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(x

^{2}+ 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) \[-3^{21/5}\]

Explanation: Put |x|

^{3/5}= y , so that (1) becomes

10. Sum of the non-real roots of

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

is

a) 1

b) -1

c) -6

d) 6

Explanation: Put x

^{2}+ x = y , so that the equation (1) becomes