Question 4 :
A positive number $n$ when divided by $8$ leaves a remainder $5$. What is the remainder when $2n + 4$ is divided by 8?
Question 6 :
The remainder when $x^{3} - 6x^{2} + 11x - 6$ is divided by $x + 2$ is<br>
Question 8 :
If the polynomial $x^3-x^2+x-1$ is divided by $x-1$, then the quotient is :
Question 9 :
If on dividing a non-zero polynomial $p(x)$ by a polynomial $g (x)$, the remainder is zero, what is the relation between the degrees of $p(x)$ and $g (x)$?<br/>
Question 10 :
By Remainder Theorem find the remainder, when $ p(x)$ is divided by $g(x)$, where$p(x) = x^3-  3x^2 + 4x + 50\ and\ g(x) = x-  3$.<br/>
Question 11 :
State whether the statement is True or False.Expand: $(2a+b)^3 $ is equal to $8a^3+12a^2b+6ab^2+b^3 $.<br/>
Question 12 :
State whether the statement is True or False.Evaluate: $(7x+\dfrac{2}{3}y)(7x-\dfrac{2}{3}y)$ is equal to $49x^2-\dfrac{4}{9}y^2$.<br/>
Question 13 :
Without actually calculating the cubes, find the value of :$\left ( \dfrac{1}{2} \right )^{3}+\left ( \dfrac{1}{3} \right )^{3}-\left ( \dfrac{5}{6} \right )^{3}$<br/>
Question 14 :
If $(x -2)$ is one factor of $x^2 +ax-6 = 0$ and  $ x^2 -9x + b= 0 $ then a + b = ____
Question 15 :
If $2x+1$ is a factor of $(3b+2)x^3 + (b-1)$, then find the value of b.  
Question 16 :
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 17 :
Without actually calculating the cubes, find the value of each of the following:$(28)^3+(-15)^3+(-13)^3$<br/>
Question 19 :
The remainder when $x^{2}+ 2x + 1$ is divided by $(x+1)$ is<br>
Question 20 :
The value of (a - b)(a$^2$ + ab + b$^2$) is
Question 23 :
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 25 :
Given $\boxed { \begin{matrix} A \\ B \end{matrix} } ={A}^{2}+{B}^{2}+2AB$, what is $A+B$, if $\boxed { \begin{matrix} A \\ B \end{matrix} } =9$?
Question 26 :
If $x\ne -5$ , then the expression $\cfrac{3x}{x+5}\div \cfrac {6}{4x+20}$ can be simplified to
Question 27 :
What should be subtracted from $p^2-6p + 7$ so that the remainder may exactly be divisible by $(p - 1)$?
Question 28 :
Find the value of $k$, if $x-1$ is a factor of $p(x)$ in the following cases:$p(x)=kx^2-\sqrt 2x+1$<br/>
Question 29 :
If the polynomials $2x^{3} + ax^{2} + 3x - 5$ and $x^{3} + x^{2} - 4x + a$ leave the same remainder when divided by $x - 2$, find the value of $a$
Question 30 :
If $a + b + c = 0$ then $a^3 + b^3 + c^3$ is
Question 33 :
If both $x + 1$ and $x-  1$ are factors of $ax^3 + x^2+  2a + b = 0$, find the values of $a$ and $b$ respectively.
Question 34 :
If x+2 is a factor of $ \displaystyle \left \{ \left ( x+1 \right )^{5}+(2x+k)^{3} \right \} $, then the value of 'k' is 
Question 38 :
The degree of the remainder is always less than the degree of the divisor.
Question 40 :
If x -a is a factor of $x^3 -3x^2a + 2a^2x + b$, then the value of b is
Question 41 :
If $p(x) = x^3-3x^2+6x-4$ and $p\left (\dfrac{\sqrt{3}}{2}\right) = 0$ then by factor theorem the corresponding factor of $p(x)$ is <br/>
Question 43 :
The value of$ \displaystyle \frac{(1.5)^{2}+(4.7)^{3}+(3.8)^{3}-3\times 1.5\times 4.7\times 3.8}{(1.5)^{2}+(4.7)^{2}+(3.8)^{2}-1.5\times 4.7-4.7\times 3.8-1.5\times 3.8} $
Question 44 :
If on division of a polynomial p (x) by a polynomial g (x), the quotient is zero, what is the relation between the degrees of p (x) and g (x) ?<br/>
Question 47 :
If $(x-1)$ is a factor of $x^2 + 2x - k$, then find $k$.
Question 48 :
If $ a^2+b^2=10 $ and $ ab=3 $, then find $ a-b $. 
Question 49 :
If $\displaystyle x \neq 0$ and $\displaystyle 3x + \dfrac{1}{3x} = 8$, find the value of :<br/>$\displaystyle 27x^{3} + \dfrac{1}{27x^{3}}$<br/>
Question 50 :
The polynomials $ax^3 + 3x^2 - 13$ and $ 2x^3 -5x+a$ are divided by $x+2$ if the remainder in each case is the same, find the value of $a$.<br/>
Question 51 :
How many pairs of natural numbers are there so that difference of the square of the first tois 60 ?<br>(Note : If (a,b) is a pair satisfying , we will not consider (b,a) as a pair)
Question 52 :
If $a + b + c = 9$ and $ab + bc + ca = 26$, then find $a^{2} + b^{2} + c^{2}$
Question 54 :
A polynomial when divided by $\displaystyle \left ( x-6 \right )$ gives a quotient $\displaystyle x^{2}+2x-13$ and leaves a remainder $-8$. Then polynomial is
Question 55 :
Workout the following divisions<br/>$54lmn (l + m) (m + n) (n + 1) \div 81mn (l + m) (n + l)$
Question 56 :
The remainder when $f(x) = 3x^4+2x^3-\displaystyle \frac{x^2}{3}-\frac{x}{9} + \frac{2}{27}$ is divided by $g(x)=x+\displaystyle \frac{2}{3}$ is :
Question 57 :
If $x - 3$ is a factor of $x^{2} - ax - 15$, then $a =$
Question 58 :
State True or False.$\left [x-\dfrac {2}{3}y\right ]^3$ <br> $ \ = \ x^3-\dfrac8{27}y^3-2x^2y+\dfrac43xy^2$ <br/>
Question 59 :
If $\displaystyle a^{2}+b^{2}+2c^{2}-4a+2c-2ab+5=0$ then the possible value of $a +b - c$ is:
Question 63 :
If $(x-1)$ is a factor of $kx^3-4kx^2+4kx-1$, the the value of $k$ is<br/>
Question 65 :
Let $r(x)$ be the remainder when the polynomial $x^{135} + x^{125} - x^{115} + x^{5} + 1$ is divided by $x^{3} - x$. Then
Question 66 :
If $(x-1)$ and $(x-2)$ are factors of $x^4-(p-3)x^3-(3p-5)x^2+(2p-9)x+6$ then the value of $p$ is:
Question 68 :
If $\displaystyle 2 \left ( x^{2} + 1 \right ) = 5x$, find $\displaystyle x - \dfrac{1}{x}$<br/>
Question 71 :
When $(x^3-2x^2+px-q)$ is divided by $(x^2-2x-3)$, the remainder is $(x-6)$. The values of $p$ and $q$ respectively are ____.
Question 72 :
If ${a}^{1/3}+{b}^{1/3}+{c}^{1/3}=0$, then the value of $\left ( a+b+c \right )^{3}$ will be:<br>
Question 73 :
If ${ x }^{ 3 }-{ ax }^{ 2 }+bx-6\quad is\quad exactly\quad divisible\quad by\quad { x }^{ 2 }-5x+6.\quad then\quad \frac { a }{ b } \quad is$
Question 74 :
State true or false:<br/>If $\displaystyle a + \dfrac{1}{a} = p$ and $\displaystyle a \neq 0$; then<br/>$\displaystyle a^{3} + \dfrac{1}{a^{3}} = p\left ( p^{2} - 3 \right )$<br/>
Question 76 :
For a polynomial $p\left( x \right) $, the value of $p\left( 3 \right) $ is $-2$. Which of the following must be true about $p\left( x \right) $?
Question 77 :
$ \left (x + 1 \right )$ is a factor of the polynomial
Question 78 :
If $2x^3 + 4x^2 + 2ax + b$ is exactly divisible by $x^2 - 1$, then the value of a and b respectively will be
Question 79 :
$\root 3 \of a + \root 3 \of b + \root 3 \of c = 0\;{\text{then}}{\left( {a + b + c} \right)^3} $
Question 80 :
Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm. $x^3-3x+1, x^5-4x^3+x^2+3x+1$<br>
Question 81 :
If $x + 1$ is a factor of the polynomial $2x^2 + kx,$ then the value of $k$ is
Question 83 :
Determine all the zeros of $x^4-x^3-8x^2+2x+12$ if two of its zeros are $\sqrt 2$ and $-\sqrt 2$<br>
Question 84 :
Given that $a(a + b) =36$ and $b(a +b) = 64$, where $a$ and $b$ are positive, $(a -b)$ equals:
Question 87 :
Find the reminder when ${x^3} + 3{x^2} + 3x + 1$ is divided by $x + \pi $
Question 88 :
Factorise : ${ (ax+by) }^{ 2 }+{ (2bx-2ay) }^{ 2 }-6abxy$
Question 90 :
If $a + b + c = 0$, the value of $\dfrac{a^{2}}{bc} + \dfrac{b^{2}}{ca} + \dfrac{c^{2}}{ab}$ is $(abc \neq 0)$<br>
Question 93 :
On dividing $f(x)$ by a polynomial $x-1-x^2$, the quotient $q(x)$ and remainder $r(x)$ are $(x-2)$ and $3$ respectively. Then $f(x)$ is<br/>
Question 95 :
Let $p(x)=ax^2+bx+c, q(x)=lx^2+mx+n. $ If $p(1)-q(1)= 0$, $p(2)-q(2)=1 $ and $p(3)-q(3)=4$, then $p(4)-q(4)$ equals 
Question 97 :
If $x + y = 5, x^3+ y^3 = 35$, then $x -y$ is equal to
Question 99 :
Apply the division algorithm to find the quotient and remainder on dividing f(x) by g(x) as given below:<br>(i) $f(x) =x^3-6x^2+11x-6, g(x) = x+2$<br>
Question 100 :
The sum of two numbers is 7 and the sum of their cubes is 133, find the sum of their squares.
Question 101 :
The equationd $x^{x^{x^{+}}} = 2$ is satisfied when $x$ is equal to
Question 102 :
Assertion: Let $\displaystyle f\left ( x \right )=6x^{4}+5x^{3}-38x^{2}+5x+6 $ then all four roots of $\displaystyle f\left ( x \right )=0 $ are real & distinct out of which two are positive & two are negative.
Reason: $\displaystyle f\left ( x \right ) $ has two changes in sign in given order as well as when $x$ is replaced by $-x$.
Question 103 :
Let P be a non zero polynomial such that $P(1 + x) = P(1 - x)$ for all real x, and $P(1) = 0$. Let m be the largest integer such that $(x - 1)^m$ divides $P(x)$ for all such $P(x)$. Then m equals
Question 104 :
Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $
Question 105 :
If $\displaystyle { \left( n+1 \right)  }^{ 3 }-{ n }^{ 3 }=-n$ , then which of the following can be the value of $n$ ?
Question 106 :
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 107 :
The value of $x+y+z$ if ${x}^{2}+{y}^{2}+{z}^{2} = 18$ and $xy + yz + zx = 9$ is
Question 108 :
Square root of $\dfrac {x^{2}}{y^{2}} + \dfrac {y^{2}}{4x^{2}} - \dfrac {x}{y} + \dfrac {y}{2x} - \dfrac {3}{4}$ is $\dfrac {x}{y} - \dfrac {1}{2} - \dfrac {y}{2x}$
Question 109 :
If $(x - 2)$ and $(x - 3)$ are two factors of $\displaystyle x^{3}+ax+b$, then find the remainder when $\displaystyle x^{3}+ax+b$ is divided by $x - 5$.
Question 111 :
If the roots of ${x^4} + q{x^2} + kx + 225 = 0$ are in arithmetic progression, then the value of q is
Question 112 :
Number of real solutions of $\sqrt { 2 x - 4 } - \sqrt { x + 5 } = 1$...
Question 113 :
If$\displaystyle x=a\left ( b-c \right );y=b\left ( c-a \right );z=c\left ( a-b \right )$ then$\displaystyle \left ( \frac{x}{a} \right )^{3}+\left ( \frac{y}{b} \right )^{3}+\left ( \frac{z}{c} \right )^{3}$ is equal to
Question 115 :
Which of the following should be added to $\displaystyle 9x^{3}+6x^{2}+x+2$ so that the sum is divisible by $(3x + 1)$?
Question 117 :
Which of the following is a factor of the polynomial $-2{x}^{2}+7x-6$?
Question 118 :
If $\displaystyle { n }^{ 3 }-{ n }^{ 2 }=n-1$, then which of the following can be the value of $ n$?
Question 119 :
When a number P is divided by 4 it leaves remainder 3. If twice of the number P is divided by the same divisor 4, then what will be the remainder?
Question 121 :
If $ax^{3}+ bx^{2}+ c x + d$ is divided by $x - 2$, then the remainder is equal<br>
Question 123 :
If n is an integer, what is the remainder when $5x^{2n + 1}- 10x^{2n} + 3x^{2n-1} + 5$ is divided by x + 1?
Question 124 :
Find the value of the reminder obtained when $6x^4 + 5x^3 - 2x + 8$ is divided by $x-\dfrac{1}{2}$.
Question 128 :
If $\displaystyle { \left( n+1 \right)  }^{ 3 }-{ \left( n-1 \right)  }^{ 3 }=n+2$, then which of the following can be the value of $n$ ?<br/>
Question 129 :
Which of the following is the remainder when $z\left({5z}^{2}-80\right)$ is divided by $5z\left(z-4\right)$:
Question 130 :
When a positive integer $y$ is divided by $47,$ the remainder is $11$. Therefore, when $\displaystyle y^{2}$ is divided by $47$, the remainder will be 
Question 131 :
If $8a-64b-c=24\sqrt [ 3 ]{ abc } $, where a, b, $c\neq 0$, then which of the following can be true ?
Question 132 :
If ${x^2} - 3x + 2$ is a factor of $f(x) = {x^4} - p{x^2} + q$ ,then $(p,q) = $
Question 133 :
$\displaystyle \frac{x^{-1}}{x^{-1} + y^{-1}} + \frac{x^{-1}}{x^{-1} - y^{-1}}$ is equal to
Question 134 :
The polynomials $p\left( x \right) = k{x^3} + 3{x^2} - 3$ and $Q\left( x \right) = 2{x^3} - 5x + k$, when divided by (x - 4) leave the same remainder. The value of K is
Question 135 :
Let p(x) be a quadratic polynomial such that $p(0)=1$. If p(x) leaves remainder $4$ when divided by $x-1$ and it leaves remainder $6$ when divided by $x+1$; then which one is correct?
Question 136 :
Find the factor of the polynomial $P(x)= \left (12x^4+13x^3-35x^2-16x+20 \right )$ .<br/>
Question 137 :
Let $f(x)=x^6-2x^5+x^3+x^2-x-1$ and $g(x)=x^4-x^3-x^2-1$ be two polynomials. Let $a,b,c$ and $d$ be the roots of $g(x)=0$. Then the value of $f(a)+f(b)+f(c)+f(d)$ is
Question 138 :
When $2f^3 + 3f^2 - 1$ is divided by $f+2$, find the remainder.<br/>
Question 139 :
Divide $\displaystyle 10{ a }^{ 2 }{ b }^{ 2 }\left( 5x-25 \right)$ by $15ab\left( x-5 \right) $
Question 141 :
If a remainder of $4$ is obtained when $x^{3} + 2x^{2} - x - k$ is divided by $x - 2$, find the value of $k$.
