1. If \[I=\int_{0}^{1}\cot^{-1} \left(1-x+x^{2}\right)dx=k\int_{0}^{1}\tan^{-1} x dx \]
then k equals
a) 1
b) 2
c) \[\pi\]
d) \[2\pi\]
Explanation:
2. Let \[f:\left(0,\infty\right)\rightarrow\left(0,\infty\right)\] be a differential
function satisfying \[x\int_{0}^{x} \left(1-t\right)f\left(t\right)dt=\int_{0}^{x} tf\left(t\right) dt,x>0\]
and
f (1) = 1. The function f (x) is given by
a) \[\frac{1}{x^{3}}e^{1/x}\]
b) \[\frac{1}{x^{3}}e^{-1/x}\]
c) \[\frac{1}{x^{2}}e^{1-\frac{1}{x}}\]
d) \[\frac{1}{x^{3}}e^{1-\frac{1}{x}}\]
Explanation:
3. Let f (x) = x – [x], where for x \[\epsilon\] R, [x]
denotes the greatest integer \[\leq x\] . Then \[I=\int_{-2}^{2} f\left(x\right)dx\] equals
a) -2
b) -1
c) 0
d) 2
Explanation:
4. If n > 1, and \[I=\int_{0}^{\infty} \frac{dx}{\left(x+\sqrt{1+x^{2}}\right)^{n}}\]
then I equals
a) \[ \frac{n}{n^{2}-1}\]
b) \[ \frac{2n}{n^{2}-1}\]
c) \[ \frac{n}{2\left(n^{2}-1\right)}\]
d) \[\sqrt{n^{2}-1}\]
Explanation:
5. If [x] denotes the greatest integer \[\leq x\] and
\[n\epsilon N\] , then value of \[I_{n}=\int_{0}^{n^{2}} \left[\sqrt{x}\right]dx\]
is
a) \[\frac{1}{6}\left(n-1\right)n\left(4n+1\right)\]
b) \[\frac{1}{6}\left(n-1\right)n^{2}\left(2n+1\right)\]
c) \[\frac{1}{6}\left(n-1\right)\left(n\right)\left(2n-1\right)\]
d) \[\frac{1}{6}\left(n-1\right)\left(3n+5\right)\]
Explanation:
6. If \[I=\int_{0}^{1} \frac{dx}{\left(1+x\right)\left(2x+x\right)\sqrt{x\left(1-x\right)}}\]
then I equals
a) \[2\pi\]
b) \[\pi\]
c) \[\frac{\pi}{2}\]
d) \[\frac{\pi}{\sqrt{6}}\left(\sqrt{3}-1\right)\]
Explanation:
7. For a > 0, \[f\left(t\right)=\int_{0}^{\pi}\left(ax-t\sin x\right) ^{2}dx\]
\[\min_{0\leq t\leq 1} f\left(t\right)\] is
a) \[\frac{a\pi^{3}}{3}-\frac{1}{2}\pi a^{2}\]
b) \[\frac{a\pi^{3}}{3}-\frac{3}{2}\pi a^{2}\]
c) \[\frac{a\pi^{3}}{3}+\frac{1}{2}\pi a^{2}\]
d) \[\frac{a\pi^{3}}{3}-\pi a^{2}\]
Explanation:
8. If \[I=\int_{-1}^{2}\mid x\sin\pi x \mid dx\]
then I equals
a) \[1/\pi\]
b) \[2/\pi\]
c) \[4/\pi\]
d) \[5/\pi\]
Explanation:
9. Let \[I=\int_{0}^{\pi}\mid \sin x+\sin 2x +\sin 3x\mid dx\]
then
a) 6I = 5
b) 6I = 11
c) 6I = 17
d) 6I = 19
Explanation:
10. If \[I=\int_{\alpha}^{\beta}\left[\log \log x+\frac{1}{\left(\log x\right)^{2}}\right] dx\]
then I equals
a) \[\alpha\log \log\alpha-\beta\log\log\beta\]
b) \[\frac{1}{\alpha}-\frac{1}{\beta}+\log\log\alpha-\log\log\beta\]
c) \[\frac{\beta-\alpha}{\alpha\beta}+\alpha\log\log\alpha-\beta\log\log\beta\]
d) \[\beta \log\log\beta-\alpha\log\left(\log\alpha\right)-\frac{\beta}{\log\beta}+\frac{\alpha}{\log\alpha}\]
Explanation:
