1. Let a, b, c, p, q real numbers and x, y, z be three numbers satisfying the system of equations
$\frac{x}{a}+\frac{y}{a-p}+\frac{z}{a-q}=1$
$\frac{x}{b}+\frac{y}{b-p}+\frac{z}{b-q}=1$
$\frac{x}{c}+\frac{y}{c-p}+\frac{z}{c-q}=1$
then
a) x + y + z = a + b + c – p – q
b) $x=\frac{abc}{pq}$
c) $y=\frac{\left(a-p\right)\left(b-p\right)\left(c-p\right)}{p\left(p-q\right)}$
d) All of the above

Explanation: Note that a, b, c are roots of

2. Let $\alpha$ be a repeated root of $p\left(x\right)=x^{3}+3ax^{2}+3bx+c=0$      , then
a) $\alpha$ is a root of $p\left(x\right)=x^{2}+2ax+b=0$
b) $\alpha=\frac{c-ab }{2\left(a^{2}-b\right)}$
c) $\alpha$ is a root of $ax^{2}+2bx+c=0$
d) All of the above

Explanation:

3. Let $\alpha,\beta,\gamma$  be roots of $x^{3}-px^{2}+qx-r=0$      , then
a) equation whose $\alpha^{2}-\beta\gamma, \beta^{2}-\gamma\alpha,\gamma^{2}-\alpha\beta$      is $x^{3}+\left(3q-p^{2}\right)x^{2}+q\left(3q-p^{2}\right)x+q^{3}-p^{3}r=0$
b) a permutation of $\alpha,\beta,\gamma$   is in G.P. if $q^{3}=p^{3} r$
c) Square of one of the roots will be additive inverse of the product of the other two if$q^{3}=p^{3} r$
d) Both a and b

Explanation:

4. Let $\alpha,\beta,\gamma$  be roots of $x^{3}+px+q=0$      , then
a) an equation whose roots are $\alpha^{3},\beta^{3},\gamma^{3}$   is $x^{3}+3qx^{2}+\left(3q^{2}+p^{3}\right)y+q^{3}=0$
b) $\alpha^{3}+\beta^{3}+\gamma^{3}=3\alpha\beta\gamma$
c) an equation whose roots are $\beta^{3}+\gamma^{3},\gamma^{3}+\alpha^{3},\alpha^{3}+\beta^{3}$      is $x^{3}+6qx^{2}+\left(p^{3}+12q^{2}\right)x+3p^{3}q+8q^{3}=0$
d) All of the above

Explanation:

5. If the equations $x^{2}+bx+c=0$    and $bx^{2}+cx+1=0$    have a common root then
a) b + c + 1 = 0
b) $b^{2}+c^{2}+1=bc$
c) $\left(b-c\right)^{2}+\left(b-1\right)^{2}+\left(c-1\right)^{2}=0$
d) Both a and c

Explanation: If $\alpha$ is a common root of the two equations,

6. Let $\alpha$ and $\beta$ be two distinct real numbers and p(x) be a quadratic polynomial such that $p\left(\alpha\right)=\alpha$   and $p\left(\beta\right)=\beta$   then
a) p(p(x)) – x = 0 has at least two real roots.
b) $\alpha$ and $\beta$ are roots of p(p(x)) – x = 0
c) p(p(x)) = x for each $x \epsilon R$
d) Both a and b

Explanation: Use p(p( $\alpha$ )) = p( $\alpha$ ) = $\alpha$ and p(p( $\beta$ )) = p( $\beta$ ) = $\beta$

7. Suppose $m \epsilon R$  . The quadratic equation
$x^{2}-\left(m-3\right)x+m=0$               (1)
has
a) real distinct roots if and only if m $\epsilon \left(-\infty,1\right)\cup\left(9,\infty\right)$
b) both positive roots if and only if m $\epsilon$ [9, $\infty$  )
c) both negative roots if and only if m $\epsilon$ (0,1]
d) All of the above

Explanation: (1) will have real and distinct roots

8. Let $\alpha , \beta$  be the roots of $ax^{2}+2bx+c=0$     and $\gamma ,\delta$  be the roots of $px^{2}+2qx+r=0$     . If $\alpha,\beta,\gamma,\delta$  are in
a) A.P. then $\frac{b^{2}-ac}{a^{2}}=\frac{q^{2}-pr }{p^{2}}$
b) G.P. then $\frac{b^{2}}{ac}=\frac{q^{2}}{pr}$
c) H.P. then $\frac{b^{2}-ac}{c^{2}}=\frac{q^{2}-pr }{r^{2}}$
d) All of the above

Explanation:

9. If $a\left(p+q\right)^{2}+2bpq+c=0$     and $a\left(p+r\right)^{2}+2bpr +c=0$     , then
a) $q+r=\frac{2\left(a+b\right)p}{a}$
b) $qr=p^{2}+\frac{c}{a}$
c) $\mid q-r\mid=\frac{2}{\mid a\mid}\sqrt{\left(2a+b\right)bp^{2}-ac}$
d) All of the above

10. The number or real roots of $\left(x+3\right)^{4}+\left(x+5\right)^{4}=16$