Question 1 :
State the following statement is True or FalseThe length of a rectangle ($x$) exceeds its breadth by $3$ cm. The area of a rectangle is $70$ sq.cm, then the equation is $x\, (x\, -\, 3)\, =\, 70$.<br/>
Question 2 :
Solve the following quadratic equation by factorization :<br>$a(x^2 \, + \, 1) \, - \, x \, (a^2 \, + \, 1) \, = \, 0$
Question 3 :
Choose the best possible option.<br>$\displaystyle{ x }^{ 2 }+\frac { 1 }{ 4{ x }^{ 2 } } -8=0$ is a quadraticequation.<br>
Question 4 :
The number of solutions of the equation,$2\left\{ x \right\} ^{ 2 }+5\left\{ x \right\} -3=0$ is
Question 5 :
If $\alpha \epsilon \left( -1,1 \right) $ then roots of the quadratic equation $\left( a-1 \right) { x }^{ 2 }+ax+\sqrt { 1-{ a }^{ 2 } } =0$ are
Question 10 :
Is the following equation quadratic?$(x\, +\, 3) (x\, -\, 4)\, =\, 0$
Question 11 :
If $\displaystyle \frac{5x+6}{\left ( 2+x \right )\left ( 1-x \right )}=\frac{a}{2+x}+\frac{b}{1-x}$, then the values of a and b respectively are
Question 12 :
Applying zero product rule for the equation $x^{2}- ax - 30 = 0$ is $x = 10$, then $a =$ _____.<br/>
Question 13 :
The quadratic polynomial whose sum of zeroes is $3$ and product of zeroes is $- 2$ is:<br/>
Question 15 :
A quadratic equation in $x$ is $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and the other condition is<br/>
Question 17 :
If $x^{2} - 4x + 2 = 0$, then the value of $4x^{2} + 2x + \dfrac {4}{x} + \dfrac {16}{x^{2}}$ is
Question 18 :
Which of the following is a quadratic polynomial in one variable?<br>
Question 19 :
The factors of the equation, $k(x + 1)(2x + 1) = 0$, find the value of $k$.<br/>
Question 21 :
The length of a rectangular verandah is $3\:m$ more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter. Taking $x$ as the breadth of the verandah, write an equation in $x$ that represents the above statement.
Question 22 :
The discriminant of $x^2 - 3x + k = 0$ is 1 then the value of $k = .............$
Question 23 :
Before Robert Norman worked on 'Dip and Field Concept', his predecessor thought that the tendency of the magnetic needle to swing towards the poles was due to a point attractive. However, Norman showed with the help of experiment that nothing like point attractive exists. Instead, he argued that magnetic power lies is lodestone. Which one of the following is the problem on which Norman and others worked?
Question 24 :
Is the following equation a quadratic equation?$\displaystyle 3x + \frac{1}{x} - 8 = 0$
Question 27 :
Obtain a quadratic equation whose roots are reciprocals of the roots of the equation $x^2-3x - 4 =0$.
Question 28 :
If $x - 4$ is one of the factor of $x^{2} - kx + 2k$, where $k$ is a constant, then the value of $k$ is
Question 29 :
A quadratic equation $ax^2 + bx+c=0$ has two distinct real roots, if<br>
Question 33 :
Choose best possible option.<br>$\displaystyle\left( x+\frac { 1 }{ 2 } \right) \left( \frac { 3x }{ 2 } +1 \right) =\frac { 6 }{ 2 } \left( x-1 \right) \left( x-2 \right)$ is quadratic.<br>
Question 34 :
The nature of the roots of a quadratic equation is determined by the:<br>
Question 35 :
If $p$ and $q$ are the roots of the equation $x^2-30x+221=0$, what is the value of $p^3+q^3$ ?
Question 36 :
State the following statement is True or False<br/>The sum of a natural number $x$ and its reciprocal is $\displaystyle \frac{37}{6}$, then the equation is $x\, +\, \displaystyle \frac{1}{x}\, =\, \displaystyle \frac{37}{6}$.<br/>
Question 37 :
The following equation is a qudratic equation. $16x^2 \, - \, 3 \, = \, (2x \, + \, 5)(5x \, - \, 3)$
Question 38 :
In a rectangle the breadth is one unit less than the length and the area is $12$ sq.units. Find the length of the rectangle.
Question 39 :
The mentioned equation is in which form?$z\, -\, \cfrac{7}{z}\, =\, 4z\, +\, 5$
Question 41 :
Check whether the following is a quadratic equation.$(x - 3) (2x + 1) = x (x + 5)$<br/>
Question 42 :
If the roots of the equation $5{x}^{2}-7x+k=0$ are mutually reciprocal then $k=$
Question 43 :
If c is small in comparision with l then ${\left( {\frac{l}{{l + c}}} \right)^{\frac{1}{2}}} + {\left( {\frac{l}{{l - c}}} \right)^{\frac{1}{2}}} = $
Question 44 :
Check whether the given equation is a quadratic equation or not.<br/>${ x }^{ 2 }+2\sqrt { x } -3$
Question 45 :
If $3$ is one of the roots $x^2-mx+15=0$. Choose the correct options -<br/>
Question 46 :
The least integer $'c'$ which makes the roots of the equation $x^2+3x+2c$ imaginary is
Question 49 :
State the following statement is True or False<br/>The product of two numbers $y$ and $(y - 3)$ is $42$, then the equation formed can be represented as $y\, (y\, -\, 3)\, =\, 42$<br/>
Question 51 :
The value of k for which the equation $x^{2} - 4x + k = 0 $ has equal roots is<br/>
Question 52 :
Which of the following equations has no solution for $a$ ?
Question 53 :
If $a,b,c$ are positive real numbers, then the number of real roots of the equation$ ax^{2}+b\left |x \right |+c=0 $is
Question 54 :
Find the value of K so that sum of the roots of the equations $3x^2 + (2x - 11) x K - 5 = 0$ is equal to the product of the roots.
Question 55 :
The quadratic equations $x^{2}-6x+a=0$ and $x^{2}-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4: 3$. Then the common root is<br><br>
Question 56 :
If $\alpha , \beta , \gamma $ are the real roots of the equation $x^{3}-3px^{2}+3qx-1=0$, then the centroid of the triangle with vertices $\displaystyle \left ( \alpha , \frac{1}{\alpha } \right )$, $\left ( \beta , \dfrac{1}{\beta } \right )$ and $\displaystyle \left ( \gamma , \frac{1}{\gamma } \right )$ is at the point
Question 57 :
What is the smallest integral value of $k$ such that $2x (kx - 4) - x^{2} + 6 = 0$ has no real roots?
Question 59 :
The discrimination of the equation $x^2 + 2x\sqrt3 + 3 = 0$ is zero. Hence, its roots are:
Question 60 :
Assertion: If $a + b + c = 0$ and $a, b, c$ are rational, then the roots of the equation $(b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0$ are rational .
Reason: Discriminant of $(b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0$ is a perfect square .
Question 61 :
The ______ product rule says that when the product of two terms is zero, then either of the terms is equal to zero.<br>
Question 62 :
Suppose $a,b,c \in R$, $a \ne 0$ and $4a - 6b + 9c < 0\,$ and $a{x^2} + bx + c = 0$ does not have real roots, then
Question 63 :
Minimum possible number of positive root of the quadratic equation${x^2} - (1 + \lambda )x + \lambda - 2 = 0, \in R:$
Question 64 :
If the roots of the quadratic equation $x^2+6x+b=0$ are real and distinct and they differ by atmost $4$ then the least value of $b$ is-
Question 65 :
If the roots of the equation $ x^{2} -15-m(2x-8)=0 $ are equal, then $m =$
Question 66 :
The value of $'k'$ for which the roots of equation $(x - 1)(x - 5) + k = 0$ differ by 2 is<br>
Question 67 :
Assertion (A): The roots of $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$  are real<br/>Reason (R): A quadratic equation with non-negative discriminant has real roots .<br/>
Question 68 :
$\alpha$ and $\beta$ are the roots of the equation $ax^{2}+bx+c=0$ and $\alpha^{4}, \beta^{4}$ are the roots of the equation $lx^{2}<br>+mx+n=0(\alpha, \beta$ are real and distinct.) Let $f(x)=a^{2}lx^{2}-4aclx+2c^{2}l+a^{2}m=0$, then<br>Roots of $f(x)=0$ are<br>
Question 69 :
If roots of the equation $12x^2 + mx + 5 = 0$ are in the ratio $3 : 2$, then $m =$
Question 70 :
Find the values of $k$ for the following quadratic equation, so that they have two real and equal roots:$4x^2 - 2(k + 1)x + (k + 4) = 0$
Question 71 :
If $b_1b_2=2(c_1+c_2)$, then at least one of the equations $x^2+b_1x+c_1=0$ and $x^2+b_2x+c_2=0$ has
Question 72 :
Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals? Why?<br><br>
Question 73 :
If $\alpha, \beta$ are the roots of the equation $2x^2 + 4x-5=0$, the equation whose roots are the reciprocals of $2\alpha -3$ and $2 \beta -3$ is<br>
Question 74 :
If (x-a)(x-b)+(x-b)(x-a)=0 has equal roots, then the relation between a, b is 
Question 75 :
The quadratic $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be equal to
Question 76 :
If the coefficient of $x^2$ and the constant term of a quadratic equation have opposite signs, then the quadratic equation has _______ roots.<br/>
Question 77 :
The roots of the equation $(b+c)x^2-(a+b+c)x+a=0$ $(a,b,c\ \epsilon \Q, b+c \neq a)$ are
Question 78 :
The set of values of k for which the given quadratic equation has real roots<br/>$2x^2$ + kx +2 = 0 is k $\leq$ 9
Question 79 :
If the equation $\displaystyle \lambda x^{2}-2x+3= 0$has positive roots for some real$\displaystyle \lambda $, then
Question 80 :
If $(x-a)(x-5)+2=0$ has only integral roots where $ a\in I,$ then value of 'a' can be:<br>
Question 81 :
If $a, b$ and $c$ are non-zero real numbers and $a{z}^{2}+bz+c+i=0$ has purely imaginary roots, then $a$ is equal to
Question 82 :
$x^2-(m-3)x+m=0\:\:(m \in R)$ be a quadratic equation. Find the value of $m$ for which, at least one root is greater than $2$.
Question 84 :
If $p, q$ are odd integers, then the roots of the equation $2px^{2} + (2p + q) x + q = 0$ are
Question 85 :
If $x=5+2\sqrt{6}$, then the value of ${ \left( \sqrt { x } -\cfrac { 1 }{ \sqrt { x } } \right) }^{ 2 }$ is _____
Question 86 :
The factors of the equation, $(x + k)\left (x + \dfrac{1}{2}\right) = 0$, find the value of $k$.<br/>
Question 87 :
If the roots of the equation $\displaystyle x^{2}+px-6=0$ are $6$ and $-1$ then the value of $p$ is
Question 88 :
Determine the value of $k$ for which the $x = -a$ is a solution of the equation $\displaystyle x^{2}-2\left ( a+b \right )x+3k=0 $<br/>
Question 89 :
Assertion: If $a$ and $b$ are integers and the roots of $x^2+ax+b=0$ are rational then they must be integers.
Reason: If the coefficient of $x^2$ in a quadratic equation is unity then its roots must be integers.
Question 90 :
The given equation has real roots. State true or false: $8x^2 + 2x -3 = 0$<br/>
Question 91 :
If $a, b, c \in  Q, $ then roots of $ax^2 + 2(a + b)x (3a + 2b) = 0$ are<br/>
Question 92 :
I. lf one root of the equation $5x^{2}+13x+k=0$ is the reciprocal of the other, then $k=5$<br>II. lf the roots of the equation $a(b-c)x^{2}+b(c-a)x+c(a-b)=0$ are equal, then $a, b,c$ are in H.P.<br>Which of the above statement is true?<br>
Question 94 :
Find the value of $k$ for the following quadratic equation, so that they have two real and equal roots:$x^2 - 2(k + 1)x + k^2 = 0$
Question 95 :
The value of k for which the roots are real and equal of the following equation<br/>$x^2$ - 4kx + k = 0 are k = 0, $\dfrac{1}{4}$
Question 96 :
The roots of the equation $ax^2 + bx + c = 0$ will be imaginary if
Question 97 :
I: If $a,b,c$ are real, the roots of $(b-c)x^{2}+(c-a)x+(a-b)=0$ are real and equal, then $a, b, c$ are in A.P.<br>II: If $a, b, c$ are real andthe roots of$(a^{2}+b^{2})x^{2}-2b(a+c)x+b^{2}+c^{2}=0$ are real and equal, then $a, b,c$ are in H.P.<br>Which of the above statement(s) is(are) true?<br>
Question 98 :
If the roots of the quadratic equation $x^2 - 4x - \log_3 a = 0$ are real, then the least value of $a$ is
Question 99 :
Each of the equations $3x^2 - 2 = 25, (2x - 1)^2 = (x - 1)^2, \sqrt{x^2 - 7} = \sqrt{x - 1}$ has
Question 100 :
If one of roots of $x^2+ ax + 4 = 0$ is twice the other root, then the value of 'a' is .
Question 102 :
The difference between two positive integers is $13$ and their product is $140$. Find the two integers.<br/>
Question 104 :
Equation $x^2 - x + q = 0$ has imaginary roots if 
Question 105 :
Given expression is $x^{2} - 3xb + 5 = 0$. If $x = 1$ is a solution, what is $b$?
Question 106 :
The quadratic equation $p(x) =0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x)) =0$ has
Question 107 :
The condition that the roots of the equation $\displaystyle ax^{2}+bx+c=0$ be such that one root is $n$ times the other is 
Question 108 :
Let $a,b,c$ be real and $ { ax }^{ 2 }+bx+c=0$ has two real roots, $\alpha$ and $\beta$ where $\alpha <-1$ and $\beta > 1$, then $ 1+\dfrac { c }{ a } +\left| \dfrac { b }{ a } \right| < 0$ is
Question 110 :
If one root of the equation $a{ x }^{ 2 } + bx + c = 0$ be the square of the other, then the value of${ b }^{ 3 } + { a }^{ 2 }c + a{ c }^{ 2 } $ is<br>
Question 111 :
The number of integral values of $a$ for which the quadratic expression $(x-a)(x-10)+1$ can be factored as a product $(x+\alpha)(x+\beta)$ of two factors $\alpha, \beta, \in I$, is
Question 112 :
If $m_1$ and $m_2$ are the roots of the equation $x^2+\left(\sqrt{3}+2\right)x+\left(\sqrt{3}-1\right)=0$, then the area of the triangle formed by the lines $y=m_1x,y=m_2x$ and $y=2$ is :
Question 113 :
The roots of the equation$\displaystyle \left ( x-a \right )\left ( x-b \right )+\left ( x-b \right )\left ( x-c \right )+\left ( x-c \right )\left ( x-a \right )=0$ are
Question 114 :
lf $\mathrm{a},\ \mathrm{b},\ \mathrm{c}$ are in G.P. then the equations $\mathrm{a}\mathrm{x}^{2}+2\mathrm{b}\mathrm{x}+\mathrm{c}=0$ and $\mathrm{d}\mathrm{x}^{2}+2\mathrm{e}\mathrm{x}+\mathrm{f}=0$ have a common root if $\dfrac { d }{ a } ,\dfrac { e }{ b } ,\dfrac { f }{ c } $ are in <br/>
Question 115 :
If $\tan \dfrac {\alpha}{2}$, and $\tan \dfrac {\beta}{2}$ are the roots of $8x^{2} - 26x + 15 = 0$, then $\cos (\alpha + \beta)$ is equal to
Question 116 :
A tradesman finds that by selling a bicycle for Rs. 75, which he had bought for Rs. $x$, he gained $x$%. Find the value of $x$.
Question 117 :
The number of values of $\displaystyle k$for which$\displaystyle \left ( x^{2} - \left ( k - 2 \right )x + k^{2} \right ) \left ( x^{2} + kx + \left ( 2k - 1 \right ) \right )$is a perfect square
Question 118 :
If the equation $\displaystyle\frac{x^{2}-bx}{ax-c}=\frac{m-1}{m+1}$has roots equal in magnitude but opposite in sign, then $m=$<br>
Question 119 :
Divide 15 into 2 parts such that the product of 2 numbers is 56.
Question 120 :
If $22^3 +23^3+24^3+.........+88^3 $is divided by 110 then the remainder will be
Question 121 :
If the equations ${x}^{2}+ax+12=0$, ${x}^{2}+bx+15=0$ and ${x}^{2}+(a+b)x+36=0$ have a common root then the possible values of $a,b$ is (are)
Question 122 :
$(a^{4}-1)x^{2}-(a^{2}+1)(sin^{-1}sin^{3} 2)x+(cos^{-1}cos2)(a^{2}-1) =0$.<br>.Find the set of values of a so that above equation have roots of opposite in sign.<br><br><br>
Question 123 :
If $x = 3t, y = 1/ 2(t + 1)$, then the value of $t$ for which $x = 2y$ is
Question 124 :
Find the roots of equation:<br>$\displaystyle{ x }^{ 2 }-\frac { 1 }{ 12 } x-\frac { 1 }{ 12 } =0$<br>
Question 125 :
The value of $a$ for which the equation $a ^ { 2 } + 2 a + \csc ^ { 2 } \pi ( a + x ) = 0$ has a solution, is/are
Question 126 :
Number of positive integral values of $b$ for which both roots of the quadratic equation $\displaystyle x^2 + bx - 16 = 0$ are integers, is 
Question 127 :
If $|2x + 3|\le 9$ and $2x + 3 < 0$, then
Question 128 :
Consider the quadratic equation $(1+m)x^2-2(1+3m)x+(1+8m)=0$, (where $m \in R-\left \{-1\right \})$, then the set of values of $'m'$ such that the given quadratic equation has both roots positive are,
Question 129 : <p>State the following statement is True or False</p><p>If the roots of the equation $x^2\,+\,px\,+\,q\,=\,0$ differ by $1$, then $p^2\,=\,1\,+\,4q$</p>
Question 130 :
The real number $k$ for which the equation $2x^ {3}+3x+k=0$ has two distinct real roots in $[0,1]$
Question 131 :
The coefficient of $x$ in the equation $x^2+px+q=0$ was wrongly written as $17$ in place of$13$ and the roots thus found was $-2$ and $-15$.<br>Then the roots of the correct equation are
Question 132 :
The rectangular fence is enclosed with an area $16$cm$^{2}$. The width of the field is $6$ cm longer than the length of the fields. What are the dimensions of the field?<br/>
Question 133 :
If both the roots of the equation$\displaystyle x^{2}-6ax+2-2a+9a^{2}=0$ exceed $3$, then
Question 134 :
If one of the roots of $\displaystyle x^{2}+f(a)x+a=0$ is equal to third power of the other for all real $a$, then 
Question 135 :
If $\alpha$ and $\beta$ are roots of the equation $a{ x }^{ 2 }+bx+c=0$ then the equation whose roots are $\alpha +\frac { 1 }{ \beta }$ are $\beta +\frac { 1 }{ \alpha }$ is
Question 137 :
The equation $\displaystyle 9y^{2}(m+3)+6(m-3)y+(m+3)=0 $, where $m$ is real has real roots then 
Question 140 :
The total cost price of certain number of books is $450$. By selling the books at $50$ each, a profit equal to the cost price of $2$ books is made. Find the approximate number of books.<br/>
Question 141 :
Assertion: If the roots of the equations $x^2-bx+c=0$ and $x^2-cx+b=0$ differ by the same quantity, then $b+c$ is equal to $-4$
Reason: If $\alpha,\beta$ are the roots of the equation $Ax^2+Bx+C=0,$ then $\displaystyle \alpha -\beta =\frac { \sqrt { { B }^{ 2 }-4AC }  }{ A } $
Question 142 :
By increasing the speed of the car by $10 km/hr$ the time of journey for a distance of $72 km$ is reduced by $36$ minutes. Find the original speed of the car.
Question 143 :
If $a, b$ and $c$ are in arithmetic progression, then the roots of the equation $ax^{2} - 2bx + c = 0$ are 
Question 144 :
If one of the roots of the quadratic equation $a{ x }^{ 2 }-bx+a=0$ is $6$, then the value of $\cfrac { b }{ a } $ is equal to
Question 145 :
If each pair of the following three equations $ { x }^{ 2 }+ax+b=0$, ${ x }^{ 2 }+cx+d=0$, ${ x }^{ 2 }+ex+f=0$ has exactly one root in common, then <br/>
Question 146 :
If the roots of the quadratic equation $x^2+6x+b=0$ are real and distinct and they differ by at most $4$, then the range of values of $b$ is:
Question 148 :
If $\alpha \,\& \beta $ are  roots if the equation ${x^2} + 5x - 5 = 0$, then evaluate $\dfrac{1}{{{{(\alpha  + 1)}^3}}} + \dfrac{1}{{{{(\beta  + 1)}^3}}}$
Question 149 :
If $x^2-10ax-11b=0$ has roots $c$ and $d$, then, $x^2-10x-11d=0$ has roots $a$ and $b$, then $a+b+c+d=$
Question 150 :
Two roots of $4x^3 + 8x^2+ Kx - 18 = 0$ are equal numerically but opposite in sign. Find the value of K.
