Inequalities of Integrals

Q.

What is the integration of $dt$?

Q. What does the term exp( something) mean?

Q.

$co{t}^{-1}\left(\frac{xy+1}{x-y}\right)+co{t}^{-1}\left(\frac{yz+1}{y-z}\right)+co{t}^{-1}\left(\frac{zx+1}{z-x}\right)=?$

1. $0$

2. $1$

3. $co{t}^{-1}\left(x\right)+co{t}^{-1}\left(y\right)+co{t}^{-1}\left(z\right)$

4. None of these

Q.

$\int \frac{dx}{\sqrt{1-e^{2x}}}=$

1. $\log \left|e^{-x}+\sqrt{e^{-2x}–1}\right|+c$

2. $\log \left|e^x+\sqrt{e^{2x}–1}\right|+c$

3. $-\log \left|e^{-x}+\sqrt{e^{-2x}–1}\right|+c$

4. $-\log \left|e^{-2x}+\sqrt{e^{-2x}–1}\right|+c$

5. $\log \left|e^{-2x}+\sqrt{e^{-2x}–1}\right|+c$

Q.

Evaluate the Integral $\int {\mathrm{e}}^{\mathrm{x}}\left[{\sec }^2\left(\mathrm{x}\right)+\hspace{0.17em}\tan \left(\mathrm{x}\right)\right]\mathrm{d}x$.

Q. The value of ${\int }_2^3\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}.dx$ is equal to

1. $1$
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. $\frac{1}{3}$

Q.

If $\frac{cosx}{a}=\frac{sinx}{b}$ then |acos2x + bsin2x| is equal to

1. 

2. |a|

3. 

4. 

Q. Let p(x) be a function defined on R such that p’(x) = p’(1–x), for all x $ϵ$ [0, 1], p(0) = 1 and p(1) = 41. Then ${\int }_0^{1}p\left(x\right)dx$ equals

1. 21
2. 41
3. 42
4. $\sqrt{41}$

Q. Let $f\left(x\right)=(x^3\sec x+3)\sqrt{16-x^2},$ then $f\left(x\right)$ is

1. an odd function
2. an even function
3. neither odd nor even function
4. both even and odd function

Q. Evaluate the following determinant:
(i) $\left|\begin{array}{ccc}1& 3& 5\\ 2& 6& 10\\ 31& 11& 38\end{array}\right|$

(ii) $\left|\begin{array}{ccc}67& 19& 21\\ 39& 13& 14\\ 81& 24& 26\end{array}\right|$

(iii) $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$

(iv) $\left|\begin{array}{ccc}1& -3& 2\\ 4& -1& 2\\ 3& 5& 2\end{array}\right|$

(v) $\left|\begin{array}{ccc}1& 4& 9\\ 4& 9& 16\\ 9& 16& 25\end{array}\right|$

(vi) $\left|\begin{array}{ccc}6& -3& 2\\ 2& -1& 2\\ -10& 5& 2\end{array}\right|$

(vii) $\left|\begin{array}{cccc}1& 3& 9& 27\\ 3& 9& 27& 1\\ 9& 27& 1& 3\\ 27& 1& 3& 9\end{array}\right|$

(viii)

Q.

The number of proper divisors of $2^p·6^q·{15}^r$ is equal to?

1. $\left(p+q+1\right)(q+r+1)(r+1)$

2. $\left(p+q+1\right)(q+r+1)(r+1)-2$

3. $\left(p+q\right)(q+r)r-2$

4. None of these

Q.

The value of ${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\left[\left(\cos \right(3x)+1)\right]}{(2\cos x-1)}dx=$

1. $2$

2. $1$

3. $\frac{1}{2}$

4. $0$

Q.

f:N-->N f(×)=x-(-1)^x then f is

One one

Onto

Many one

Into

Q. Find the domain and range of $f\left(x\right)={\sin }^{-1}\left(\log \right[x\left]\right)+\log \left({\sin }^{-1}\right[x\left]\right),$ where $[\cdot ]$ denotes the greatest integer function

1. domain $\in [1,2)$
2. domain $\in [-1,2)$
3. Range of $f\left(x\right)=\in \left\{\log \frac{\pi }{4}\right\}$
4. Range of $f\left(x\right)=\in \left\{\log \frac{\pi }{2}\right\}$

Q.

If then x is equal to

1. 
   

2. ab

3. 
   

4. 2ab

Q. If $3x+2y=1$ acts as a tangent to $y=f\left(x\right)$ at $x=\frac{1}{2}$ and if $p=\underset{x\to 0}{lim}\frac{x(x-1)}{f\left(\frac{e^{2x}}{2}\right)-f\left(\frac{e^{-2x}}{2}\right)},$ then $\sum _{r=1}^{\infty }{p}^r=$______

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{6}$
4. $\frac{1}{7}$

Q. When n!+1 is divided by any natural number between 2and n then the remainder obtained is
1)1
2)2
, 3)3
4)4

Q. Evaluate the limit: $\underset{x\to 2}{lim}\frac{x^3-8}{x^2-4}$

Q. ${\int }_0^{\frac{\pi }{2}}log\left(cosx\right)dx=$

1. $\pi log2$
2. $-\frac{\pi }{2}log2$
3. $-\frac{\pi }{2}2log2$
4. $none$

Q. If the binary operation ⊙ is defined on the set Q⁺ of all positive rational numbers by $a\odot b=\frac{ab}{4}.\mathrm{Then},3\odot \left(\frac{1}{5}\odot \frac{1}{2}\right)$ is equal to
(a) $\frac{3}{160}$

(b) $\frac{5}{160}$

(c) $\frac{3}{10}$

(d) $\frac{3}{40}$

Q. Consider the integrals $I_1=\underset{0}{\overset{1}{\int }}{e}^{-x}{\cos }^{2}xdx,I_2=\underset{0}{\overset{1}{\int }}{e}^{-x^2}\cos xdx$
$I_3=\underset{0}{\overset{1}{\int }}{e}^{-x^2}dx\text{and}{I}_4=\underset{0}{\overset{1}{\int }}{e}^{-x^2/2}dx.$ Which of the following is/are correct?

1. $I_2>I_1$
2. $I_2<I_3$
3. $I_2<I_1$
4. $I_4$ has largest value.

Q. The function$f\left(x\right)=\left[x\right]\cos \frac{2x-1}{2}\pi$ , where [ ] denotes the greatest integer function is discontinuous at

1. no $x$
2. all $x$
3. all integer points
4. $x$ which is not integer

Q.

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}tanxdx.$

Q.

$Find\frac{d}{dx}\left(2^{co{s}^{2}x}\right)$

Q. Assertion :The function $f\left(x\right)=\frac{\sin x}{x}$ is decreasing in the interval $(0,\frac{\pi }{2})$ Reason: $\tan x>x$ for $0<x<\frac{\pi }{2}$

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is incorrect but Reason is correct
4. Both Assertion and Reason are incorrect

Q. $\int \cos \left(\log x\right)dx=F\left(x\right)+c,$ where $c$ is an arbitrary constant. Here $F\left(x\right)=$

1. $x\left[\cos \right(\log x)-\sin (\log x\left)\right]$
2. $\frac{x}{2}\left[\cos \right(\log x)+\sin (\log x\left)\right]$
3. $\frac{x}{2}\left[\cos \right(\log x)-\sin (\log x\left)\right]$
4. $x\left[\cos \right(\log x)+\sin (\log x\left)\right]$

Q.

$co{s}^{-1}x\{-cos\left(\frac{-13\pi }{6}\right)\}$ is equal to

1. $-\frac{\pi }{6}$
   

2. $\frac{5\pi }{6}$
   

3. $\frac{\pi }{6}$
   

4. None

Q. show that $\cos {38}^{\circ }\cos {52}^{\circ }-\sin {38}^{\circ }\sin {52}^{\circ }=0$

Q.

If f: R → R be given by, then fof(x) is

(A) (B) x³ (C) x (D) (3 − x³)

Q. Let f(x), g(x), h(x) be continous in [0, 2a] and satisfies $f(2a-x)=f\left(x\right),g(2a-x)=g\left(x\right),h\left(x\right)+h(2a-x)=3,f(2a-x)g(2a-x)=f\left(x\right)g\left(x\right)then{}^{}{\int }_0^{2a}f\left(x\right)g\left(x\right)h\left(x\right)dx=$

1. $3{\int }_0^{a}f\left(x\right)g\left(x\right)dx$
2. ${\int }_0^{2a}f\left(x\right)g\left(x\right)dx$
3. ${\int }_0^{a}f\left(x\right)g\left(x\right)dx$
4. $2{\int }_0^{a}f\left(x\right)g\left(x\right)dx$