1.If \[I=\int_{0}^{1/\sqrt{3}} \frac{dx}{\left(1+x^{2}\right)\sqrt{1-x^{2}}}\]
then I is equal to
a) \[\pi/2\]
b) \[\pi/2\sqrt{2}\]
c) \[\pi/4\sqrt{2}\]
d) \[\pi/4\]
Explanation:
2. Let f be a continuous function on R satisfying
f (x + y) = f (x) + f (y) for all x, \[y\epsilon R\] with f (1)
= 2 and g be a function satisfying \[f\left(x\right)+g\left(x\right)=e^{x}\] then the
value of the integral \[\int_{0}^{1} f\left(x\right)g\left(x\right)dx\] is
a) 1/e – 4
b) \[\frac{1}{4}(e – 2)\]
c) 2/3
d) (1/2) (e – 3)
Explanation:
3. A line tangent to the graph of the function y
= f (x) at the point x = a forms an angle \[\pi/3\] with
y-axis and at x = b and angle \[\pi/4\] with x-axis then
\[\int_{a}^{b} f''\left(x\right)dx\] is
a) \[1/\sqrt{3}-1\]
b) \[\pi/12\]
c) \[-\pi/12\]
d) \[\sqrt{3}-1\]
Explanation:
4. The value of \[\int_{1}^{e} \left(\frac{\tan^{-1}x}{x}+\frac{\log x}{1+x^{2}}\right)dx\]
is
a) tan e
b) \[\tan^{-1}e\]
c) \[\tan^{-1}(1/e)\]
d) \[2\tan^{-1}e\]
Explanation:
5. If a continuous function f on [0, a] satisfy
f (x) f (a – x) = 1, a > 0 then \[\int_{0}^{a}\frac{dx}{1+f\left(x\right)}\]
is
equal to
a) 0
b) a
c) a/2
d) none of these
Explanation:
6. Let f (x), g(x) and h(x) be continuous function on
[0, a] such that f (x) = f (a – x), g(x) = – g(a – x), 3h(x)
– 4h(a – x) = 5 then \[\int_{0}^{a}\] f(x) g(x) h(x) dx is equal to
a) 1
b) 0
c) a
d) -1
Explanation:
7. The value of \[\int_{0}^{\pi}\frac{dx}{1-2\alpha \cos x+\alpha^{2}}\]
is
a) \[\frac{\pi}{1+\alpha^{2}} if a>1\]
b) \[\frac{\pi}{\alpha^{2}-1} if a>1\]
c) \[\frac{\pi}{1+\alpha^{2}} if a<1\]
d) \[\frac{\pi}{\alpha^{2}-1} if a<1\]
Explanation:
8. The value of \[\int_{0}^{1} \mid \sin 2\pi x\mid dx\]
is equal to
a) 0
b) \[-1/\pi\]
c) \[1/\pi\]
d) \[2/\pi\]
Explanation:
9. Let f (x) and g(x) be two functions satisfying \[f\left(x^{2}\right)+g\left(4-x\right)=4x^{3},g\left(4-x\right)+g\left(x\right)=0,\]
then the value of \[\int_{-4}^{4} f\left(x^{2}\right)dx\] is
a) 512
b) 64
c) 256
d) 0
Explanation:
10. If \[f\left(x\right)=\int_{1}^{x} \frac{\log t}{t+1}dt\] and \[f\left(x\right)+f\left(1/x\right)=k\left(\log x\right)^{2},\]
then k equal to
a) 1
b) 1/2
c) 1/4
d) 1/3
Explanation:
