The equation $x ^ { 2 } + b x + c = 0$ has nonzero real coefficients satisfying $b ^ { 2 } > 4 c$. Moreover, exactly one of $b$ and $c$ is irrational. Consider the solutions $p$ and $q$ of this equation.
(A) Both $p$ and $q$ must be rational.
(B) Both $p$ and $q$ must be irrational.
(C) One of $p$ and $q$ is rational and the other irrational.
(D) We cannot conclude anything about rationality of $p$ and $q$ unless we know $b$ and $c$.

a) Show that if $a$ and $b$ are irrational numbers that are roots of a quadratic with rational coefficients, then $(a-b)^2$ is not a perfect square of any rational number.
b) i) If $a = r \pm \sqrt{s}$ is a quadratic surd, find a rational $x$ such that $a + x$ is irrational but $a_n = (r + (r^2 - s)) \pm \sqrt{s} \notin \mathbb{Q}$. If $a$ is not a surd, take $x = -a$.
ii) Find $y$ such that the required condition holds.

Consider a quadratic equation $ax^2 + 2bx + c = 0$, where $a, b$ and $c$ are positive real numbers. If the equation has no real roots, then which of the following is true?
(A) $a, b, c$ cannot be in AP or HP, but can be in GP.
(B) $a, b, c$ cannot be in GP or HP, but can be in AP.
(C) $a, b, c$ cannot be in AP or GP, but can be in HP.
(D) $a, b, c$ cannot be in AP, GP or HP.

If $a , b , c$ are distinct odd natural numbers, then the number of rational roots of the polynomial $a x ^ { 2 } + b x + c$
(A) must be 0 .
(B) must be 1 .
(C) must be 2 .
(D) cannot be determined from the given data.

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

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

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

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
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { N } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { P } } \right) y ^ { 2 } = 0 .$$
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