Question 1 :
Use the product $ (a+b)(a-b) = a^2-b^2$ to evaluate:<br>$21\times 19 $
Question 4 :
What is $\dfrac {x^{2} - 3x + 2}{x^{2} - 5x + 6} \div \dfrac {x^{2} - 5x + 4}{x^{2} - 7x + 12}$ equal to
Question 6 :
If $x+y -z = 4$ and $x^2+y^2+ z^2=50$, find the value of $xy -yz-zx$
Question 7 :
<b></b>If $ a^2+b^2=10 $ and $ ab=3 $, then find $ a+b $. 
Question 10 :
The value of $m$, in order that ${ x }^{ 2 }-mx-2$ is the quotient when $3{ x }^{ 3 }+3{ x }^{ 2 }-4$ is divided by $x+2$, is
Question 11 :
Use the factor theorem and state whether $g(x)$ is a factor of $p(x)$ in the following case. State True Or False:$p(x)=x^3-4x^2+x+6; \ g(x)=x-3$<br/>
Question 13 :
The value of $k$ for which $x - k$ is a factor of $x^{3} - kx^{2} + 2x + k + 4$ is<br/>
Question 14 :
What must be added to $x^3-3x^2-12x + 19$, so that the result is exactly divisible by $x^2 + x-6$?
Question 18 :
A positive number $n$ when divided by $8$ leaves a remainder $5$. What is the remainder when $2n + 4$ is divided by 8?
Question 19 :
What is the degree of the remainder atmost, when a fourth degree polynomial is divided by a quadratic polynomial?
Question 20 :
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 21 :
If ${ x }^{ 3 }+ax-28$ is exactly divisible by $x-4$, then the value of $a$ is
Question 22 :
If the polynomial $3x^4-4x^3-3x-1$ is divided by $x-1$, then the remainder is :
Question 23 :
State whether the following statement  is true or false.$(x-1)$ is a factor of ${x}^{3}-27{x}^{2}+8x$.
Question 24 :
One factor of $x^4 + x^2-20$ is $x^2+ 5$. The other factor is
Question 26 :
The product of $x^2y$ and $\cfrac{x}{y}$ is equal to the quotient obtained when $x^2$ is divided by ____.<br/>
Question 28 :
If $a + b + c = 0$ then $a^3 + b^3 + c^3$ is
Question 29 :
Find the value of 'a' if (x-2) is factor of $2x^3-6x^2+5x+a$.
Question 30 :
If $(x -2)$ is a factor of $x^2 + 4x -2k$, then the value of k is
Question 31 :
If $kx^3 + 9x^2+4x -10 $ divided by $x+3$ leaves a remainder $5$, then the value of $k$ will be 
Question 32 :
The value of (a - b)(a$^2$ + ab + b$^2$) is
Question 33 :
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 34 :
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 35 :
If (x -1) is a factor of polynomial f(x) but not of g(x), then it must be a factor of
Question 36 :
State whether the statement is True or False.The square of $(2x+\dfrac{1}{x}+1) $ is equal to $4x^2+\dfrac{1}{x^2}+5+\dfrac{2}{x}+4x $.<br/>
Question 37 :
What should be subtracted from $p^2-6p + 7$ so that the remainder may exactly be divisible by $(p - 1)$?
Question 39 :
Let $f(x)$ be polynomial in $x$ of degree not less than $1$ and $'a'$ be a real number. If $f(x)$ is divided by $(x-a)$, then the remainder is $f(a)$. If $(x-a)$ is a factor of $f(x)$, then $f(a) = 0$. Find the remainder of $x^4+x^3-x^2+2x+3$ when divided by $x-3.$
Question 40 :
If $\left (x + \dfrac {1}{x}\right ) = 2\sqrt {3}$, then the value of $\left (x^{3} + \dfrac {1}{x^{3}}\right )$ is
Question 41 :
The remainder when $x^{6} - 3x^{5} + 2x^{2} + 8$ is divided by $x - 3$ is<br>
Question 43 :
Use factor theorem to verify that the following polynomial q(x) is a factor of p(x) $p(x)=x^5-x^4-4x^2-2x+4, \ q(x)=x-2$<br/>
Question 45 :
The coefficient of $x$ in the expansion of $(x + 3)^3$ is
Question 46 :
State True or False.If $x^2-1$ is a factor of $ax^4+bx^3+cx^2+dx+e$, then $a+c+e=b+d=0$<br/>
Question 47 :
If $2x+1$ is a factor of $(3b+2)x^3 + (b-1)$, then find the value of b.  
Question 49 :
State whether the statement is True or False.$(3x-2y)^3 $ is equal to $27x^3-54x^2y+36xy^2-8y^3$.<br/>
Question 50 :
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 51 :
When $4g^3 - 3g^2 + g + k$ is divided by $g - 2$, the remainder is $27$. Find the value of $k$.
Question 52 :
If $-1$ is a root of $x^4+ax^3-19x^2-46x+20$, then the value of $a$ is
Question 54 :
When $3x^{3} + 2x^{2} + 2x + k$ is divided by $x + 2$, the remainder is $4$. Calculate the value of $k$.
Question 55 :
$\displaystyle \left ( p-q \right )^{3}-\left ( p+q \right )^{3}$ is equal to
Question 56 :
Factorize:<br/>$4x^{2} - 12ax - y^{2} - z^{2} - 2yz + 9a^{2}$
Question 57 :
If $a, b, c$ are all non-zero and $a + b + c = 0$, then $\dfrac {a^2}{bc} +\dfrac {b^2}{ca} +\dfrac {c^2}{ab} =$
Question 59 :
If $x - 3$ is a factor of $x^{2} - ax - 15$, then $a =$
Question 64 :
Determine which of the following polynomials has $x - 2$, a factor :<br/>(i) $3x^2+6x-24$<br/>(ii) $4x^2+x-2$<br/>
Question 67 :
A number when divided by 5 leaves 3 as remainder. if the square of the same number is divided by 5, the remainder obtained is
Question 68 :
A two digit number is such that the product of its digit is $18$. When $63$ is subtracted from the number, the digits interchange their places. Find the number.
Question 69 :
Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder. <br/>$p(x)=x^3-3x^2+5x-3$<br/>$ g(x)=x^2-2$<br/>
Question 70 :
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 71 :
If both $x - 2$ and  $x - \dfrac {1}{2}$ are factors of $px^2 + 5x + r$. Which of the conditions hold true?
Question 75 :
If $\displaystyle f\left ( x \right )=x^{4}-2x^{3}+3x^{2}-ax+b $ is a polynomial such that when it is divide by $(x - 1)$ and $(x + 1)$ the remainders are $5$ and $19$ respectively the remainder when $f(x)$ is divisible by $(x - 2)$ is 
Question 76 :
The value of $k$ for which $(x - 1)$ is a factor of $x^{3} - kx^{2} + 11x - 6$ is<br/>
Question 77 :
Find the remainder when $p(x) = -x+3x^2-1$ is divided by $x+1$.
Question 79 :
The remainder obtained when the polynomial $x^{4}-3 x^3+9x^{2}-27x+81$ is divided by $(x-3)$ is:<br/>
Question 82 :
In the real number system, the equation<br>$\sqrt { x+3-4\sqrt { x-1 } } +\sqrt { x+8-6\sqrt { x-1 } } =1$
Question 83 :
If $p(x)$ and $g(x)$ are any two polynomials with $g(x)\neq 0$, then we can find polynomial $q(x)$ and $r(x)$ such that $p(x) = q(x) g(x) + r(x)$ where<br/>
Question 84 :
If $x + 1$ is a factor of the polynomial $2x^2 + kx,$ then the value of $k$ is
Question 85 :
If $f(x) = 16x^2+51x+35$, then one of the factors of $f(x)$ is :
Question 91 :
The simplified value of the experession$\displaystyle \left ( a+b-c \right )^{2}+2\left ( a+b-c \right )\left ( a-b+c \right )+\left ( a-b+c \right )^{2}$ is
Question 92 :
$12{\left( {x + 3y} \right)^4} - 6{\left( {x + 3y} \right)^3}$ is factorized then its simplified form is
Question 93 :
Divide the polynomial $p(x)$ by the polynomial $g(x)$ and find the quotient and remainder. <br/>$p(x)=x^4-3x^2+4x+5$<br/>$g(x)=x^2+1-x$<br/>
Question 95 :
If $\dfrac{x^{a^2}}{x^{b^2}} = x^{16}, x > 1,$ and $a+b=2$, what is the value of $a-b$?
Question 96 :
The remainders of polynomial f(x) when dividedby x-1, x-2 are 2,3 then the remainder of f(x)when divided by (x-1) (x-2) is
Question 98 :
Number of integers $n$ such that the number $1+n$ is a divisor of the number $1 + {n^2}$ is
Question 99 :
When a number is divided by $13$, the remainder is $11$. When the same number is divided by $17$, the remainder is $9$. What is the number ?
Question 103 :
Find the factor of the polynomial $P(x)= \left (12x^4+13x^3-35x^2-16x+20 \right )$ .<br/>
Question 104 :
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 105 :
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 106 :
$\displaystyle \frac{x^{-1}}{x^{-1} + y^{-1}} + \frac{x^{-1}}{x^{-1} - y^{-1}}$ is equal to
Question 107 :
If ${x^2} - 3x + 2$ is a factor of $f(x) = {x^4} - p{x^2} + q$ ,then $(p,q) = $
Question 108 :
Which of the following is the remainder when $z\left({5z}^{2}-80\right)$ is divided by $5z\left(z-4\right)$:
Question 110 :
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 111 :
If a remainder of $4$ is obtained when $x^{3} + 2x^{2} - x - k$ is divided by $x - 2$, find the value of $k$.
Question 112 :
The equationd $x^{x^{x^{+}}} = 2$ is satisfied when $x$ is equal to
Question 113 :
If $\displaystyle { n }^{ 3 }-{ n }^{ 2 }=n-1$, then which of the following can be the value of $ n$?
Question 114 :
Divide $\displaystyle 10{ a }^{ 2 }{ b }^{ 2 }\left( 5x-25 \right)$ by $15ab\left( x-5 \right) $
Question 115 :
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 116 :
When $2f^3 + 3f^2 - 1$ is divided by $f+2$, find the remainder.<br/>
Question 117 :
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 118 :
Find the value of the reminder obtained when $6x^4 + 5x^3 - 2x + 8$ is divided by $x-\dfrac{1}{2}$.
Question 119 :
Number of real solutions of $\sqrt { 2 x - 4 } - \sqrt { x + 5 } = 1$...
Question 122 :
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 123 :
Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $
Question 124 :
If $8a-64b-c=24\sqrt [ 3 ]{ abc } $, where a, b, $c\neq 0$, then which of the following can be true ?
Question 125 :
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 126 :
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 127 :
If the roots of ${x^4} + q{x^2} + kx + 225 = 0$ are in arithmetic progression, then the value of q is
Question 128 :
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 129 :
The value of $x+y+z$ if ${x}^{2}+{y}^{2}+{z}^{2} = 18$ and $xy + yz + zx = 9$ is
Question 130 :
If $\displaystyle { \left( n+1 \right)  }^{ 3 }-{ n }^{ 3 }=-n$ , then which of the following can be the value of $n$ ?
Question 131 :
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 132 :
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 133 :
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 134 :
Which of the following is a factor of the polynomial $-2{x}^{2}+7x-6$?
Question 137 :
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 139 :
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 141 :
If $ax^{3}+ bx^{2}+ c x + d$ is divided by $x - 2$, then the remainder is equal<br>
