Question 1 :
$2\times 2\times 2\times 3\times 3\times 13 = 2^{3} \times 3^{2} \times 13$ is equal to
Question 2 :
A number $x$ when divided by $7$  leaves a remainder $1$ and another number $y$ when divided by $7$  leaves the remainder $2$. What will be the remainder if $x+y$ is divided by $7$?
Question 4 :
Euclids division lemma can be used to find the $...........$ of any two positive integers and to show the common properties of numbers.
Question 5 :
................. states the possibility of the prime factorization of any natural number is unique. The numbers can be multiplied in any order.
Question 6 :
Which of the following irrational number lies between $\dfrac{3}{5}$ and $\dfrac{9}{10}$
Question 7 :
For three irrational numbers $p,q$ and $r$ then $p.(q+r)$ can be
Question 8 :
The LCM of 54 90 and a third number is 1890 and their HCF is 18 The third number is
Question 9 :
Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion$\displaystyle \frac{7}{210}$
Question 10 :
State True or False:$4\, - \,5\sqrt 2 $ is irrational if $\sqrt 2 $ is irrational.
Question 12 :
............. states that for any two positive integers $a$ and $b$ we can find two whole numbers $q$ and $r$ such that $a = b \times q + r$ where $0 \leq r < b .$
Question 15 :
If $a=\sqrt{11}+\sqrt{3}, b =\sqrt{12}+\sqrt{2}, c=\sqrt{6}+\sqrt{4}$, then which of the following holds true ?<br/>
Question 17 :
Assertion: The denominator of $34.12345$ is of the form $2^n \times 5^m$, where $m, n$ are non-negative integers.
Reason: $34.12345$ is a terminating decimal fraction.
Question 18 :
State whether the following statement is true or not:$\left( 3+\sqrt { 5 }  \right) $ is an irrational number. 
Question 20 :
What is the HCF of $4x^{3} + 3x^{2}y - 9xy^{2} + 2y^{3}$ and $x^{2} + xy - 2y^{2}$?
Question 22 :
Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion$\displaystyle \frac{15}{1600}$
Question 25 :
A rectangular veranda is of dimension $18$m $72$cm $\times 13$ m $20$ cm. Square tiles of the same dimensions are used to cover it. Find the least number of such tiles.
Question 26 :
State whether the following statement is true or false.The following number is irrational<br/>$6+\sqrt {2}$
Question 27 :
To get the terminating decimal expansion of a rational number $\dfrac{p}{q}$. if $q = 2^m 5^n$ then m and n must belong to .................
Question 28 :
The greatest number that will divided $398, 436$ and $542$ leaving $7,11$ and $14$ remainders, respectively, is
Question 29 :
Assertion: $\displaystyle \frac{13}{3125}$ is a terminating decimal fraction.
Reason: If $q=2^n \cdot 5^m$ where $n, m$ are non-negative integers, then $\displaystyle \frac{p}{q}$ is a terminating decimal fraction.
Question 30 :
The number of possible pairs of number, whose product is 5400 and the HCF is 30 is<br>
Question 32 :
According to Euclid's division algorithm, HCF of any two positive integers a and b with a > b is obtained by applying Euclid's division lemma to a and b to find q and r such that $a = bq + r$, where r must satisfy<br/>
Question 34 :
Determine the HCF of $a^2 - 25, a^2 -2a -35$ and $a^2+12a+35$
Question 35 :
State the following statement is True or False<br>35.251252253...is an irrational number<br>
Question 36 :
In a division sum the divisor is $12$  times the quotient and  $5$  times the remainder. If the remainder is  $48$  then what is the dividend?
Question 37 :
Use Euclid's division algorithm to find the HCF of :$196$ and $38220$
Question 39 :
We need blocks to build a building. In the same way _______ are basic blocks to form all natural numbers .
Question 40 :
For finding the greatest common divisor of two given integers. A method based on the division algorithm is used called ............
Question 42 :
Using fundamental theorem of Arithmetic find L.C.M. and H.C.F of $816$ and $170$.
Question 43 :
If $a=107,b=13$ using Euclid's division algorithm find the values of $q$ and $r$ such that $a=bq+r$
Question 44 :
Fundamental theorem of arithmetic is also called as ______ Factorization Theorem.
Question 45 :
Euclid's division lemma states that for two positive integers a and b, there exist unique integers q and r such that $a = bq + r$, where r must satisfy<br>
Question 46 :
Without actually dividing find which of the following are terminating decimals.
Question 54 :
Mark the correct alternative of the following.<br>The HCF of $100$ and $101$ is _________.<br>
Question 56 :
 One and only one out of  $n, n + 4, n + 8, n + 12\  and \ n + 16 $ is ......(where n is any positive integer)<br/>
Question 57 :
State true or false of the following.<br>The predecessor of a two digit number cannot be a single digit number.<br>
Question 59 :
A number when divided by $114$ leaves the remainder $21.$ If the same number is divided by $19$ the remainder will be
Question 61 :
There are five odd numbers $1, 3, 5, 7, 9$. What is the HCF of these odd numbers?
Question 62 :
State whether the given statement is True or False :<br/>$5-2\sqrt { 3 } $ is an irrational number.
Question 63 :
State whether the given statement is True or False :<br/>$\sqrt { 3 } +\sqrt { 4 } $ is an irrational number.
Question 64 :
In a question on division the divisor is  $7$  times the quotient and  $3 $ times the remainder. If the remainder is  $28$  then what is the dividend?
Question 65 :
If the square of an odd positive integer can be of the form $6q + 1 $ or  $6q + 3$ for some $ q$ then q belongs to:<br/>
Question 66 :
Assuming  that x,y,z  are positive real numbers,simplify the following :<br/>$ (\sqrt{x})^{-2/3}\sqrt{y^{4}}\div \sqrt{xy^{-1/2}} $<br/>
Question 67 :
The H.C.F. of two expressions is x and their L.C.M is $ \displaystyle x^{3}-9x  $  IF one of the expression is $ \displaystyle x^{2}+3x  $  then,the other expression is 
Question 68 :
State whether True or False :<br/>All the following numbers are irrationals.<br/>(i) $\dfrac { 2 }{ \sqrt { 7 }  } $ (ii) $\dfrac { 3 }{ 2\sqrt { 5 }  }$ (iii) $4+\sqrt { 2 } $ (iv) $5\sqrt { 2 } $
Question 70 :
Using the theory that any positive odd integers are of the form $4 q + 1$ or $4 q + 3$ where $q$ is a positive integer. If quotient is $4$, dividend is $19$ what will be the remainder?
Question 71 :
If the H.C.F. of $A$ and $B$ is $24$ and that of $C$ and $D$ is $56,$ then the H.C.F. of $A, B, C$ and $D$ is
Question 72 :
State whether the given statement is True or False :<br/>$3+\sqrt { 2 } $ is an irrational number.
Question 73 :
The greatest number which divides $134$ and $167$ leaving $2$ as remainder in each case is
Question 74 :
The value of $\sqrt { 1+2\sqrt { 1+2\sqrt { 1+2+.... } } }$ is
Question 76 :
The H. C. F. of $252$, $324$ and $594$ is ____________.
Question 77 :
Find the dividend which when a number is divided by $45$ and the quotient was $21$ and remainder is $14.$
Question 79 :
If HCF of numbers $408$ and $1032$ can be expressed in the form of $1032x -408 \times 5$, then find the value of $x$.
Question 80 :
State whether the given statement is true/false:$\sqrt{p} + \sqrt{q}$, is irrational, where <i>p,q</i> are primes.
Question 81 :
The given pair of number $ 231, 396$ are __________ .<br/>
Question 82 :
If HCF of $210$ and $55$ is of the form $(210) (5) + 55 y$, then the value of $y$ is :<br/>
Question 83 :
Write whether every positive integer can be of the form $4q + 2$, where $q$ is an integer.<br/>
Question 84 :
If $x=6+2\sqrt {6}$, then what is the value of $\sqrt { x-1 } +\cfrac { 1 }{ \sqrt { x-1 } } $?
Question 86 :
State true or false of the following.<br>If a and b are natural numbers and $a < b$, than there is a natural number c such that $a < c < b$.<br>
Question 88 :
$n$  is a whole number which when divided by  $4$  gives  $3 $ as remainder. What will be the remainder when  $2n$  is divided by $4$ ?<br/>
Question 89 :
In a question on division if four times the divisor is added to the dividend then how will the new remainder change in comparison with the original remainder?
Question 90 :
 The square of any positive odd integer for some integer $ m$ is of the form <br/>
Question 91 :
The divisor when the quotient, dividend and the remainder are respectively $547, 171282$ and $71$ is equal to 
Question 92 :
If these numbers form positive odd integer 6q+1, or 6q+3 or 6q+5 for some q then q belongs to:<br/>
Question 93 :
In algebra $a \times b$ means $ab$, but in arithmetic $3 \times 5$ is
Question 94 :
The decimal expansion of the rational number $\displaystyle\frac{23}{2^{3}5^{2}}$, will terminate after how many places of decimal?
Question 95 :
Sum of digits of the smallest number by which $1440$ should be multiplied so that it becomes a perfect cube is
Question 96 :
Say true or false:A positive integer is of the form $3q + 1,$ $q$  being a natural number, then you write its square in any form other than  $3m + 1$, i.e.,$ 3m $ or $3m + 2$  for some integer $m$.<br/>
Question 98 :
State whether the given statement is True or False :<br/>$4-5\sqrt { 2 } $ is an irrational number.<br/>
Question 99 :
Three ropes are $7\ m, 12\ m\ 95\ cm$ and $3\ m\ 85\ cm$ long. What is the greatest possible length that can be used to measure these ropes?
Question 100 :
State whether the given statement is True or False :<br/>The number $6+\sqrt { 2 } $ is irrational.
Question 101 :
When a natural number x is divided by 5, the remainder is 2. When a natural number y is divided by 5, the remainder is 4. The remainder is z when x+y is divided by 5. The value of $\dfrac { 2z-5 }{ 3 } $ is
Question 104 :
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 105 :
Factorise the expressions and divide them as directed.$4yz(z^2 + 6z-  16)\div  2y(z + 8)$<br/>
Question 106 :
Find the expression which is equivalent to : $\displaystyle \frac { { x }^{ 3 }+{ x }^{ 2 } }{ { x }^{ 4 }+{ x }^{ 3 } } $?
Question 107 :
The remainder when$4{a^3} - 12{a^2} + 14a - 3$ is divided by $2a-1$, is
Question 108 :
If $a\ne 2$, which of the following is equal to $\cfrac { b\left( { a }^{ 2 }-4 \right) }{ ab-2b } $?
Question 109 :
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 110 :
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 112 :
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 113 :
If $x\ne -5$ , then the expression $\cfrac{3x}{x+5}\div \cfrac {6}{4x+20}$ can be simplified to
Question 114 :
Factorise the expressions and divide them as directed.$12xy(9x^2-  16y^2)\div  4xy(3x + 4y)$
Question 117 :
If $\alpha , \beta$ are the zeros of the polynomials $f(x) = x^2+x+1 $ then $\dfrac{1}{\alpha}+\dfrac{1}{\beta}=$________.
Question 118 :
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 119 :
Simplify:$20(y + 4) (y^2 + 5y + 3) \div 5(y + 4)$<br/>
Question 120 :
What is $\dfrac {x^{2} - 3x + 2}{x^{2} - 5x + 6} \div \dfrac {x^{2} - 5x + 4}{x^{2} - 7x + 12}$ equal to
Question 125 :
What is the remainder, when<br>$(4{x^3} - 3{x^2} + 2x - 1)$ is divided by (x+2)?<br>
Question 126 :
State whether True or False.Divide: $x^2 + 3x -54 $ by $ x-6 $, then the answer is $x+9$.<br/>
Question 127 :
Divide the first expression by the second. Write the quotient and the remainder.<br/>$a^2-b^2 ; a-b$
Question 130 :
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 131 :
Work out the following divisions.<br/>$96abc(3a -12) (5b +30)\div  144(a-  4) (b+  6)$<br/>
Question 132 :
Is $(3x^{2} + 5xy + 4y^{2})$ a factor of $ 9x^{4} + 3x^{3}y + 16x^{2} y^{2} + 24xy^{3}  + 32y^{4}$?<br/>
Question 134 :
$\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 136 :
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 $.
Question 138 :
Divide:$\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 139 :
The degree of the remainder is always less than the degree of the divisor.
Question 141 :
If $\alpha , \beta $ are the roots of the equation $ax^{2}+bx+c=0$, find the value of $\alpha ^{2}+\beta ^{2}$.
Question 142 :
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 144 :
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 145 :
Simplify:Find$\ x(x + 1) (x + 2) (x + 3) \div  x(x + 1)$<br/>
Question 147 :
What must be added to $x^3-3x^2-12x + 19$, so that the result is exactly divisible by $x^2 + x-6$?
Question 148 :
The common quantity that must be added to each term of $a^{2}:b^{2}$ to make itequal to $a:b$ is:
Question 150 :
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 151 :
The product of the roots of the quadratic equation $2x^{2}-8x+3=0$ is
Question 152 :
If a polynomial $3x^4- 15x^3+14x^2+2px-q$ is exactly divisible by $x^2-5x +6$ then find the value of $p$ & $q.$
Question 153 :
If equation $\displaystyle { x }^{ 2 }+8x+p=0$ has real and distinct roots then
Question 155 :
The sum of the reciprocals of the roots of the equation$\displaystyle \frac{2009}{2010}x+1+\frac{1}{x}=0$ is
Question 156 :
If one factor of the polynomial $x ^ { 3 } + 4 x ^ { 2 } - 3 x - 18$ is $x + 3,$ then the other factor is
Question 157 :
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 159 :
If one of the zeros of a quadratic polynomial of the form $x^2 + ax + b$ is the negative of the other, then it<br>
Question 160 :
Workout the following divisions<br/>$36(x + 4) (x^2 + 7x + 10) \div 9(x + 4)$
Question 161 :
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 162 :
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 163 :
Divide :$\displaystyle \left[ { x }^{ 4 }-{ \left( y+z \right)  }^{ 4 }\right] \ by \left[{ x }^{ 2 }+{ \left( y+z \right)  }^{ 2 }\right]$
Question 164 :
If $\alpha$ and $\beta$ are the roots of $x^2-pX +1=0$ and $\gamma$ is a root of $X^2+pX+1=0$, then $(\alpha+\gamma)(\beta+\gamma)$ is
Question 165 :
If $\displaystyle \alpha $ and $\displaystyle \beta  $ are roots of the polynomial $\displaystyle f\left ( x \right )= x^{2}-5x+k$ such that $\displaystyle \alpha -\beta = 1$, then value of $k$ is equal to<br/>
Question 166 :
If equation $\displaystyle p{ x }^{ 2 }+9x+3=0$ has real roots, then find value of $p$.<br/>
Question 168 :
The sum of reciprocals of the roots of the equation $ax^2+bx+c=0$, $a, b,$ $c\ne0$, is _______.
Question 171 :
Find the polynomial which when divided by $3x + 4$, equals $2x^{2} + 5x - 3$ with a remainder of $3$
Question 173 :
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 174 :
A quadratic equation with rational coefficients has both roots real and irrational, ifthe discriminant is
Question 176 :
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 177 :
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 178 :
Let $\alpha , \beta \in \mathrm { R }$. If $\alpha , \beta ^ { 2 }$ are the roots of quadratic equation $x ^ { 2 } - p x + 1 = 0$ and $\alpha ^ { 2 } , \beta$ equation $x ^ { 2 } - q x + 8 = 0$, then the value r if $\frac { r } { 8 }$ is the arithmetic means of p and q, is
Question 180 :
If $p$ and $q$ are the roots of the equation $ax^2 +bx +c =0$, then the value of $\dfrac {p}{q}+\dfrac {q}{p}$ is<br/>
Question 181 :
Quadratic polynomial having sum of it's zeros is 5 and product of it's zeros is - 14 is<br/>
Question 182 :
$\alpha $ and $\beta $ are the roots of ${ x }^{ 2 }+2x+C=0$. If ${ \alpha  }^{ 3 }+{ \beta  }^{ 3 }=4$, then the value of $C$ is
Question 183 :
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 184 :
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 185 :
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 186 :
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 190 :
$mx^2+(m-1)x +2=0$ has roots on either side of x=1 the m $\in$
Question 191 :
Simplify: $\displaystyle 7\left( 4x+5 \right) \left( 2x+6 \right) \div \left( 4x+5 \right) $
Question 192 :
If the ratio of the roots of ${x}^{2}+bx+c=0$ is equal to the ratio of the roots of ${x}^{2}+px+q=0$ then ${p}^{2}c-{b}^{2}q=$
Question 194 :
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 195 :
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 196 :
When $(x^{3} - x^{2} - 5x - 3)$ is divided by $(x - 3)$, the remainder is
Question 197 :
If the equation$\displaystyle{ px }^{ 2 }+2x+p=0$ hastwo distinct roots if.
Question 198 :
Workout the following divisions<br/>$54lmn (l + m) (m + n) (n + 1) \div 81mn (l + m) (n + l)$
Question 199 :
Write whether the following statement is true or false. Justify your answer.A quadratic equation with integral coefficients has integral roots.
Question 200 :
Find the value of $p$ in the equation, where the roots are real,<br/>$\displaystyle 5{ x }^{ 2 }+3x-p=0$<br/>
Question 202 :
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 203 :
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 204 :
Find the value of p for which the given equation has real roots.<br>$\displaystyle8p{ x }^{ 2 }-9x+3=0$<br>
Question 205 :
$\displaystyle \frac{x^{-1}}{x^{-1} + y^{-1}} + \frac{x^{-1}}{x^{-1} - y^{-1}}$ is equal to
Question 206 :
Simplify: $\displaystyle \frac { 49\left( { x }^{ 4 }-2{ x }^{ 3 }-15{ x }^{ 2 } \right)  }{ 14x\left( x-5 \right)  } $
Question 208 :
Evaluate: $\displaystyle \frac { 35\left( x-3 \right) \left( { x }^{ 2 }+2x+4 \right)  }{ 7\left( x-3 \right)  } $
Question 209 :
Divide $\displaystyle 10{ a }^{ 2 }{ b }^{ 2 }\left( 5x-25 \right)$ by $15ab\left( x-5 \right) $
Question 210 :
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 211 :
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 212 :
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 213 :
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 214 :
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 215 :
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 216 :
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 217 :
The number of integers $n$ for which $3x^3-25x+n=0$ has three real roots is$?$<br/>
Question 218 :
Simplify: $\displaystyle \frac { 45\left( { a }^{ 4 }-3{ a }^{ 3 }-28{ a }^{ 2 } \right)  }{ 9a\left( a+4 \right)  } $
Question 219 :
$x_1$ 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 220 :
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 221 :
The difference of the roots of$\displaystyle 2y^{2}-ky+16=0$ is 1/3 Find k
Question 222 :
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 223 :
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 226 :
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 227 :
Divide $\displaystyle x\left( x+1 \right) \left( x+2 \right) \left( x+3 \right)$ by $\left( x+3 \right) \left( x+2 \right) $
