6. What is the sum of the roots of the equation $\dfrac{1}{x^2} + \dfrac{1}{(1-x)^2} = \dfrac{16}{9}$?
(1) $1$ (2) $1.75$ (3) $2$ (4) $2.25$

The sum of all natural numbers $a$ such that $a ^ { 2 } - 16 a + 67$ is a perfect square is:
(A) 10
(B) 12
(C) 16
(D) 22.

The number of integers $n$ for which the cubic equation $X ^ { 3 } - X + n = 0$ has 3 distinct integer solutions is:
(A) 0
(B) 1
(C) 2
(D) infinite.

The number of all integer solutions of the equation $x ^ { 2 } + y ^ { 2 } + x - y = 2021$ is
(A) 5 .
(B) 7 .
(C) 1 .
(D) 0 .

Let $a , b$ and $c$ be three real numbers. Then the equation $\frac { 1 } { x - a } + \frac { 1 } { x - b } + \frac { 1 } { x - c } = 0$
(A) always have real roots.
(B) can have real or complex roots depending on the values of $a , b$ and $c$.
(C) always have real and equal roots.
(D) always have real roots, which are not necessarily equal.

Suppose that the equations $x ^ { 2 } + b x + c a = 0$ and $x ^ { 2 } + c x + a b = 0$ have exactly one common non-zero root. Then
(A) $a + b + c = 0$.
(B) the two roots which are not common must necessarily be real.
(C) the two roots which are not common may not be real.
(D) the two roots which are not common are either both real or both not real.

29. If $\vec { a } , \vec { b }$ and $\vec { c }$ are unit coplanar vectors, then the scalar triple product
$$[ 2 \vec { a } - \vec { b } , 2 \vec { b } - \vec { c } , 2 \vec { c } - \vec { a } ] =$$
(A) 0
(B) 1
(C) $- \sqrt { } 3$
(D) $\sqrt { } 3$

32. For the equation $3 \times 2 + p x + 3 = 0 , p > 0$, if one of the root is square of the other, then $p$ is equal to :
(A) $1 / 3$
(B) 1
(C) 3
(D) $2 / 3$

3. Let- $1 < \mathrm { p } < 1$. Show that the equation $4 \mathrm { x } 2 - 3 \mathrm { x } - \mathrm { p } = 0$ has a unique root in the interval [1/2, 1] and identify it.

11. (a) Let $\mathrm { a } , \mathrm { b } , \mathrm { c }$ be real numbers with $\mathrm { a } ^ { 1 } 0$ and let $\mathrm { a } , \mathrm { b }$ be the roots of the equation $\mathrm { ax } 2 + \mathrm { bx } + \mathrm { c } = 0$. Express the roots of $\mathrm { a } 2 \times 2 + \mathrm { abcx } + \mathrm { c } 3 = 0$ in terms of $\mathrm { a } , \mathrm { b }$.
(b) Let $\mathrm { a } , \mathrm { b } , \mathrm { c }$ be real numbers with $\mathrm { a } 2 + \mathrm { b } 2 + \mathrm { c } 2 = 1$. Show that the equation
$$\left| \begin{array} { c c c } a x - b y - c & b x + a y & c x + a \\ b x + a y & - a x + b y - c & c y + b \\ c x + a & c y + b & - a x - b y + c \end{array} \right| = 0 \text { represents a straight line. }$$

1. (a) Let P be a point on the ellipse $\mathrm { x } 2 / \mathrm { a } 2 + \mathrm { y } 2 / \mathrm { b } 2 = 1,0 < \mathrm { b } < \mathrm { a }$. Let the line parallel to-qxis passing through P meet the circle $\mathrm { x } 2 + \mathrm { y } 2 = \mathrm { a } 2$ at the point Q such that P and Q are on the same side of - xxis . For two positive real numbers r and s , find the locus of the point R on PQ such that $\mathrm { PR } : \mathrm { RQ } = \mathrm { r } : \mathrm { s }$ as P varies over the ellipse.
   (b) If D is the area of a triangle with side lengths $\mathrm { a } , \mathrm { b } , \mathrm { c }$ then show that $\mathrm { D } < 1 / 4 \sqrt { } ( ( \mathrm { a } + \mathrm { b } + \mathrm { c } ) \mathrm { abc } )$.

Also show that the equality occurs in the above inequality if and only if $\mathrm { a } = \mathrm { b } = \mathrm { c }$.

18. If
$$\left[ \begin{array} { l l l } 4 a ^ { 2 } & 4 a & 1 \\ 4 b ^ { 2 } & 4 b & 1 \\ 4 c ^ { 2 } & 4 c & 1 \end{array} \right] \left[ \begin{array} { c } f ( - 1 ) \\ f ( 1 ) \\ f ( 2 ) \end{array} \right] = \left[ \begin{array} { l l l } 3 a ^ { 2 } & + & 3 a \\ 3 b ^ { 2 } & + & 3 b \\ 3 c ^ { 2 } & + & 3 c \end{array} \right] ,$$
$f ( x )$ is a quadratic function and its maximum value occurs at a point $V$. $A$ is a point of intersection of $y = f ( x )$ with $x$-axis and point $B$ is such that chord $A B$ subtends a right angle at V . Find the area enclosed by $\mathrm { f } ( \mathrm { x } )$ and chord AB .

Suppose $a , b$ denote the distinct real roots of the quadratic polynomial $x ^ { 2 } + 20 x - 2020$ and suppose $c , d$ denote the distinct complex roots of the quadratic polynomial $x ^ { 2 } - 20 x + 2020$. Then the value of
$$a c ( a - c ) + a d ( a - d ) + b c ( b - c ) + b d ( b - d )$$
is
(A) 0
(B) 8000
(C) 8080
(D) 16000

If the sum of the square of the roots of the equation $x^{2} - (\sin\alpha - 2)x - (1+\sin\alpha) = 0$ is least, then $\alpha$ is equal to
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{2}$

If $p$ and $q$ are non-zero real numbers and $\alpha ^ { 3 } + \beta ^ { 3 } = - p , \alpha \beta = q$, then a quadratic equation whose roots are $\frac { \alpha ^ { 2 } } { \beta } , \frac { \beta ^ { 2 } } { \alpha }$ is :
(1) $p x ^ { 2 } - q x + p ^ { 2 } = 0$
(2) $q x ^ { 2 } + p x + q ^ { 2 } = 0$
(3) $p x ^ { 2 } + q x + p ^ { 2 } = 0$
(4) $q x ^ { 2 } - p x + q ^ { 2 } = 0$

The equation $\sqrt { 3 x ^ { 2 } + x + 5 } = x - 3$, where $x$ is real, has
(1) no solution
(2) exactly four solutions
(3) exactly one solution
(4) exactly two solutions

Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0$, $p \neq 0$. If $p$, $q$, $r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is:
(1) $\frac{\sqrt{61}}{9}$
(2) $\frac{2\sqrt{17}}{9}$
(3) $\frac{\sqrt{34}}{9}$
(4) $\frac{2\sqrt{13}}{9}$

The sum of all real values of $x$ satisfying the equation $(x^2 - 5x + 5)^{x^2 + 4x - 60} = 1$ is: (1) 3 (2) $-4$ (3) 6 (4) 5

If, for a positive integer $n$, the quadratic equation,
$$x(x + 1) + (x + 1)(x + 2) + \ldots + (x + \overline{n-1})(x + n) = 10n$$
has two consecutive integral solutions, then $n$ is equal to:
(1) 12
(2) 9
(3) 10
(4) 11

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy,\ \forall x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ 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 $\tan A$ and $\tan B$ are the roots of the quadratic equation, $3 x ^ { 2 } - 10 x - 25 = 0$ then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cdot \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is
(1) 25
(2) - 25
(3) - 10
(4) 10

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x ^ { 2 } + ( 3 - \lambda ) x + 2 = \lambda$ has the least value is:
(1) 2
(2) $\frac { 4 } { 9 }$
(3) $\frac { 15 } { 8 }$
(4) 1

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

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) $\frac{27}{16}$