Let $a , b , c \in \mathbb { R }$. If $f ( x ) = a x ^ { 2 } + b x + c$ is such that $a + b + c = 3$ and $f ( x + y ) = f ( x ) + f ( y ) + x y$, $\forall x , y \in \mathbb { R }$, then $\sum _ { n = 1 } ^ { 10 } f ( n )$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255

Let $p , q$ and $r$ be real numbers ( $p \neq q , r \neq 0$ ), such that the roots of the equation $\frac { 1 } { x + p } + \frac { 1 } { x + q } = \frac { 1 } { r }$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to
(1) $p ^ { 2 } + q ^ { 2 }$
(2) $\frac { p ^ { 2 } + q ^ { 2 } } { 2 }$
(3) $2 \left( p ^ { 2 } + q ^ { 2 } \right)$
(4) $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is :
(1) $4 \sqrt { 2 }$
(2) 20
(3) $2 \sqrt { 5 }$
(4) $2 \sqrt { 7 }$

The sum of the solutions of the equation $\sqrt{x} - 2 + \sqrt{x}\sqrt{x} - 4 + 2 = 0, x > 0$ is equal to
(1) 10
(2) 9
(3) 12
(4) 4

If $m$ is chosen in the quadratic equation $\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 3 x + \left( m ^ { 2 } + 1 \right) ^ { 2 } = 0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
(1) $4 \sqrt { 3 }$
(2) $10 \sqrt { 5 }$
(3) $8 \sqrt { 3 }$
(4) $8 \sqrt { 5 }$

The number of real roots of the equation, $e ^ { 4 x } + e ^ { 3 x } - 4 e ^ { 2 x } + e ^ { x } + 1 = 0$ is:
(1) 1
(2) 3
(3) 2
(4) 4

Let $f ( x )$ be a quadratic polynomial such that $f ( - 1 ) + f ( 2 ) = 0$. If one of the roots of $f ( x ) = 0$ is 3 , then its other root lies in
(1) $( - 1,0 )$
(2) $( 1,3 )$
(3) $( - 3 , - 1 )$
(4) $( 0,1 )$

Let $[ \mathrm { t } ]$ denote the greatest integer $\leq \mathrm { t }$. Then the equation in $\mathrm { x } , [ \mathrm { x } ] ^ { 2 } + 2 [ \mathrm { x } + 2 ] - 7 = 0$ has :
(1) exactly two solutions
(2) exactly four integral solutions
(3) no integral solution
(4) infinitely many solutions

Let $\lambda \neq 0$ be in $R$. If $\alpha$ and $\beta$ are the roots of the equation, $x ^ { 2 } - x + 2 \lambda = 0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3 x ^ { 2 } - 10 x + 27 \lambda = 0$, then $\frac { \beta \gamma } { \lambda }$ is equal to:
(1) 27
(2) 18
(3) 9
(4) 36

The product of the roots of the equation $9 x ^ { 2 } - 18 | x | + 5 = 0$ is :
(1) $\frac { 5 } { 9 }$
(2) $\frac { 25 } { 81 }$
(3) $\frac { 5 } { 27 }$
(4) $\frac { 25 } { 9 }$

If $\alpha$ and $\beta$ are the roots of the equation, $7x^2 - 3x - 2 = 0$, then the value of $\frac{\alpha}{1-\alpha^2} + \frac{\beta}{1-\beta^2}$ is equal to:
(1) $\frac{27}{32}$
(2) $\frac{1}{24}$
(3) $\frac{3}{8}$
(4) $\frac{27}{16}$

If $\alpha$ and $\beta$ be two roots of the equation $x ^ { 2 } - 64 x + 256 = 0$. Then the value of $\left( \frac { \alpha ^ { 3 } } { \beta ^ { 5 } } \right) ^ { \frac { 1 } { 8 } } + \left( \frac { \beta ^ { 3 } } { \alpha ^ { 5 } } \right) ^ { \frac { 1 } { 8 } }$ is :
(1) 2
(2) 3
(3) 1
(4) 4

If $\alpha$ and $\beta$ are the roots of the equation $2\mathrm{x}(2\mathrm{x}+1)=1$, then $\beta$ is equal to:
(1) $2\alpha(\alpha+1)$
(2) $-2\alpha(\alpha+1)$
(3) $2\alpha(\alpha-1)$
(4) $2\alpha^{2}$

Let $a , b \in R , a \neq 0$ be such that the equation, $a x ^ { 2 } - 2 b x + 5 = 0$ has a repeated root $\alpha$, which is also a root of the equation, $x ^ { 2 } - 2 b x - 10 = 0$. If $\beta$ is the other root of this equation, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to:
(1) 25
(2) 26
(3) 28
(4) 24

Let $p$ and $q$ be two positive numbers such that $p + q = 2$ and $p ^ { 4 } + q ^ { 4 } = 272$. Then $p$ and $q$ are roots of the equation:
(1) $x ^ { 2 } - 2 x + 2 = 0$
(2) $x ^ { 2 } - 2 x + 8 = 0$
(3) $x ^ { 2 } - 2 x + 136 = 0$
(4) $x ^ { 2 } - 2 x + 16 = 0$

$\operatorname { cosec } 18 ^ { \circ }$ is a root of the equation:
(1) $x ^ { 2 } - 2 x - 4 = 0$
(2) $4 x ^ { 2 } + 2 x - 1 = 0$
(3) $x ^ { 2 } + 2 x - 4 = 0$
(4) $x ^ { 2 } - 2 x + 4 = 0$

Let $P ( x ) = x ^ { 2 } + bx + c$ be a quadratic polynomial with real coefficients such that $\int _ { 0 } ^ { 1 } P ( x ) d x = 1$ and $P ( x )$ leaves remainder 5 when it is divided by $( x - 2 )$. Then the value of $9 ( b + c )$ is equal to:
(1) 9
(2) 15
(3) 7
(4) 11

The minimum value of the sum of the squares of the roots of $x ^ { 2 } + 3 - a x = 2 a - 1$ is
(1) 6
(2) 4
(3) 5
(4) 8

If $\alpha , \beta$ are the roots of the equation $x ^ { 2 } - \left( 5 + 3 ^ { \sqrt { \log _ { 3 } 5 } } - 5 ^ { \sqrt { \log _ { 5 } 3 } } \right) x + 3 \left( 3 ^ { \left( \log _ { 3 } 5 \right) ^ { \frac { 1 } { 3 } } } - 5 ^ { \left( \log _ { 5 } 3 \right) ^ { \frac { 2 } { 3 } } } - 1 \right) = 0$ then the equation, whose roots are $\alpha + \frac { 1 } { \beta }$ and $\beta + \frac { 1 } { \alpha }$,
(1) $3 x ^ { 2 } - 20 x - 12 = 0$
(2) $3 x ^ { 2 } - 10 x - 4 = 0$
(3) $3 x ^ { 2 } - 10 x + 2 = 0$
(4) $3 x ^ { 2 } - 20 x + 16 = 0$

Let $a, b \in R$ be such that the equation $ax^2 - 2bx + 15 = 0$ has repeated root $\alpha$ and if $\alpha$ and $\beta$ are the roots of the equation $x^2 - 2bx + 21 = 0$, then $\alpha^2 + \beta^2$ is equal to:
(1) 37
(2) 58
(3) 68
(4) 92

$\alpha = \sin 36 ^ { \circ }$ is a root of which of the following equation
(1) $16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0$
(2) $16 x ^ { 4 } + 20 x ^ { 2 } + 5 = 0$
(3) $10 x ^ { 4 } - 10 x ^ { 2 } - 5 = 0$
(4) $16 x ^ { 4 } - 10 x ^ { 2 } + 5 = 0$

Let $f ( x ) = a x ^ { 2 } + b x + c$ be such that $f ( 1 ) = 3 , f ( - 2 ) = \lambda$ and $f ( 3 ) = 4$. If $f ( 0 ) + f ( 1 ) + f ( - 2 ) + f ( 3 ) = 14$, then $\lambda$ is equal to
(1) $- 4$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 23 } { 2 }$
(4) $4$

If for some $p , q , r \in R$, all have positive sign, one of the roots of the equation $\left( p ^ { 2 } + q ^ { 2 } \right) x ^ { 2 } - 2 q ( p + r ) x + q ^ { 2 } + r ^ { 2 } = 0$ is also a root of the equation $x ^ { 2 } + 2 x - 8 = 0$, then $\frac { q ^ { 2 } + r ^ { 2 } } { p ^ { 2 } }$ is equal to $\_\_\_\_$.

Let $f ( x ) = 2 x ^ { 2 } - x - 1$ and $S = \{ n \in \mathbb { Z } : | f ( n ) | \leq 800 \}$. Then, the value of $\sum _ { n \in S } f ( n )$ is equal to $\_\_\_\_$ .

The number of real solutions of the equation $3 \left( \mathrm { x } ^ { 2 } + \frac { 1 } { \mathrm { x } ^ { 2 } } \right) - 2 \left( \mathrm { x } + \frac { 1 } { \mathrm { x } } \right) + 5 = 0$, is
(1) 4
(2) 0
(3) 3
(4) 2