1.If \[0<\alpha<\pi/2\] then the value of
\[\int_{0}^{\alpha}\frac{dx}{1-\cos x\cos\alpha}\]
is
a) \[\pi/\alpha\]
b) \[\pi/\sin\alpha\]
c) \[\pi/\cos\alpha\]
d) \[\pi/2\alpha\]
Explanation:
2. If \[I_{1}=\int_{x}^{1}\frac{dt}{1+t^{2}}\] and \[I_{2}=\int_{1}^{1/x}\frac{dt}{1+t^{2}}\]
for x>0, then
a) \[I_{1}=I_{2}\]
b) \[I_{1}>I_{2}\]
c) \[I_{2}>I_{1}\]
d) \[I_{2}=\left(\pi/2\right)-\tan^{-1}x\]
Explanation:
3. The solution of the equation \[\int_{\sqrt{2}}^{x}\frac{dx}{x\sqrt{x^{2}-1}}=\frac{\pi}{12}\]
is given by
a) 1
b) 2
c) 3
d) \[\sqrt{3}\]
Explanation:
4. The mean value of the function \[f\left(x\right)=\frac{2}{e^{x}+1}\]
on the interval [0, 2] is
a) \[\log\frac{2}{e^{2}+1}\]
b) \[1+log\frac{2}{e^{2}+1}\]
c) \[2+\log\frac{2}{e^{2}+1}\]
d) \[2+\log\left(e^{2}+1\right)\]
Explanation:
5. If \[I_{1}=\int_{x}^{1} \frac{dt}{1+t^{2}}\] and \[I_{2}=\int_{1}^{1/x} \frac{dt}{1+t^{2}}\] for x > 0 then
a) \[I_{1}=I_{2}\]
b) \[I_{1}>I_{2}\]
c) \[I_{2}=\left(\pi/4\right)-\tan^{-1}x\]
d) Both a and c
Explanation:
6. The points of extremum of \[f\left(x\right)=\int_{1}^{x}e^{-t^{2}/2}\left(1-t^{2}\right)dt\]
are
a) x = 1
b) x = -1
c) x = 2
d) Both a and b
Explanation:
7. Let \[f\left(x\right)=\int_{x^{2}}^{x^{3}}\frac{dt}{\log t},x> 0\]
then
a) f' (x) = - 1/6 log x
b) f is an increasing function
c) f is an increasing function on \[\left(1 ,\infty\right)\]
d) Both b and c
Explanation:
8. If \[I=\int_{0}^{1/2}\frac{d x}{\sqrt{1-x^{2n}}}\]
for n \[\geq1\] , the value of I is
a) less than 1
b) more than 1/2
c) more than 1
d) Both a and b
Explanation:
9.The value of \[\int_{3}^{4}\frac{d x}{\sqrt[3]{\log x}}\]
is
a) less than one
b) greater than 1/2
c) less than two
d) Both a and b
Explanation:
10. The absolute value of \[\int_{10}^{19}\frac{\sin x}{1+x^{8}}dx\]
is
a) less than \[10^{-7}\]
b) more than \[10^{-7}\]
c) less than \[10^{-6}\]
d) Both a and c
Explanation:
