The values of ' $a$ ' for which one root of the equation $x ^ { 2 } - ( a + 1 ) x + a ^ { 2 } + a - 8 = 0$ exceeds 2 and the other is lesser than 2 , are given by :
(1) $3 < a < 10$
(2) $a \geq 10$
(3) $- 2 < a < 3$
(4) $a \leq - 2$

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0, a, b, c \in R$, have a common root, then $a : b : c$ is:
(1) $1 : 3 : 2$
(2) $3 : 1 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

The number of all possible positive integral value of $\alpha$ for which the roots of the quadratic equation $6x^2 - 11x + \alpha = 0$ are rational numbers is:
(1) 5
(2) 3
(3) 4
(4) 2

Consider the quadratic equation $( c - 5 ) x ^ { 2 } - 2 c x + ( c - 4 ) = 0 , c \neq 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $( 0,2 )$ and its other root lies in the interval $( 2,3 )$. Then the number of elements in $S$ is
(1) 11
(2) 12
(3) 18
(4) 10

The number of integral values of $m$ for which the quadratic expression $( 1 + 2 m ) x ^ { 2 } - 2 ( 1 + 3 m ) x + 4 ( 1 + m ) , x \in R$ is always positive, is
(1) 7
(2) 3
(3) 6
(4) 8

Let $p , q \in Q$. If $2 - \sqrt { 3 }$ is a root of the quadratic equation $x ^ { 2 } + p x + q = 0$, then
(1) $p ^ { 2 } - 4 q + 12 = 0$
(2) $q ^ { 2 } + 4 p + 14 = 0$
(3) $p ^ { 2 } - 4 q - 12 = 0$
(4) $q ^ { 2 } - 4 p - 16 = 0$

The number of integral values of $m$ for which the equation, $1 + m ^ { 2 } x ^ { 2 } - 21 + 3 m x + 1 + 8 m = 0$ has no real root, is
(1) 2
(2) 3
(3) Infinitely many
(4) 1

Consider the two sets: $A = \left\{ m \in R : \right.$ both the roots of $x ^ { 2 } - ( m + 1 ) x + m + 4 = 0$ are real $\}$ and $B = [ - 3,5 )$
Which of the following is not true?
(1) $A - B = ( - \infty , - 3 ) \cup ( 5 , \infty )$
(2) $A \cap B = \{ - 3 \}$
(3) $B - A = ( - 3,5 )$
(4) $A \cup B = R$

The set of all real values of $\lambda$ for which the quadratic equation $\left( \lambda ^ { 2 } + 1 \right) x ^ { 2 } - 4 \lambda x + 2 = 0$ always have exactly one root in the interval $( 0,1 )$ is :
(1) $( - 3 , - 1 )$
(2) $( 0,2 )$
(3) $( 1,3 ]$
(4) $( 2,4 ]$

The integer $k$, for which the inequality $x ^ { 2 } - 2 ( 3 k - 1 ) x + 8 k ^ { 2 } - 7 > 0$ is valid for every $x$ in $R$ is:
(1) 4
(2) 2
(3) 3
(4) 0

The set of all values of $k > - 1$, for which the equation $\left( 3 x ^ { 2 } + 4 x + 3 \right) ^ { 2 } - ( k + 1 ) \left( 3 x ^ { 2 } + 4 x + 3 \right) \left( 3 x ^ { 2 } + 4 x + 2 \right) + k \left( 3 x ^ { 2 } + 4 x + 2 \right) ^ { 2 } = 0$ has real roots, is: (1) $\left[ - \frac { 1 } { 2 } , 1 \right)$ (2) $\left( 1 , \frac { 5 } { 2 } \right]$ (3) $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right] - \{ 1 \}$ (4) $[ 2,3 )$

The probability of selecting integers $a \in [ - 5,30 ]$ such that $x ^ { 2 } + 2 ( a + 4 ) x - 5 a + 64 > 0$, for all $x \in R$, is:
(1) $\frac { 7 } { 36 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 4 }$

The coefficients $a , b$ and $c$ of the quadratic equation, $a x ^ { 2 } + b x + c = 0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is:
(1) $\frac { 1 } { 72 }$
(2) $\frac { 1 } { 36 }$
(3) $\frac { 1 } { 54 }$
(4) $\frac { 5 } { 216 }$

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x + \sqrt { 6 } = 0$ and $\frac { 1 } { \alpha ^ { 2 } } + 1 , \frac { 1 } { \beta ^ { 2 } } + 1$ be the roots of the equation $x ^ { 2 } + a x + b = 0$. Then the roots of the equation $x ^ { 2 } - ( a + b - 2 ) x + ( a + b + 2 ) = 0$ are :
(1) non-real complex numbers
(2) real and both negative
(3) real and both positive
(4) real and exactly one of them is positive

If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x ^ { 2 } + \alpha x + \beta > 0$, for all $x \in R$, is
(1) $\frac { 17 } { 36 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 19 } { 36 }$

Let $S = \left\{ \sin ^ { 2 } 2 \theta : \left( \sin ^ { 4 } \theta + \cos ^ { 4 } \theta \right) x ^ { 2 } + ( \sin 2 \theta ) x + \left( \sin ^ { 6 } \theta + \cos ^ { 6 } \theta \right) = 0 \right.$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3 \left( ( \alpha - 2 ) ^ { 2 } + ( \beta - 1 ) ^ { 2 } \right)$ equals $\_\_\_\_$

If the set of all $\mathrm { a } \in \mathbf { R }$, for which the equation $2 x ^ { 2 } + ( a - 5 ) x + 15 = 3 \mathrm { a }$ has no real root, is the interval $( \alpha , \beta )$, and $X = \{ x \in Z : \alpha < x < \beta \}$, then $\sum _ { x \in X } x ^ { 2 }$ is equal to :
(1) 2109
(2) 2129
(3) 2119
(4) 2139

If the equation $\mathrm { a } ( \mathrm { b} - \mathrm { c } ) \mathrm { x } ^ { 2 } + \mathrm { b } ( \mathrm { c } - \mathrm { a } ) \mathrm { x } + \mathrm { c } ( \mathrm { a } - \mathrm { b } ) = 0$ has equal roots, where $\mathrm { a } + \mathrm { c } = 15$ and $\mathrm { b } = \frac { 36 } { 5 }$, then $a ^ { 2 } + c ^ { 2 }$ is equal to

Q2 Consider the following three conditions (a), (b) and (c) on two real numbers $x$ and $y$:
(a) $x+y=5$ and $xy=3$,
(b) $x+y=5$ and $x^2+y^2=19$,
(c) $x^2+y^2=19$ and $xy=3$.
(1) Using the equality $x^2+y^2=(x+y)^2 - \square\mathbf{F}\, xy$, we see that
$$\text{condition (b) gives } xy = \mathbf{G},$$ $$\text{condition (c) gives } x+y = \mathbf{H} \text{ or } x+y = \mathbf{IJ}.$$
(2) For each of the following $\mathbf{K} \sim \mathbf{M}$, choose the most appropriate answer from among the choices (0)$\sim$(3) below.
(i) (a) is $\mathbf{K}$ for (b).
(ii) (b) is $\mathbf{L}$ for (c).
(iii) (c) is $\mathbf{M}$ for (a).
(0) a necessary and sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary condition but not a sufficient condition
(3) neither a necessary condition nor a sufficient condition

Q2 Consider the following three conditions (a), (b) and (c) on two real numbers $x$ and $y$:
(a) $x+y=5$ and $xy=3$,
(b) $x+y=5$ and $x^2+y^2=19$,
(c) $x^2+y^2=19$ and $xy=3$.
(1) Using the equality $x^2+y^2=(x+y)^2-\square\mathbf{F}\,xy$, we see that
condition (b) gives $xy=\mathbf{G}$,
condition (c) gives $x+y=\mathbf{H}$ or $x+y=\mathbf{IJ}$.
(2) For each of the following $\mathbf{K}\sim\mathbf{M}$, choose the most appropriate answer from among the choices (0)$\sim$(3) below.
(i) (a) is $\mathbf{K}$ for (b).
(ii) (b) is $\mathbf{L}$ for (c).
(iii) (c) is $\mathbf{M}$ for (a).
(0) a necessary and sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary condition but not a sufficient condition
(3) neither a necessary condition nor a sufficient condition

Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.

Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.

Let $a$ be a real number. Consider the quadratic expressions in $x$
$$\begin{aligned} & A = x^2 + ax + 1 \\ & B = x^2 + (a+3)x + 4 \end{aligned}$$
(1) The range of values taken by $a$ such that there exists a real number $x$ satisfying $A + B = 0$ is
$$a \leq -\sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \text{ or } \sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \leq a.$$
(2) The range of values taken by $a$ such that there exists a real number $x$ satisfying $AB = 0$ is
$$a \leq \mathbf{EF} \text{ or } \mathbf{G} \leq a.$$
(3) There exists a real number $x$ satisfying $A^2 + B^2 = 0$ only when $a = \mathbf{H}$. In this case $x = \mathbf{IJ}$.

For each of A $\sim$ D in questions (1)$\sim$(4) below, choose the appropriate answer from among (0) $\sim$ (3) of each question. For $\mathbf { E } \sim \mathbf { G }$ in question (5), put the correct number.
Suppose that $a , b$ and $c$ are integers, and $a > 0$. Also, suppose that the graph of a quadratic function $y = a x ^ { 2 } - 2 b x + c$ intersects the $x$-axis and all points of intersection are in the interval $0 < x < 1$.
(1) The relationship between $a$ and $b$ is A. (0) $a > b$
(1) $a < b$
(2) $a = b$
(3) indeterminate
(2) The conditions on $b$ and $c$ are $\mathbf { B }$. (0) $b < 0 , c < 0$
(1) $b < 0 , c > 0$
(2) $b > 0 , c < 0$
(3) $b > 0 , c > 0$
(3) The relationship between $2 b$ and $a + c$ is $\mathbf { C }$. (0) $2 b > a + c$
(1) $2 b < a + c$
(2) $2 b = a + c$
(3) indeterminate
(4) The relationship between $b$ and $c$ is $\mathbf { D }$. (0) $b > c$
(1) $b < c$
(2) $b = c$
(3) indeterminate
(5) The smallest integer which $a$ can take is $\mathbf { E }$. In this case, the value of $b$ is $\mathbf { F }$, and the value of $c$ is $\mathbf { G }$.

For $\mathbf { H } , \mathbf { I }$ in question (1), and for $\mathbf { J }$, $\mathbf { K }$ in question (2), choose the appropriate answer from among (0) $\sim$ (3) at the bottom of this page.
For $\mathbf { L } \sim \mathbf { R }$ in question (3), enter the appropriate number.
Consider the following three possible conditions on two real numbers $x$ and $y$:
$p : x$ and $y$ satisfy the equation $( x + y ) ^ { 2 } = a \left( x ^ { 2 } + y ^ { 2 } \right) + b x y$, where $a$ and $b$ are real constants.
$$\begin{aligned} & q : x = 0 \text { and } y = 0 . \\ & r : x = 0 \text { or } y = 0 . \end{aligned}$$
(1) Suppose that in condition $p$, $a = b = 1$. Then $p$ is $\mathbf { H }$ for $q$, and $p$ is $\mathbf { I }$ for $r$.
(2) Suppose that in condition $p$, $a = b = 2$. Then $p$ is $\mathbf { J }$ for $q$, and $p$ is $\mathbf { K }$ for $r$.
(3) If in condition $p$ we set $a = 2$, we can transform the equation in $p$ into
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { L } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { L } } \right) y ^ { 2 } = 0 .$$
Hence $p$ is a necessary and sufficient condition for $q$ if and only if $b$ satisfies
$$\mathbf { Q } < b < \mathbf { R } .$$
(0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition