1. The value of \[\int_{-\pi/2}^{\pi/2}\log \left(\frac{2-\sin\theta}{2+\sin\theta}\right)d\theta\]
is
a) 0
b) 1
c) 2
d) none of these
Explanation:
2. The area bounded by \[y=x^{2}\] and \[y=1-x^{2}\]
is
a) \[\sqrt{8}/3\]
b) 16/3
c) 32/3
d) 2
Explanation:
3. The value of the integral \[\int_{0}^{\pi^{2}/4}\sin \sqrt{x} dx\]
is
a) 1
b) 1/2
c) 3/2
d) 2
Explanation:
4. The difference between the greatest and least values
of the function \[f\left(x\right)=\int_{0}^{x}\left(t+1\right) dt\]
on [2, 3] is
a) 3
b) 2
c) 7/2
d) 11/2
Explanation:
5. The area of the figure bounded by the lines x = 0 , \[x=\pi/2\] , \[f\left(x\right)=\sin x\] and
\[g\left(x\right)=\cos x\] is
a) \[2\left(\sqrt{2}-1\right)\]
b) \[\sqrt{3}-1\]
c) \[2\left(\sqrt{3}-1\right)\]
d) \[2\left(\sqrt{2}+1\right)\]
Explanation:
6. The value of \[\lim_{x \rightarrow \infty}\frac{\int_{0}^{x}e^{2x}dx}{\int_{0}^{x}e^{2x^{2}}dx}\]
is
a) 1
b) 2
c) 3
d) 0
Explanation:
7. The value of the integral \[\int_{0}^{3}\frac{dx}{\sqrt{x+1}+\sqrt{5x+1}}\]
is
a) 11/15
b) 14/15
c) 2/5
d) \[\frac{1}{2}\left(1+\frac{1}{2}\log2\right)\]
Explanation:
8. The value of the integral
\[\int_{0}^{\pi/4}\frac{\sin x+\cos x}{3+\sin 2x}dx\]
is
a) log 2
b) log 3
c) (1/4) log 3
d) (1/9) log 3.
Explanation:
9. The area of the figure bounded by \[y^{2}=2x+1\] and x – y – 1 = 0 is
a) 2/3
b) 4/3
c) 8/3
d) 11/3
Explanation:
10. A polynomial P is positive for x > 0 and the area
of the region bounded by P(x), the x-axis, and the
vertical lines x = 0 and x = k is \[k^{2}\left(k+3\right)/3.\]
The
polynomial P(x) is
a) \[x^{2}+x+1\]
b) \[x^{2}+2x+1\]
c) \[x^{2}+2x\]
d) \[x^{3}+1\]
Explanation:
