Question 1 :
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 2 :
The degree of the remainder is always less than the degree of the divisor.
Question 3 :
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 4 :
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 5 :
What is the remainder, when<br>$(4{x^3} - 3{x^2} + 2x - 1)$ is divided by (x+2)?<br>
Question 6 :
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 7 :
State whether True or False.Divide: $x^2 + 3x -54 $ by $ x-6 $, then the answer is $x+9$.<br/>
Question 8 :
If $a\ne 2$, which of the following is equal to $\cfrac { b\left( { a }^{ 2 }-4 \right) }{ ab-2b } $?
Question 9 :
If $x\ne -5$ , then the expression $\cfrac{3x}{x+5}\div \cfrac {6}{4x+20}$ can be simplified to
Question 11 :
The product of the roots of the quadratic equation $2x^{2}-8x+3=0$ is
Question 13 :
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 19 :
The common quantity that must be added to each term of $a^{2}:b^{2}$ to make itequal to $a:b$ is:
Question 20 :
If $\alpha , \beta $ are the roots of the equation $ax^{2}+bx+c=0$, find the value of $\alpha ^{2}+\beta ^{2}$.
Question 21 :
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 22 :
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 23 :
Choose the correct answer from the alternatives given.<br>If the expression $2x^2$ + 14x - 15 is divided by (x - 4). then the remainder is
Question 25 :
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 27 :
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 28 :
The remainder when$4{a^3} - 12{a^2} + 14a - 3$ is divided by $2a-1$, is
Question 29 :
State whether True or False.Divide : $a^2 +7a + 12 $ by $  a + 4 $, then the answer is $a+3$.<br/>
Question 31 :
Factorise the expressions and divide them as directed.$4yz(z^2 + 6z-  16)\div  2y(z + 8)$<br/>
Question 32 :
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 33 :
Work out the following divisions.$10y(6y + 21) \div 5(2y + 7)$<br/>
Question 35 :
Find the expression which is equivalent to : $\displaystyle \frac { { x }^{ 3 }+{ x }^{ 2 } }{ { x }^{ 4 }+{ x }^{ 3 } } $?
Question 36 :
If the roots of ${ x }^{ 2 }-2mx+{ m }^{ 2 }-1=0$ lie between $-2$ and $4$, then
Question 37 :
Simplify:$20(y + 4) (y^2 + 5y + 3) \div 5(y + 4)$<br/>
Question 39 :
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 40 :
Is $(3x^{2} + 5xy + 4y^{2})$ a factor of $ 9x^{4} + 3x^{3}y + 16x^{2} y^{2} + 24xy^{3}  + 32y^{4}$?<br/>
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 45 :
Factorise the expressions and divide them as directed.$12xy(9x^2-  16y^2)\div  4xy(3x + 4y)$
Question 46 :
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 47 :
What is $\dfrac {x^{2} - 3x + 2}{x^{2} - 5x + 6} \div \dfrac {x^{2} - 5x + 4}{x^{2} - 7x + 12}$ equal to
Question 48 :
What must be added to $x^3-3x^2-12x + 19$, so that the result is exactly divisible by $x^2 + x-6$?
Question 50 :
Work out the following divisions.<br/>$96abc(3a -12) (5b +30)\div  144(a-  4) (b+  6)$<br/>
Question 51 :
If $p, q$ are the distinct roots of the equation $x^2 + px + q = 0$, then
Question 53 :
The condition that one root is twice the other root of the quadratic equation$\displaystyle x^{2}+px+q=0$ is
Question 54 :
If $\alpha$ and $\beta$ be two zeros of the quadratic polynomial $ax^2+bx+c$, then $\dfrac {\alpha^2}{\beta}+\dfrac {\beta^2}{\alpha}$ is equal to<br/>
Question 55 :
If a and b are such that the quadratic equation$\displaystyle ax^{2}-5x+c=0$ has 10 as the sum of the root and also as the product of the roots find a and b respectively
Question 56 :
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 58 :
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 61 :
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 63 :
If $\alpha, \beta$ be the zeros of the quadratic polynomial $2-3x-x^2$, then $\alpha+\beta=$<br>
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 :
If $\alpha$ and $\beta$ are the zeros of the polynomial $f(x)=6x^2-3-7x$, then $(\alpha+1)(\beta+1)$ is equal to<br/>
Question 67 :
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 68 :
Consider the equation ${x^2} + 2x - n = 0$, where $n \in N$ and $n \in \left[ {5,100} \right]$. Total number of different values of 'n' so that the given equation has integral roots, is
Question 69 :
If $\alpha$ and $\beta$ are the roots of $x^2 - ax + b^2 = 0$, then $\alpha^2 + \beta^2$ is equal to
Question 70 :
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 71 :
If a root of the equations${x^2} + px + 12 = 0$ is 4 ,while the roots of the equation ${x^2} + px + q = 0$ , are the same ,then the value of q will be
Question 72 :
If one factor of the polynomial $x ^ { 3 } + 4 x ^ { 2 } - 3 x - 18$ is $x + 3,$ then the other factor is
Question 73 :
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 74 :
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 75 :
Find all values of a for which the equation $x^4+(a−1)x^3+x^2+(a−1)x+1=0$ possesses at least two distinct negative roots.
Question 76 :
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 77 :
Evaluate :$\displaystyle \frac { 60pqr\left( { p }^{ 2 }+{ q }^{ 2 } \right) \left( { q }^{ 2 }+{ r }^{ 2 } \right) \left( { r }^{ 2 }+{ p }^{ 2 } \right) }{ 30pq\left( { p }^{ 2 }+{ q }^{ 2 } \right) \left( { r }^{ 2 }+{ p }^{ 2 } \right) }$
Question 78 :
Using long division method, divide the polynomial$4p^3-4p^2+6p -\displaystyle \frac{5}{2}$ by $2p-1$
Question 79 :
If $\alpha$ and $\beta$ are the zeros of the polynomial $f(x)=5x^2+4x-9$ then evaluate the following: $\alpha^4-\beta^4$<br/>
Question 80 :
If$\displaystyle \alpha ,\beta$ are the roots of the equation$\displaystyle { x }^{ 2 }-x-4=0$, find the value of$\displaystyle \frac { 1 }{ \alpha } +\frac { 1 }{ \beta } -\alpha \beta$.
Question 81 :
If the equation$\displaystyle{ px }^{ 2 }+2x+p=0$ hastwo distinct roots if.
Question 82 :
Write whether the following statement is true or false. Justify your answer.A quadratic equation with integral coefficients has integral roots.
Question 83 :
Divide: $(6a^{5}+ 8a^{4}+ 8a^{3} +2a^{2}+26a +35)$ by $(2a^{2} + 3a +5)$<br/>Answer: $3a^{3} - 3a^{2} + a +7$
Question 85 :
Find the polynomial which when divided by $3x + 4$, equals $2x^{2} + 5x - 3$ with a remainder of $3$
Question 86 :
$\left[2x\right]-2\left[x\right]=\lambda$ where $\left[.\right]$ represents greatest integer function and $\left\{.\right\}$ represents fractional part of a real number then 
Question 89 :
Find the zeros of the quadratic polynomial $f(x) = x^2-3x -28$ and verify the relationships between the zeros and the coefficients.
Question 92 :
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 93 :
The sum and product of zeros of the quadratic polynomial are - 5 and 3 respectively the quadratic polynomial is equal to<br>
Question 94 :
Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and their coefficients.$48y^2-13y-1$<br/>
Question 96 :
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 97 :
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<br/>
Question 99 :
Evaluate: $96 abc (3a -12)(5b -30) \div 144 (a -4) (b -6)$
Question 101 :
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 102 :
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 103 :
$x_1$ 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 104 :
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 105 :
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 107 :
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 108 :
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 110 :
Simplify: $\displaystyle \frac { 45\left( { a }^{ 4 }-3{ a }^{ 3 }-28{ a }^{ 2 } \right)  }{ 9a\left( a+4 \right)  } $
Question 111 :
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 112 :
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 113 :
Find the value of p for which the given equation has real roots.<br>$\displaystyle8p{ x }^{ 2 }-9x+3=0$<br>
Question 114 :
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 115 :
Divide $\displaystyle 10{ a }^{ 2 }{ b }^{ 2 }\left( 5x-25 \right)$ by $15ab\left( x-5 \right) $
Question 116 :
Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $
Question 117 :
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 118 :
Evaluate: $\displaystyle \frac { 35\left( x-3 \right) \left( { x }^{ 2 }+2x+4 \right)  }{ 7\left( x-3 \right)  } $
Question 119 :
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 120 :
$\displaystyle \frac{x^{-1}}{x^{-1} + y^{-1}} + \frac{x^{-1}}{x^{-1} - y^{-1}}$ is equal to
Question 121 :
The difference of the roots of$\displaystyle 2y^{2}-ky+16=0$ is 1/3 Find k
Question 122 :
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 123 :
The number of integers $n$ for which $3x^3-25x+n=0$ has three real roots is$?$<br/>
Question 124 :
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 126 :
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>
