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) \[-3^{21/5}\]
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 x2 + x = y , so that the equation (1) becomes
