Question 1 :
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<br/>$3x^{2}-4xy + 8y^{2}$
Question 2 :
If the quotient of $\displaystyle x^4 - 11x^3 + 44x^2 - 76x +48$. When divided by $(x^2 - 7x +12)$ is $Ax^2 + Bx + C$, then the descending order of A, B, C is
Question 3 :
Simplify:Find$\ x(x + 1) (x + 2) (x + 3) \div  x(x + 1)$<br/>
Question 4 :
$\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 5 :
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 6 :
If the roots of ${ x }^{ 2 }-2mx+{ m }^{ 2 }-1=0$ lie between $-2$ and $4$, then
Question 8 :
Factorise the expressions and divide them as directed.$12xy(9x^2-  16y^2)\div  4xy(3x + 4y)$
Question 9 :
Find the expression which is equivalent to : $\displaystyle \frac { { x }^{ 3 }+{ x }^{ 2 } }{ { x }^{ 4 }+{ x }^{ 3 } } $?
Question 10 :
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 13 :
The degree of the remainder is always less than the degree of the divisor.
Question 14 :
If a polynomial $p(x)$ is divided by $x - a$ then remainder is<br/>
Question 18 :
If $\alpha , \beta $ are the roots of the equation $ax^{2}+bx+c=0$, find the value of $\alpha ^{2}+\beta ^{2}$.
Question 19 :
Divide the first expression by the second. Write the quotient and the remainder.<br/>$\displaystyle x^2-\frac{1}{4x^2}; x-\frac{1}{2x}$
Question 20 :
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and their coefficients.$2s^2-(1+2\sqrt 2)s+\sqrt 2$<br/>
Question 22 :
Divide:$\left ( 15y^{4}- 16y^{3} + 9y^{2} - \cfrac{1}{3}y - \cfrac{50}{9} \right )$ by $(3y-2)$Answer: $5y^{3} + 2y^{2} - \cfrac{13}{3}y + \cfrac{25}{9}$
Question 23 :
Work out the following divisions.<br/>$96abc(3a -12) (5b +30)\div  144(a-  4) (b+  6)$<br/>
Question 24 :
What must be added to $x^3-3x^2-12x + 19$, so that the result is exactly divisible by $x^2 + x-6$?
Question 25 :
Divide the first expression by the second. Write the quotient and the remainder.<br/>$a^2-b^2 ; a-b$
Question 27 :
If $x\ne -5$ , then the expression $\cfrac{3x}{x+5}\div \cfrac {6}{4x+20}$ can be simplified to
Question 29 :
If $\alpha$ and $\beta$ are the zeroes of the polynomial $4x^{2} + 3x + 7$, then $\dfrac{1}{\alpha }+\dfrac{1}{\beta }$ is equal to:<br/>
Question 30 :
If $a\ne 2$, which of the following is equal to $\cfrac { b\left( { a }^{ 2 }-4 \right) }{ ab-2b } $?
Question 31 :
What is $\dfrac {x^{2} - 3x + 2}{x^{2} - 5x + 6} \div \dfrac {x^{2} - 5x + 4}{x^{2} - 7x + 12}$ equal to
Question 32 :
Factorise the expressions and divide them as directed.$4yz(z^2 + 6z-  16)\div  2y(z + 8)$<br/>
Question 33 :
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 34 :
State whether True or False.Divide: $x^2 + 3x -54 $ by $ x-6 $, then the answer is $x+9$.<br/>
Question 36 :
What is the remainder, when<br>$(4{x^3} - 3{x^2} + 2x - 1)$ is divided by (x+2)?<br>
Question 37 :
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 38 :
State whether True or False.Divide : $a^2 +7a + 12 $ by $  a + 4 $, then the answer is $a+3$.<br/>
Question 41 :
The product of the roots of the quadratic equation $2x^{2}-8x+3=0$ is
Question 43 :
If $\alpha , \beta$ are the zeros of the polynomials $f(x) = x^2+x+1 $ then $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=$________.
Question 44 :
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 48 :
Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and their coefficients.$49x^2-81$<br/>
Question 49 :
State whether true or false:Divide: $4a^2 + 12ab + 91b^2 -25c^2 $ by $ 2a + 3b + 5c $, then the answer is $2a+3b+5c$.<br/>
Question 50 :
Work out the following divisions.$10y(6y + 21) \div 5(2y + 7)$<br/>
Question 51 :
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< $ <br> $r<s $are in arithmetic progression, then the values of $A $and $B $are
Question 52 :
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 55 :
If a and b are the roots of the quadratic equation$\displaystyle { 2x }^{ 2 }-6x+3=0$, find the value of<br>$\displaystyle { a }^{ 3 }+{ b }^{ 3 }-3ab\left( { a }^{ 2 }+{ b }^{ 2 } \right) -3ab\left( a+b \right)$.<br>
Question 57 :
Workout the following divisions<br/>$a(a + 1) (a + 2) (a + 3) \div a(a + 3)$
Question 59 :
If equation $\displaystyle p{ x }^{ 2 }+9x+3=0$ has real roots, then find value of $p$.<br/>
Question 60 :
If $a, b$ are the roots of $x^2 + px + 1 = 0$ and $c, d$ are the roots of $x^2 + qx + 1 = 0,$ the value of $E = (a - c)(b - c)(a + d) ( b + d)$ is
Question 61 :
Find the Quotient and the Remainder when the first polynomial is divided by the second.<br/>$(x^3+1) $ by $(x+1)$
Question 62 :
Divide the following and write your answer in lowest terms: $\dfrac{2x^2+5x-3}{2x^2+9x+9}\div \dfrac{2x^2+x-1}{2x^2+x-3}$
Question 63 :
Find the zeros of the quadratic polynomial $f(x) = x^2-3x -28$ and verify the relationships between the zeros and the coefficients.
Question 64 :
The product of the roots of the equation $(x -2)^2 -3(x -2) + 2 = 0$ is
Question 65 :
If $\alpha$ and $\beta$ be two zeros of the quadratic polynomial $ax^2+bx+c$, then $\dfrac {1}{\alpha^3}+\dfrac {1}{\beta^3}$ is equal to <br/>
Question 66 :
The area of a rectangle is $\displaystyle 12y^{4}+28y^{3}-5y^{2}$. If its length is $\displaystyle 6y^{3}-y^{2}$, then its width is
Question 67 :
If the roots of the equation $\dfrac{x^{2}-bx}{ax-c}=\dfrac{m-1}{m+1}$ are equal but opposite in sign, then the value of $m$ will be
Question 68 :
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 69 :
Let $P(x) = 1 + x + x^2 +x^3+x^4+x^5$. What is the remainder when $P(x^{12})$ is divided by $P(x)$?
Question 70 :
Evaluate: $96 abc (3a -12)(5b -30) \div 144 (a -4) (b -6)$
Question 73 :
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 75 :
Divide the following and write your answer in lowest terms: $\dfrac{x^2+11x+28}{x^2-4x-77}\div \dfrac {x^2+7x+12}{x^2-2x-15}$
Question 76 :
Simplify: $\displaystyle \frac { 20xyz\left( 4x+5y+6z \right)  }{ xz\left( 40x+50y+60z \right)}$
Question 78 :
Find the polynomial which when divided by $3x + 4$, equals $2x^{2} + 5x - 3$ with a remainder of $3$
Question 79 :
Divide $\displaystyle 4{ x }^{ 2 }{ y }^{ 2 }\left( 6x-24 \right) \div 4xy\left( x-4 \right) $
Question 81 :
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 82 :
Evaluate: $\displaystyle \frac { x\left( 8{ x }^{ 2 }-32 \right)  }{ 8x\left( x-4 \right)  } $
Question 85 :
Divide the following and write your answer in lowest terms: $\dfrac{x^2-4x-5}{x^2-25}\div \dfrac{x^2-3x-10}{x^2+7x+10}$
Question 86 :
If $x^4 \, + \, 2x^3 \, - \, 3x^2 \, + \, x \, - \, 1$ is divided by $x - 2$. then the remainder is
Question 87 :
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 89 :
Divide $\displaystyle 8\left( 3x+4 \right) \left( 8x+9 \right) $ by $\displaystyle \left( 3x+4 \right) $
Question 90 :
Find the value of $p$ in the equation, where the roots are real,<br/>$\displaystyle 5{ x }^{ 2 }+3x-p=0$<br/>
Question 91 :
If the sum of two numbers is $9$ and the sum of their squares is $41$, then the numbers are<br/>
Question 92 :
If one of the zeros of the quadratic polynomial $2x^2 + px + 4$ is 2, find the other zero. Also find the value of p<br>
Question 93 :
If the roots of the equation $ a x^{2}+b x+c=0 $are of the form $(k+1) / k $and $(k+2) /(k+1), $then $(a+b+c)^{2} $is equal to
Question 94 :
The value of $\displaystyle \frac { 28xy\left( y-5 \right) \left( y+4 \right)  }{ 14y\left( y-5 \right)}$ is 
Question 95 :
A quadratic equation with rational coefficients has both roots real and irrational, ifthe discriminant is
Question 97 :
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 98 :
Let $ p $ and $ q $ be real numbers such that $ p \neq 0, p^{3} \neq q $ and $ p^{3} $ $ \neq-q . $ If $ \alpha $ and $ \beta $ are non-zero complex numbers satisfying and $ \alpha+\beta=-p $ and $ \alpha^{3}+\beta^{3}=q, $ then a quadratic equation having $ \dfrac{\alpha}{\beta} $ and $ \dfrac{\beta}{\alpha} $ as its roots is
Question 99 :
Divide the following and write your answer in lowest terms: $\dfrac{x^2-36}{x^2-49}\div \dfrac{x+6}{x+7}$
Question 100 :
Workout the following divisions<br/>$36(x + 4) (x^2 + 7x + 10) \div 9(x + 4)$
Question 101 :
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 102 :
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 103 :
Divide $\displaystyle 10{ a }^{ 2 }{ b }^{ 2 }\left( 5x-25 \right)$ by $15ab\left( x-5 \right) $
Question 104 :
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 105 :
Evaluate: $\displaystyle \frac { 35\left( x-3 \right) \left( { x }^{ 2 }+2x+4 \right)  }{ 7\left( x-3 \right)  } $
Question 107 :
Simplify: $\displaystyle \frac { 45\left( { a }^{ 4 }-3{ a }^{ 3 }-28{ a }^{ 2 } \right)  }{ 9a\left( a+4 \right)  } $
Question 108 :
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 109 :
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 110 :
If the equation<br>$\displaystyle\left( { p }^{ 2 }+{ q }^{ 2 } \right) { x }^{ 2 }-2\left( pr+qs \right) x+{ r }^{ 2 }+{ s }^{ 2 }=0$ has equal rootsthen<br>
Question 111 :
Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $
Question 112 :
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 113 :
The number of integers $n$ for which $3x^3-25x+n=0$ has three real roots is$?$<br/>
Question 116 :
If$\alpha ,\beta $ are roots of the equation $2x^{2}+6x+b=0$ where $b<0$, then find least integral value of$\displaystyle \left ( \dfrac{\alpha ^{2}}{\beta }+\dfrac{\beta ^{2}}{\alpha } \right )$.<br>
Question 117 :
If $\cos{\cfrac{\pi}{7}},\cos{\cfrac{3\pi}{7}},\cos{\cfrac{5\pi}{7}}$ are the roots of the equation $8{x}^{3}-4{x}^{2}-4x+1=0$<br>The value of $\sec{\cfrac{\pi}{7}}+\sec{\cfrac{3\pi}{7}}+\sec{\cfrac{5\pi}{7}}=$
Question 118 :
State the following statement is True or False<br/>The zeros of the polynomial $(x - 2) (x^{2} + 4x + 3)$ are $2,-1 and -3$
Question 119 :
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 $s$. <br/>
Question 120 :
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 121 :
$x_1$ 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<br/>
Question 122 :
$\displaystyle \frac{x^{-1}}{x^{-1} + y^{-1}} + \frac{x^{-1}}{x^{-1} - y^{-1}}$ is equal to
Question 123 :
Let $f(x)=2{ x }^{ 2 }+5x+1$. If we write $f(x)$ as<br>$f(x)=a(x+1)(x-2)+b(x-2)(x-1)+c(x-1)(x+1)$ for real numbers $a,b,c$ then
Question 124 :
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 125 :
The difference of the roots of$\displaystyle 2y^{2}-ky+16=0$ is 1/3 Find k
