687 : A square is inscribed in a circle of radius $7\: cm$. Find area of the square.

Question 688 : A sector of a circle with sectorial angle of$\displaystyle 36^{\circ} $ has an area of 15.4 sq cm The length of the arc of the sector is

Question 689 : If the area of the circle be $ \displaystyle 154 cm^{2},$ then its radius is equal to:

Question 690 : A horse is tied to a pole fixed at one corner of a $50 m \times 50 m$ square field of grass by means of a $20 m$ long rope. What is the area to the nearest whole number of that part of the field which the horse can graze?

Question 691 : A rope by which a cow is tethered is in reased from 16m to 23m. How much additional ground does it have to graze now?

Question 692 : If the diameter of a circle is increased by 200% then its area is increased by

Question 693 : The diameter of a circle is divided into n equal parts.On each part a semicircle is constructed. as n becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length:

Question 694 : If the circumference of a circle be 8.8 m then its radius is equal to -

Question 695 : A circular disc of radius 10 cm is divided into sectors with angles $ \displaystyle 120^{\circ}   $ and  $ \displaystyle 150^{\circ}   $ then  the ratio of the areas of two sectors is

Question 696 : Say true or false:A sector is a region between the chord and its corresponding arc.

Question 697 : The area of a circle is $24.64$. $\displaystyle m^{2}$ What is the circumference of the circle ?

Question 698 : If the radius and arc length of a sector are 17 cm and 27 cm respectively, then the perimeter is

Question 700 : If the radius of a circle increased by 20% then the corresponding increase in the area of circle is ................

Question 701 : The area of a sector is 1/18th of the area of the circle The sectorial angle is

Question 703 : What is the length of arc AB making angle of $126^0$ at center of radius $8$?

Question 704 : The radius of a circle whose area is equal to the sum of the areas of two circles where radii are 5 cm and 12 cm is

Question 705 : A wire of length $36$ cm is bent in the form of a semicircle. What is the radius of the semicircle?

Question 706 : Given, $\displaystyle A = \frac{S}{360}\times \pi r^2$ $A$ is the area of setor, $ S$ is the angle measure in degrees of the sector and $r$ is the radius of the circle. Find $r$ in terms of $A$ and $S$.

Question 708 : What is the minimum radius $(>1)$ of a circle whose circumference is an integer?

Question 709 : The angle of sector with area equal to one fifth of total area of whole circle 

Question 710 : If the difference between the circumference and radius of a circle is 37 cm then its diameter is

Question 711 : The area of a sector formed by two mutually perpendicular radii in $\odot \left( 0,5cm \right) $ is ............... ${cm}^{2}$.

Question 712 : The radius of a circular wheel is $1.75\ m$. The number of revolutions that it will make in covering $11\ kms$ is:

Question 713 : If the circumference of a circle is reduced by 50 % then the area will be reduced by

Question 714 : The area of two circles are in the ratio $25 : 36$. Then the ratio of their circumference is _________.

Question 715 : If the area of a circle is $346.5 \displaystyle cm^{2}$. Its circumference is

Question 716 : The area of a sector of a circle of radius 16 cm cut off by an arc which is 18.5 cm long is

Question 717 : Area of a circle with diameter 'm' radius 'n' and circumference 'p' is

Question 718 : If the radius of a circle be r cm then its area will be equal to-

Question 719 : The minute hand of a clock is $\displaystyle \sqrt{21}$ cm long. The area described by the minute hand on the face of the clock between $7$ am and $7.05$ am is

Question 720 : Find the area of equilateral  triangle inscribed in a circle of unit radius.

Question 721 : Find the circumference of the circle with the following radius : 10 cm

Question 722 : The diameter of a wheel of a cycle is 21 cm How far will it go in 28 complete revolutions?

Question 723 : Choose the correct answer from the alternative given. A can go round a circular path $8$ times in $40$ minutes. If the diameter of the circle is increased to $10$ times the original diameter, the time required by A to go round the new path once travelling at the same speed as before is:

Question 724 : If radius of a circle is increased to twice its original length, how much will the area of the circle increase ?

Question 725 : The area of a circle is 314 sq. cm and area of its minor sector is 31.4 sq. cm. Find thearea of its major sector.

Question 726 : A man runs with the speed of $15.84\ km/hr$. He completes $12$ rounds of a circular ground in one hour, find the area of the ground in $sq. m$.

Question 727 : The area of a circle drawn with its diameter as the diagonal of a cube of side of length 1 cm each is :

Question 728 : The sides of a triangle are $5$, $12$ and$ 13$ units. A rectangle of width $10$ units is constructed equal in area to the area of the triangle. Then the perimeter of the rectangle is 

Question 729 : A sector of $120^{\circ}$ cut out from a circle has an area of $9\displaystyle \frac{3}{7}$sq cm. The radius of the circle is

Question 730 : The area of the part of the square filed in which a horse tied to a fixed pole at one comer by means of a $10\ m$ rope, can graze is 

Question 731 : In $\bigodot (P, 6)$, the length of an arc is $\pi$. Then the arc subtends an angle of measure ___at the center.

Question 732 : State whether True or False:The radius of a circle is $7 cm$, then area of the sector of this circle if the corresponding angle is: $3$ rt. angles is $115.50 \,cm^2$

Question 733 : If the difference between the circumference and diameter of a circle is $30\ cm$, then the radius of the circle must be:

Question 734 : Find the diameter of a wheel that makes $113$ revolutions to go $2 km 26dm$. $ \displaystyle \left ( \pi =\frac{22}{7} \right )$

Question 735 : The radii of two circles are in the ratio $3 : 8$. If the difference between their areas is$2695\pi \: cm^{2}$ ,find the area of the smaller circle.

Question 736 : The circumference of a circle is $31.4\ cm$ Find the radius and the area of the circle? (Take $\pi=3.14$)

Question 737 : Given radius = $11 $ cm, area of the sector is $230 $ $cm^2$. Find the length of the arc $SR$.

Question 738 : The minute hand of a clock is 14 cm long If it moves between 8:00 AM and 8:45 AM What is the area covered by it on the face of the clock?

Question 739 : The radius of a circle is $14$ m, then the circumference of a circle is 

Question 740 : The circumference of a circular field is $528\ m$. Then its radius is

Question 741 : If the radius of a circle is $\displaystyle \frac{7}{\sqrt{\pi}}$ cm, then the area of the circle is equal to

Question 742 : The area of a sector is $120\pi$ and the arc measure is $160^o$. What is the radius of the circle?

Question 744 : Find the area of a sector in radians whose central angle is $45^o$ and radius is $2$.

Question 745 : If the circumference of a circle is reduced by $50\%$ the area of the circle is reduced by:

Question 747 : If the sum of the areas of two circles with radii $R_1$ and $R_2$ is equal to the area of a circle of radius R, then

Question 748 : The diameter of a wheel that makes 113 revolutions to go 2 Km 26 decametres is$\displaystyle \left ( \pi =\frac{22}{7} \right )$

Question 749 : If the circumference of a circle increases from $4\pi $to $8\pi $,then its area is :

Question 750 : How long is the arc subtended by an angle of $\dfrac{2\pi}{3}$ radians on a circle of radius $12$ cm?

Question 751 : The diameter of a circle is $10$ cm, then find the length of the arc, when the corresponding central angle is $144^{\circ}$.$(\pi =3.14)$

Question 752 : The ratio between the diameters of two circles is $3 : 5,$ then find the ratio between their areas.

Question 753 : Assertion: If the circumference of a circle is $176$ cm, then its radius is $28$ cm. Reason: Circumference $=2\pi \times radius$

Question 754 : If the difference between the circumference and radius of a circle is 37 cm then the area of the circle is

Question 755 : Let a semicircle with centre O and diameter AB. Let P and Q be points on the semicircle and R be a point on AB extended such that OA =QR < PR. If $\widehat{POA} = 102^0$ then $\widehat{PRA} $ is

Question 756 : What is the area of a sector with a central angle of $100$ degrees and a radius of $5$? (Use $\pi = 3.14$)

Question 757 : There are two circular gardens A and B. The circumference of garden A is $1.760 km$ and the area of garden B is $25$ times the area of garden A. Find the circumference of garden B.

Question 758 : The circumference of a circular field is $308 m$, Find its Area.

Question 759 : The area of a circular plot is $3850$ square meters. What is the circumference of the plot ?

Question 761 : The area of a circle is$\displaystyle 2464\:m^{2}$, then the diameter is

Question 762 : The area of the sector of a circle, whose radius is $6$ m when the angle at the centre is $42^0$, is

Question 763 : Ratio of circumference of a circle to its radius is always $2 \pi : 1$

Question 764 : The cost of fencing a circular field at the rate of $Rs\:.240\: per\: metre$ is $Rs. \: 52,800$ . The field is to be ploughed at the rate of $Rs. 12.50 \: per \: m^{2}$. Find the cost of ploughing the field.

Question 765 : If area of circular field is $6.16 \ sq. m$, then its diameter will be

Question 766 : If the sum of the circumferences of two circles with radii $R_1$ and $ R_2$ is equal to the circumference of a circle of radius $R$, then

Question 767 : The minute hand of a clock is $7\ cm$ long. Find the area traced by it on the clock face between $4{:}15$ p.m. and $4{:}35$ p.m.

Question 768 : The area of the circle whose centre is (1,2) and which passes through the point (4,6) is

Question 769 : When the circumference and area of a circle are numerically equal then the diameter is numerically equal to

Question 770 : The circumference of a circle is $44 m$, then the area of the circle is

Question 771 : If the radius of a circle is increased $100$%, the area is increased.

Question 772 : If the difference between the circumference and radius of a circle is $37$ cm, then the area of the circle is

Question 773 : The area of a sector of a circle of radius $16$ cm cut off by an arc which is $18.5$ cm long is 

Question 774 : What is the radius of a circle whose circumference is $\pi$?

Question 775 : A wire bent in the form a square incloses an area of 484 $\displaystyle m^{2}$ but if the same wire is bent in the form of a circle the area enclosed will be

Question 776 : The area of a sector of a circle of angle $\displaystyle 60^{\circ}$ is $\displaystyle \frac{66}{7}cm^{2}$ then the area of the corresponding major sector is

Question 777 : If a bicycle wheel makes $5000$ revolution in moving $11$ km, then diameter of wheel is

Question 778 : The ratio of the slant height of two right cones of equal base is 3 : 2 then the ratio of their volumes is

Question 779 : The perimeter of a quadrant of a circle of radius $\dfrac{7}{2}$ cm is:

Question 780 : A sphere with diameter $50$ cm intersects a plane $14$ cm from the center of the sphere. What is the number of square centimeters in the area of the circle formed?

Question 781 : What is the length of the chord of a unit circle which substends an angle $\theta$ at the centre ?

Question 782 : Upper part of a vertical tree which is broken over by the winds just touches the ground and makes an angle of$ \displaystyle 30^{\circ} $ with the ground. If the length of the broken part is 20 meters , then the remaining part of the tree is of length

Question 783 : A kite is flying with the string inclined at$\displaystyle 45^{\circ}$ to the horizontal If the string is straight and 50 m long the height at which the kite is flying is

Question 784 : The angles of elevation of the top of $12$m high tower from two points in opposite directions with it are complementary. If distance of one point from its base is $16$m, then distance of second point from tower's base is?

Question 785 : If the altitude of the sun is $60^{\circ}$, the height of a tower which casts a shadow of length 30 m is :

Question 786 : The angles of elevation of an artificial satellite measured from two earth stations are $30^0$ and $40^0$ respectively. If the distance between the earth stations is 4000 km, then the height of the satellite is

Question 787 : On the level ground the angle of elevation of the top of a tower is $30^{0 }$ On moving 20 metres nearer tower, the angle of elevation is found to be $60^{0}$ The height of the towerin metres is

Question 789 : From the top of a tower $80$ metres high, the angles of depression of two points $P$ and $Q$ in the same vertical plane with the tower are $45^{0}$ and $75^{0}$ respectively, $PQ=$

Question 790 : The angle of elevation of the top of tower from the top and bottom of a building h meter high are$\displaystyle \alpha $ and$\displaystyle \beta $ then the height of tower is

Question 791 : A ladder rests against a wall at an angle $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance $a$ slides a distance $b$ down the wall making an angle $\beta$ with the horizontal. Choose the correct option-

Question 792 : The angle of elevation a vertical tower standing inside a triangular at the vertices of the field are each equal to $\theta$. If the length of the sides of the field are $30\ m,\ 50\ m$ and $70\ m$, the height of the tower is:

Question 793 : From the top of a tower, the angles of depression of two objects on the same side of the tower are found to be $\alpha $ and $\beta $ where $\alpha >\beta $.The height of the tower is $130\ m,$ $\alpha =60^o\: and\: \beta =30^o$. The distance of the extreme object from the top of the tower is

Question 794 : Two poles of equal heights are standing opposite each other on either side of the road which is $80$ m wide. From the points between them on the road, the  elevation of the top of the poles are ${60^ \circ }$ and ${30^ \circ }$ respectively. Find the height of the poles.

Question 795 : A tree breaks due to storm and the broken part bends so that the top of the trees touches the ground making an angle ${30}^{o}$ with ground. The distance between the foot of the tree to the point where the top touches the ground is $8m$. Find the height of the tree.

Question 796 : The shadow of a tower on a level plane is found to be $60$ metres longer when the sun's altitude is $30^{0}$ than that when it is $45^{0 }$. The height of the tower in metres is

Question 797 : $A$ flag staff stands upon the top of a building. $A$t a distance of 40 $m$. the angles of elevation of the tops of the flag staff and building are $60^{ }$ and $30^{0}$ then the height of the flag staff in metres is

Question 798 : The angle of elevation of the top of a tower from the top and bottom of a building of height $a$ are ${0}^{o}$ and ${45}^{o}$ respectively. If the tower and the building stand at the same level, then height of tower is:

Question 799 : A boat is rowed away from a cliff $150$ m high At the top of the the cliff the angle of depression of the boat change from $\displaystyle 60^{0}$ to $\displaystyle 45^{0}$ in $2.5$ minutes The speed of the boat (in m/sec) is

Question 800 : A ladder is placed against tower. If the ladder makes an angle of $30^{\circ}$ with the ground and reaches upto a height of 15 m of the tower; find length of the ladder.

Question 801 : A man observes the elevation of a tower to be$ \displaystyle 30^{\circ} $. After advancing 11 cm towards it, he finds that the elevation is$ \displaystyle 45^{\circ} $. The height of the tower to the nearest meter is

Question 802 : $AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. A man finds that the angle of elevation of the point A from a certain point $C$ on the ground is $60^{{o}}$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD=7$ m. From $D$ the angle of elevation of the point $A$ is $45^{{o}}$. Then the height of the pole is

Question 803 : If the given object is above the level of the observer, then the angle by which the observer raises his head is called _____.

Question 804 : If the ratio of height of a tower and the length of its shadow on the ground is $\sqrt{3}:1 $, then the angle of elevation of the sun is

Question 805 : Two points at distance x and y from the base point are on the same side of the line passing through the base pf a tower. The angle of elevation from these two points to the top of the tower are complementary. Then, the height of the tower is :

Question 806 :  A person walking along a straight road towards a hill observes at two points distance  $\sqrt{3}$ km, the angle of elevation of the hill to be $30^{0}$ and $60^{0}$. The height of the hill is   

Question 807 : The ladder resting against a vertical wall is inclined at an angle of ${30}^{o}$ to the ground. The foot of the ladder is $7.5m$ from the wall. Find the length of the ladder.

Question 808 : The angle of elevation of a jet plane from a point A on the ground is${ 60 }^{ 0 }$. After a flight of 15 seconds, the angle of elevation changes to${ 30 }^{ 0 }$. If the plane at a constant height of$1500\sqrt { 3 } m$, then the speed of jet plane is :

Question 809 : Two boats are sailing in the sea on either side of a lighthouse. At a particular time the angles of depression of the two boats, as observed from the top of the lighthouse are 45$^{\circ}$ and 30$^{\circ}$ respectively. If the lighthouse is 100m high, find the distance between the two boats.

Question 810 : The angles of elevation of the top of a vertical tower from points at distance $a$ and $b$ from the base and in the same line with it are complementary. If $a > b$, find the height of the tower.

Question 811 : A $25\ m$ long ladder is placed against a vertical wall such that the foot of the ladder is $7\ m$ from the feet of the wall. If the top of the ladder slides down by $4\ cm$, by how much distance will the foot of the ladder slide ?

Question 812 : The shadow of a flagstaff is three times as long as the shadow of the flagstaff when the sun rays meet the ground at$\displaystyle 60^{\circ}$ Find he angle between the sun rays and the ground at the time of longer shadow.

Question 813 : A vertical pole subtends an angle $\tan^{-1}\left (\dfrac {1}{2}\right )$ at a point P on the ground. If the angles subtended by the upper half and the lower half of the pole at P are respectively $\alpha$ and $\beta$ then $(\tan \alpha, \tan \beta) =$

Question 815 : The angle of elevation of stationary cloud from a point 25 ml above the lake is $ 15^0$ and the angle of depression of reflection in the lake is $45^0$ .Then the height of the cloud above the level

Question 816 : The tops of two poles of height 20 m and 14 m are connected by a wire. If the wire makes an angle of $30^o$ with horizontal, then the length of the wire is

Question 817 : Each side of square subtends an angle of $60^{o}$ at the top of a tower of $h$ meter height standing in the centre of the square. If $a$ is the length of each side of the square then which of the following is/are correct?

Question 818 : From the top of a tower $100m$ high ,the angels of depression of the bottom and the top of a building just opposite to it are observed to be ${60^ \circ }$ and ${45^ \circ }$ respectively,then height of the building is 

Question 819 : A man observes the elevation of a balloon to be $30^{0}$ at a point $A$. He then walks towards the balloon and at a certain place $B$, find the elevation to be $60^{0}$. He further walks in the direction of the balloon and finds it to be directly over him at a height of $\dfrac12\ km$, then $AB=$

Question 820 : the altitude of the sun when the length of the shadow is $7\sqrt 3m$.

Question 821 : Two flagstaffs stand on a horizontal plane. A and B are two points on the line joining their feet and between them. The angles of elevation of the tops of the flagstaff as seen from A are 30$^o$ and 60$^o$ and as seen from B are 60$^o$ and 45$^o$. If AB is 30 m, the distance between the flagstaffs in metres is

Question 822 : The angle of elevation of a Jet fighter from a point $A$ on the ground is ${60}^{o}$. After $10$ seconds flight, the angle of elevation changes to ${30}^{o}$. If the Jet is flying at a speed of $432km/hour$, find the height at which the jet is flying.

Question 823 : A man on the deck of a ship is $12m$ above water level. he observes that the angle of elevation, of the top of a cliff is ${45}^{o}$ and the angle of depression of its base is ${30}^{o}$. Calculate the distance of the cliff from the ship and the height of the cliff.

Question 824 : On the same side of a tower, two objects are located. When observed from the top of the tower, their angles of depression are $45^o$ and $60^o$. If the height of the tower is $50\sqrt 3$, then the distance between the objects is

Question 825 : A man in a boat rowing away from a light-house $100m$ high, takes $2$ minutes to change the angle of elevation of the top of the light-house form ${60}^{o}$ to ${45}^{o}$. Find the speed of the boat.

Question 826 : $\tan \theta$ increases as $\theta$ increases. If true then enter $1$ and if false then enter $0$.

Question 828 : Given $tan \theta = 1$, which of the following is not equal to tan $\theta$?

Question 830 : The expression$ \displaystyle \left (\tan \Theta +sec\Theta \right )^{2} $ is equal to

Question 832 : If $\displaystyle 5\tan \theta =4$, then find the value of $\displaystyle \frac{5\sin \theta -3\cos \theta }{5\sin \theta +2\cos \theta }$. 

Question 833 : If $\displaystyle \tan { \theta  } =\frac { 1 }{ 2 } $ and $\displaystyle \tan { \phi  } =\frac { 1 }{ 3 } $, then the value of $\displaystyle \theta +\phi $ is:

Question 835 : If $\sec{2A}=\csc{(A-42^\circ)}$ where $2A$ is acute angle then value of $A$ is

Question 836 : If$\displaystyle \cot A=\frac{12}{5}$ then the value of$\displaystyle \left ( \sin A+\cos A \right )$ $\displaystyle \times cosec$ $\displaystyle A$ is

Question 838 : Find the value of $\sin^3\left( 1099\pi -\dfrac { \pi  }{ 6 }  \right) +\cos^3\left( 50\pi -\dfrac { \pi  }{ 3 }  \right) $

Question 840 : If $\tan \theta = \dfrac {4}{3}$ then $\cos \theta$ will be

Question 841 : The value of $[\dfrac{\tan 30^{o}.\sin 60^{o}.\csc 30^{o}}{\sec 0^{o}.\cot 60^{o}.\cos 30^{o}}]^{4}$ is equal to

Question 842 : If $ \alpha \epsilon \left[ \frac { \pi  }{ 2 } ,\pi  \right] $ then the value of $\sqrt { 1+sin\alpha  } -\sqrt { 1-sin\alpha  } $ is equal to

Question 843 : IF A+B+C=$ \displaystyle 180^{\circ}  $ ,then $  tan A+tanB+tanC $ is equal to

Question 845 : Select and wire the correct answer from the given alternatives.  $\cos \left(\dfrac {3\pi}{2}+\theta \right)=$ ____

Question 846 : Find the value of $ \displaystyle  \theta , cos\theta  \sqrt{\sec ^{2}\theta -1}     = 0$

Question 847 : The given relation is $(1 + \tan a + \cos a)(\sin a - \cos a )= 2\sin a\tan a - cat\,a\cos a$

Question 849 : Solve:$\displaystyle \sin ^{4}\theta +2\cos ^{2}\theta \left ( 1-\frac{1}{\sec ^{2}\theta } \right )+\cos ^{4}\theta $

Question 850 : IF $ \displaystyle \tan \theta =\sqrt{2}    $ , then the value of $ \displaystyle \theta     $ is 

Question 853 : If $\displaystyle  \cos A+\cos ^2A=1$ then the value of $\displaystyle  \sin ^{2}A+\sin ^{4}A$ is

Question 854 : The solution of $(2 cosx-1)(3+2 cosx)=0$ in the interval $0 \leq \theta \leq 2\pi$ is-

Question 856 : If $\theta$ increases from $0^0$ to $90^o$, then the value of $\cos\theta$:

Question 858 : If $\displaystyle x=y\sin \theta \cos \phi ,y=\gamma \sin \theta \sin \phi ,z=\gamma \cos \theta $. Eliminate  $\displaystyle \theta $ and  $\displaystyle \phi $

Question 859 : Choose the correct option. Justify your choice. $\displaystyle 9{ \sec }^{ 2 }A-9{ \tan }^{ 2 }A=$

Question 860 : Simplest form of $\displaystyle \dfrac{1}{\sqrt{2 + \sqrt{2 + \sqrt{2 + 2 cos 4x}}}}$ is

Question 861 : Maximum value of the expression $\begin{vmatrix} 1+{\sin}^{2}x & {\cos}^{2}x & 4\sin2x \\ {\sin}^{2}x & 1+{\cos}^{2}x & 4\sin2x \\ {\sin}^{2}x & {\cos}^{2}x & 1+4\sin2x \end{vmatrix}=$

Question 862 : $\left( \dfrac { cosA+cosB }{ sinA-sinB }  \right) ^{ 2014 }+\left( \cfrac { sinA+sinB }{ cosA-cosB }  \right) ^{ 2014 }=...........$

Question 866 : If $3\sin\theta + 5 \cos\theta =5$, then the value of $5\sin\theta -3 \cos\theta $ are 

Question 867 : As value of $x$ increases from $0$ to $\cfrac{\pi}{2}$, the value of $\cos {x}$:

Question 868 : Value of ${ cos }^{ 2 }{ 135 }^{ \circ  }$

Question 869 : Solve : $\dfrac { 2tan{ 30 }^{ \circ  } }{ 1+{ tan }^{ 2 }{ 30 }^{ \circ  } } $

Question 870 : find whether ${ \left( \sin { \theta  } +co\sec { \theta  }  \right)  }^{ 2 }+{ \left( \cos { \theta  } +\sec { \theta  }  \right)  }^{ 2 }=7+\tan ^{ 2 }{ \theta  } +\cos ^{ 2 }{ \theta  } $ is true or false.

Question 871 : If $sin({ 90 }^{ 0 }-\theta )=\dfrac { 3 }{ 7 } $, then $cos\theta $

Question 873 : Choose and write the correct alternative. If $3 \sin \theta = 4 \cos \theta$ then $\cot \theta = ?$

Question 874 : The value of $\sqrt { 3 } \sin { x } +\cos { x } $ is max. when $x$ is equal to

Question 876 : If $\sec \theta - \cos \theta = 1$, then $\tan^{2} \theta=$

Question 877 : If $\tan\theta$ = $\dfrac{6}{8}$ and $\theta$ is acute, what is the value of $\cos\theta$?

Question 878 : If $ \displaystyle  \cos \theta -\sin \theta =\sqrt{2} \sin \theta $ , then  $ \cos \theta +\sin \theta $ is

Question 879 : $\tan 5^{\circ} \tan 25^{\circ} \tan 30^{\circ} \tan 65^{\circ} \tan 85^{\circ} =$

Question 883 : If$\displaystyle 3\tan { \theta } =4$, then$\displaystyle \sin { \theta }$ is :

Question 885 : The value of $\cot 1^{\circ} \cot 2^{\circ} .... \cot 89^{\circ}$ is .....

Question 886 : If A, B, C are the angles of a triangle, then $\cos A+\cos B+\cos C$ is equal to?

Question 888 : The value of ${ e }^{ \log _{ 10 }{ \tan { 1+ } } \log _{ 10 }{ \tan { 2+\log _{ 10 }{ \tan { 3+...+\log _{ 10 }{ \tan { 89 } } } } } } }$ is

Question 889 : If $A$ and $B$ are acute angles such that $\sin A=\sin^2 B, 2\cos^2 A=3 \cos^2 B$; then

Question 890 : Evaluate: $\displaystyle\frac{-\tan\theta \cot(90^o-\theta)+\sec\theta cosec(90^o-\theta)+\sin^235^o+\sin^255^o}{\tan 10^o\tan 20^o\tan 30^o \tan 70^o \tan 80^o}$.

Question 891 : Let$\theta \in \left( {0,{\pi \over 4}} \right)$ and${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }}$,${t_2} = {\left( {\tan \theta } \right)^{\cot \theta }}$,${t_3} = {\left( {\cot \theta } \right)^{\tan \theta }}$ then${t_4} = {\left( {\cot \theta } \right)^{\cot \theta }}$, then

Question 892 : The maximum value of $\cos x\,\left(\displaystyle \dfrac{\cos x}{1-\sin x}+\dfrac{1-\sin x}{\cos x}\right)$ is

Question 893 : If $\cos 9 \alpha = \sin  \alpha$ and $9 \alpha < 90^o$, what is the value of $\tan  5 \alpha$?

Question 894 : $\displaystyle \tan { { 5 }^{ o } } .\tan { { 40 }^{ o } } .\tan { 4{ 5 }^{ o } } .\tan { { 50 }^{ o } } .\tan { 8{ 5 }^{ o } }$ is equal to :

Question 896 : If $\alpha, \beta$ are the different values of  $3\cos \theta+4\sin \theta=\dfrac{9}{2}$ and $A=\tan\left(\dfrac{\alpha}{2}+\dfrac{\beta}{2}\right), B=\tan\dfrac{\alpha}{2}\tan\dfrac{\beta}{2}, C=\sin(\alpha+\beta),$ then which one of the option is true?

Question 898 : Find the value of $\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ} $

Question 899 : The value of $\cos { { 10 }^{ 0 } } -\sin { { 10 }^{ 0 } } $ is?

Question 900 : Evaluate: $\dfrac {\tan 35^{\circ}}{\cot 55^{\circ}} + \dfrac {\cot 78^{\circ}}{\tan 12^{\circ}} = $

Question 901 : Choose the correct answer and justify. $\quad (1+\tan\theta+\sec\theta)(1+\cot\theta - cosec\theta) = $

Question 902 : The value of $\cos \dfrac {2\pi}{7} + \cos \dfrac {4\pi}{7} + \cos \dfrac {6\pi}{7}$ is

Question 903 : Find the value of:$\sin ^{ 2 }{ 30 } \cos ^{ 2 }{ 45 } +4\tan ^{ 2 }{ 30 } +\cfrac { 1 }{ 2 } \sin ^{ 2 }{ 90 } +\cfrac { 1 }{ 2 } \cot ^{ 2 }{ 60 } $

Question 904 : If $\sec \theta + \tan \theta = p$ then $\sin \theta = \frac { p ^ { 2 } + 1 } { p ^ { 2 } - 1 }$ 

Question 905 : Check whether the statement is true/false  $\sec ^ { 2 } \theta + cosec ^ { 2 } \theta = \sec ^ { 2 } \theta \cdot \sin ^ { 2 } \theta$

Question 908 : IF tan ${ 35 }^{ \circ  }=k,$ then the value of $\dfrac { \tan{ 145 }^{ \circ  }-\tan{ 125 }^{ \circ  } }{ 1+\tan{ 145 }^{ \circ  }\tan{ 125 }^{ \circ  } } =$

Question 909 : If $\sin ^{2}\theta _{1}+\sin ^{2}\theta _{2}+\sin ^{2}\theta _{3}=0$, then which of the following is NOT the possible value of $\cos ^{2}\theta _{1}+\cos ^{2}\theta _{2}+\cos ^{2}\theta _{3}$.

Question 910 : The value of $\displaystyle \frac { \cot { { 40 }^{ o } }  }{ \tan { { 50 }^{ o } }  } -\frac { 1 }{ 2 } \left( \frac { \cos { { 35 }^{ o } }  }{ \sin { { 55 }^{ o } }  }  \right) $ is 

Question 914 : Match the following columns with the values obtained for the solution. <table class="wysiwyg-table"><tbody><tr><td>$I.$ $x\cos \theta+ y\sin \theta=a$, $x\sin \theta- y\cos \theta=b$ </td><td> $a)$ $(x^{2}-y^{2})^{2}=16xy$ </td></tr><tr><td>$II.$ $x= \sec \theta+\tan\theta$, $y=\sec\theta-\tan\theta$ </td><td>$b)$ $xy = 1$ </td></tr><tr><td>$III.$ $x\sec \theta+ y\tan \theta=a$, $x\tan \theta+ y\sec \theta=b$ </td><td>  $c)$ $x^{2}-y^{2}=a^{2}-b^{2}$ </td></tr><tr><td> $IV.$  $x=\cot\theta+\cos\theta$, $y=\cot\theta-\cos\theta$ </td><td>  $d)$ $x^{2}+y^{2}=a^{2}+b^{2}$ </td></tr></tbody></table>

Question 915 : If $\left( 1+\tan { \theta } \right) \left( 1+\tan { \phi } \right) =2$, then $\left( \theta +\phi \right) $ is equal to

Question 916 : The value of the expression$\displaystyle \frac {tan^2 20^0 - sin^2 20^0}{tan^2 20^0 . sin^2 20^0}$ is

Question 917 : If $\displaystyle x = a\cos ^{3}\theta $ and y = b $\displaystyle \sin ^{3}\theta ,$ then the value of $\displaystyle \left ( \frac{x}{a} \right )^{2/3}+\left ( \frac{y}{b} \right )^{2/3}$ is

Question 918 : If $\displaystyle x=r\sin \theta \cdot \cos \phi,$  $y=r\sin \theta \cdot \sin \phi$ and $\displaystyle z= r\cos \theta$, then the value of $\displaystyle x^{2}+y^{2}+z^{2}$ is independent of 

Question 920 : If a line in the space makes angle $a, p$ and $y$ with the coordinate axes, then $\cos\,2a\,+\cos\,2b\,+\,\cos\,2y\,+\,\sin^2\,a\,+\,\sin^2\beta\,+\,\sin^2\,y\,$ equals

Question 921 : If $\tan A = \displaystyle\dfrac{3}{4}$ and $A+B = 90^{\small\circ}$, then what is the value of $\cot B$?

Question 922 : If $\sin \theta, \cos \theta, \tan \theta$ are in $G.P$, then $\cos^{9}\theta+\cos^{6}\theta+3\cos^{5}\theta-1$ is equal to:

Question 923 : The value of   $\displaystyle \sin { \theta  } \cos { \theta  } -\frac { \sin { \theta  } \cos { \left( { 90 }^{ o }-\theta  \right)  } \cos { \theta  }  }{ \sec { \left( { 90 }^{ o }-\theta  \right)  }  } -\frac { \cos { \theta  } \sin { \left( { 90 }^{ o }-\theta  \right)  } \sin { \theta  }  }{ \text{cosec }\left( { 90 }^{ o }-\theta  \right)  } $ is :

Question 924 : If, $\displaystyle \tan \, 9^o = \frac{x}{y}$ then, value of $\displaystyle \frac{\sec^2\, 81^o}{1+\cot^2\, 81^o}$ is

Question 925 : The value of$\displaystyle \frac { \cos { \left( { 90 }^{ o }-A \right) } }{ 1+\sin { \left( { 90 }^{ o }-A \right) } } +\frac { 1+\sin { \left( { 90 }^{ o }-A \right) } }{ \cos { \left( { 90 }^{ o }-A \right) } }$ is equal to :

Question 926 : If $\displaystyle X=\tan 1^{0}+\tan 2^{0}+........+\tan 45^{0}$ and $\displaystyle y= -(\cot 46^{0}+\cot 47^{0}+.......+\cot 89^{0})$ then find the value of $(x + y)$.

Question 929 : If $x = a \cos^{3} \theta \sin^{2} \theta, y = a \sin^{3} \theta \cos^{2} \theta$ and $\dfrac {(x^{2} + y^{2})^{p}}{(xy)^{q}}(p, q\epsilon N)$ is independent of $\theta$, then

Question 930 : Assertion: Statement 1:If $\displaystyle x+y+z= xyz,$ then at most one of the numbers can be negative. Reason: Statement 2: In a triangle ABC, $\displaystyle \tan A+\tan B+\tan C= \tan A \tan B \tan C $ ,there can be at most one obtuse angle in a triangle.

Question 931 : $\cos { { 1 }^{ o } } .\cos { { 2 }^{ o } } .\cos { { 3 }^{ o } } ......\cos { { 179 }^{ o } } $ is equal to

Question 932 : In $\displaystyle A_{n}=\cos^{n}\theta+\sin^{n}\theta, n\in N$ and $\displaystyle \theta \in R$ If $\displaystyle A_{n-4}-A_{n-2}=\sin^{2}\theta\cos^{2}\theta A_{\lambda} $ , then $\displaystyle \lambda $ equals

Question 933 : The value of $\displaystyle \sum _{ r=0 }^{ 10 }{ \cos ^{ 3 }{ \dfrac { \pi r }{ 3 } } }$ is equal to:

Question 934 : The value of the expression $\displaystyle 1\, - \,\frac{{{{\sin }^2}y}}{{1\, + \cos \,y}}\, + \frac{{1\, + \cos \,y}}{{\sin \,y}}\, - \,\frac{{\sin \,y}}{{1\, - \cos \,y}}$ is equal to 

Question 936 : For all real values of $\theta$ , $\cot\theta-2 \cot 2\theta$ is equal to

Question 937 : Find the relation obtained by eliminating$\displaystyle \theta $ from the equation$\displaystyle x=a\cos \theta +b\sin \theta $ and$\displaystyle y=a\sin \theta -b\cos \theta $

Question 939 : If $\sin A, \cos A$ and $\tan A$ are in G.P. then $\cot^6 A- \cot^2A$ is equal to

Question 940 : If $\displaystyle \sin \theta+\sin ^{2} \theta +\sin ^{3}\theta= 1$ then the value of $\displaystyle \cos ^{6}\theta-4\cos ^{4}\theta+8\cos ^{2}\theta$ equals

Question 942 : If $a=\cos\alpha \cos\beta+\sin \alpha \sin\beta \cos\gamma$ $b=\cos\alpha \sin \beta-\sin\alpha \cos\beta \cos\gamma$ and $c=\sin \alpha \sin\gamma$, then $a^2+b^2+c^2$ is equal to

Question 943 : If $3 \sin\theta+ 5 \cos\theta=5$, then $5 \sin\theta-3 \cos\theta$ is equal to

Question 945 : ${\cos ^2}{48^ \circ } - {\sin ^2}{12^ \circ }$ is equal to -

Question 946 : A person on the top of tower observes scooter moving with uniform velocity towards the base of the tower he finds that the angle of depression changes from$\displaystyle 30^{\circ}$ to$\displaystyle 60^{\circ}$ in 18 minutes The Scooter will reach the base of the tower in next

Question 948 : If the quadratic equation $ax^2+bx+c=0$ ($a > 0$) has $\sec^2\theta$ and $\text{cosec}^2\theta$ as its roots, then which of the following must hold good?

Question 949 : $A$ tower of height $h$' standing at the centre of a square with sides of length $a$' makes the same angle $\alpha$ at each of the four corners then $a^{2}=$

Question 951 : If the angles of a triangle are in arithmetic progression such that $\sin (2A+B)=\dfrac{1}{2}$, then

Question 953 : In a triangle $ABC$, right angled at $C$, $a$, $b$ $c$ are the lengths of sides of triangle and hypotenuse respectively. Find the value of $\tan A+\tan B$.

Question 955 : If $\displaystyle \frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1,\frac{x}{a}\sin \theta-\frac{y}{b}\cos \theta=1,$ then eliminate $\theta $

Question 956 : If $x_{1}=1$ and $x_{n+1}=\frac{1}{x_{n}}\left ( \sqrt{1+x_{n}^{2}}-1 \right ),n\geq 1,n \in N$, then $x_{n}$ is equal to :

Question 957 : If $16\cot \theta = 12$, then $\dfrac {\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = $ _____

Question 958 : In atriangle $ABC$, $\sin A\cos B=\dfrac{1}{4}$ and $3\tan A=\tan B$ , the triangle is

Question 959 : If $\tan { \theta  } +\sin { \theta  } =m, \tan { \theta - \sin { \theta =n }  } $, then $(m^{2}-n^{2})^{2}=$.

Question 960 : If $cosec \theta -\sin \theta =m$ and $\sec \theta -\cos \theta =n$, eliminate $\theta $.

Question 961 : Let $x=(1+\sin A)(1-\sin B)(1+\sin C), y=(1-\sin A)(1-\sin B)(1-\sin C)$ and if $x=y$, then

Question 962 : If $\displaystyle \frac { \sin ^{ 4 }{ x }  }{ 2 } +\frac { \cos ^{ 4 }{ x }  }{ 3 } =\frac { 1 }{ 5 } ,$ then:

Question 963 : If $x \cos \alpha +y \sin \alpha=x \cos\beta+y \sin\beta=2a(0 < \alpha, \beta < \pi /2)$, then

Question 965 : If$\displaystyle \sin \Theta =\frac{3}{5} $ and$\displaystyle \Theta $ is acute then find the value of$\displaystyle \frac{\tan \Theta -2\cos \Theta }{3\sin \Theta +\sec \Theta }$

Question 966 : In a $\Delta ABC$, if $\cos A \cos B \cos C=\displaystyle\dfrac {\sqrt 3-1}{8}$ and $\sin A. \sin B. \sin C=\displaystyle \dfrac {3+\sqrt 3}{8}$, then- On the basis of above information, answer the following questions:The angles of $\Delta ABC$ are:

Question 968 : In $\triangle ABC$, the measure of $\angle B$ is $90^{\circ}, BC = 16$, and $AC = 20$. $\triangle DEF$ is similar to $\triangle ABC$, where vertices $D, E,$ and $F$ correspond to vertices. $A, B$, and $C$, respectively, and each side of $\triangle DEF$ is $\dfrac {1}{3}$ the length of the corresponding side of $\triangle ABC$. What is the value of $\sin F$?

Question 969 : If $2 \sec 2\alpha = \tan\beta + \cot \beta$, then one of the value of $\alpha+\beta$ is-

Question 970 : $1)$ lf $\mathrm{x}$ lies in the lst quadrant and $\cos \mathrm{x}+\cos 3\mathrm{x}=\cos 2\mathrm{x}$ then $\mathrm{x}=30^{\mathrm{o}}$ or $45^{\mathrm{o}}$ $2)\mathrm{x}\in(0,2\pi)$ and cosec $\mathrm{x}+2=0$ then $x=\displaystyle \frac{7\pi}{6},\frac{l1\pi}{6}$ $3)\mathrm{x}\in[0,2\pi]$ and $(2 \cos \mathrm{x}- \mathrm{l}) (3+2\cos \mathrm{x})=0$ then $\displaystyle \mathrm{x}=\frac{\pi}{3}$ , $\displaystyle \frac{5\pi}{3}$ Which of the above statements are correct?

Question 971 : If $\text{cosec } \theta = \dfrac {13}{5}$, then $\cos \theta = ......$

Question 973 : The value of $ \cos y \cos\left(\dfrac{\pi}{2} -x\right) - \cos \left(\dfrac{\pi}{2}-y \right)\cos x + \sin y \cos\left(\dfrac{\pi}{2}-x\right)+ \cos x \sin\left(\dfrac{\pi}{2} -y\right)$ is zero if

Question 974 : In a $\Delta ABC$, if $\cos A \cos B \cos C=\displaystyle\dfrac {\sqrt 3-1}{8}$ and $\sin A. \sin B. \sin C=\displaystyle \dfrac {3+\sqrt 3}{8}$,  then- On the basis of above information, answer the following questions:The value of $ \tan A \tan B + \tan B \tan C + \tan C \tan A$ is:

Question 975 : If $ \cos^{-1}\left ( 4x^{3}-3x \right )= 2\pi -3\cos^{-1}x $, then $ x $ lies in interval

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