1. If p > 1/2, the number of real solutions of the equation
$\sqrt{x^{2}+2px-p^{2}}+\sqrt{x^{2}-2px-p^{2}}=1$             (1)
is
a) 0
b) 1
c) 2
d) infinite

Explanation: For each x $\epsilon$ R, we have

2. If a, b, c are distinct real numbers, then number of solutions of
$\frac{x+a}{x+b}+\frac{x+b}{x+c}+\frac{x+c}{x+a}=3$
is
a) 0
b) 1
c) 2
d) infinite

Explanation:

3. Number real solutions of the equation
$\sum_{k=1}^{2019}k^{2}\mid x^{2}+\left(k+3\right)x-k-4| = 0$             (1)
is
a) 0
b) 1
c) 2
d) infinite

Explanation:

4. Suppose $a,b \epsilon$ R. Let $f\left(x\right)=3x^{2}+2ax+b$      if $\int_{-1}^{1} \mid f \left(x\right)| dx>2$     , then f (x) = 0 has
a) distinct real roots
b) equal roots
c) purely imaginary roots
d) nature of roots depend on values of a, b

Explanation:

5.If [x] = greatest integer $\leq x$ , then number of solutions of the equation $\left(x-\left[x \right]\right)\left(\frac{1}{x}+\frac{1}{\left[x\right]}\right)=2$             (1)
is
a) 0
b) 1
c) 2
d) infinite

Explanation:

6. Number of real solutions of the equations
$\sqrt{1-2x}+\sqrt{1+2x}=\sqrt{\frac{1-2x}{1+2x}}+\sqrt{\frac{1+2x}{1-2x}}$             (1)
is
a) 0
b) 1
c) 2
d) infinite

Explanation:

7. Let p and q be real numbers such that $p\neq 0,p^{3}\neq q$   and $p^{3}\neq -q$    . If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha + \beta = -p$    and $\alpha^{3} + \beta^{3} =q$    , then a quadratic equation having $\frac{\alpha}{\beta}$   and$\frac{\beta}{\alpha}$  as its roots is
a) $\left(p^{3}+q\right)x^{2}-\left(p^{3}+2q\right)x+\left(p^{3}+q\right)=0$
b) $\left(p^{3}+q\right)x^{2}-\left(p^{3}-2q\right)x+\left(p^{3}+q\right)=0$
c) $\left(p^{3}-q\right)x^{2}-\left(5p^{3}-2q\right)x+\left(p^{3}-q\right)=0$
d) $\left(p^{3}-q\right)x^{2}-\left(5p^{3}+2q\right)x+\left(p^{3}-q\right)=0$

Explanation:

8. The quadratic equation p(x) = 0 with real coefficient has purely imaginary roots .then equation p(p(x)) = 0 has
a) only purely imaginary roots
b) all real roots
c) two real and two purely imaginary roots
d) neither real nor purely imaginary roots

Explanation: As p(x) is quadratic and p(x) = 0 has purely imaginary roots, p(x) must be of the form p(x) = a(x2 + b)

9. The product of real roots of the equation
$\mid x\mid^{6/5}-26\mid x\mid^{3/5}-27=0$             (1)
is
a) $-3^{10}$
b) $-3^{12}$
c) $-3^{12/5}$
d) $-3^{21/5}$

$\left(x^{2}+x-2\right)\left(x^{2}+x-3\right)=12$             (1)