First Fundamental Theorem of Calculus

Q.

If $A=\left[\begin{array}{cc}\cos \theta & i\sin \theta \\ i\sin \theta & \cos \theta \end{array}\right]$, $\left(\theta =\frac{\pi }{24}\right)$ and $A^5=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, where $i=\surd -1$, then which one of the following is not true?

1. $a^2-d^2=0$

2. $a^2-c^2=1$

3. $0\le a^2+b^2\le 1$

4. $a^2-b^2=\frac{1}{2}$

Q.

Number of integral solutions of $\frac{[x+2]}{[x^2+1]}>\frac{1}{2}$ is / are

1. $0$

2. $1$

3. $2$

4. $3$

5. $4$

Q.

The value of${\int }_0^4|x-1|dx$ is

1. $\frac{5}{2}$

2. $5$

3. $4$

4. $1$

Q.

Find the integral of $\frac{1}{x}$.

Q.

If $f\left(x\right)$ satisfies the relation $2f\left(x\right)+f(1-x)=x^2$ for all real $x$, then $f\left(x\right)$ is

1. $\frac{x^2+2x-1}{6}$

2. $\frac{x^2+2x-1}{3}$

3. $\frac{x^2+4x-1}{3}$

4. $\frac{x^2-3x+1}{6}$

5. $\frac{x^2+3x-1}{3}$

Q.

The function $x^4–4x$ is decreasing in the interval

1. $[–1,1]$

2. $(–\infty ,1)$

3. $[1,+\infty )$

4. None of these

Q.

If $y=\sin \left(\frac{1+x^2}{1-x^2}\right)$, then $\frac{dy}{dx}=$

1. $\frac{4x}{1-x^2}\cos \left(\frac{1+x^2}{1-x^2}\right)$

2. $\frac{x}{1-x^2}\cos \left(\frac{1+x^2}{1-x^2}\right)$

3. $\frac{4x}{{\left(1-x^2\right)}^2}\cos \left(\frac{1+x^2}{1-x^2}\right)$

4. $\frac{x}{{\left(1-x^2\right)}^2}\cos \left(\frac{1+x^2}{1-x^2}\right)$

Q.

The number of solutions of ${\log }_4\left(x-1\right)={\log }_2\left(x-3\right)$

1. $3$

2. $1$

3. $2$

4. $0$

Q.

Integrate $\int {\sin }^2\left(\mathrm{x}\right)\mathrm{dx}$.

Q.

Evaluate ${\int }_0^{\frac{\mathrm{\pi }}{2}}\left[\frac{(\surd \sin x)}{(\surd \sin x+\surd \cos x)}\right]dx$

1. $\frac{\mathrm{\pi }}{4}$

2. $\frac{\mathrm{\pi }}{2}$

3. $0$

4. $1$

Q. The total number of distinct $x\in (0,1]$ for which $\underset{0}{\overset{x}{\int }}\frac{t^2}{1+t^4}dt=2x-1\text{is}$

Q.

The solution of the differential equation $\left(\frac{dy}{dx}\right)+{\sin }^{2}y=0$ is

1. $y+2\cos y=c$

2. $y-2\sin y=c$

3. $x=\cot \left(y\right)+c$

4. $y=\cot \left(x\right)+c$

Q.

If $\mathrm{y}=\mathrm{y}\left(\mathrm{x}\right)$ is the solution of the differential equation $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)+\left(\mathrm{tanx}\right)\mathrm{y}=\mathrm{sinx},0\le \mathrm{\pi }\le \frac{\mathrm{\pi }}{3},$ with$\mathrm{y}\left(0\right)=0$, then $y\left(\frac{\mathrm{\pi }}{4}\right)$ equal to:

1. ${\log }_{e}2$

2. $\left(\frac{1}{2}\right){\log }_{e}2$

3. $\left(\frac{1}{2\sqrt{2}}\right){\log }_{e}2$

4. $\left(\frac{1}{4}\right){\log }_{e}2$

Q.

The solution of differential equation$x\left(\frac{dy}{dx}\right)=y-x{\cos }^2\left(\frac{y}{x}\right)$ is

1. $\tan \left(\frac{y}{x}\right)+\log x=c$

2. $\log \left(\frac{x}{y}\right)+\log x=c$

3. $\log \left(\frac{x}{y}\right)+\tan x=c$

4. $\tan \left(\frac{x}{y}\right)+\tan x=c$

Q.

If ${\sin }^{-1}(1-x)-2{\sin }^{-1}x=\frac{\pi }{2}$, then $x$ is equal to:

1. $0,-\frac{1}{2}$

2. $0,\frac{1}{2}$

3. $0$

4. None of these

Q.

If $x-y=5$ and $xy=12,$ find the value of $x^2+y^2.$

Q.

For real numbers $\alpha ,\beta ,\gamma and\delta ,$ if

$\int \frac{\left(x^2-1\right)+{\tan }^{-1}\left({\displaystyle \frac{x^2+1}{x}}\right)dx}{\left(x^4+3x^2+1\right){\tan }^{-1}\left(\frac{x^2+1}{x}\right)}=\alpha {\log }_e\left({\tan }^{-1}\left(\frac{x^2+1}{x}\right)\right)+\beta {\tan }^{-1}\left[\gamma \left(\frac{x^2+1}{x}\right)\right]+\delta {\tan }^{-1}\left(\frac{x^2+1}{x}\right)$

where $C$ is an arbitrary constant, then the value of $10(\alpha +\beta \gamma +\delta )$ is equal to

Q.

Solving $3-2yi=9^x-7i$, where $i^2=-1$ for $xandy$ real, we get

1. $x=0.5,y=3.5$

2. $x=5,y=3$

3. $x=\frac{1}{2},y=7$

4. $x=0,y=\frac{(3+7i)}{2i}$

Q.

If $2{\sin }^{-1}x-{\cos }^{-1}x=\frac{\pi }{2}$, then $x=$

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

2. $\frac{-1}{\surd 2}$

3. $\frac{–\surd 3}{2}$

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

Q. The imaginary part of ${(3+2\sqrt{-54})}^{1/2}-{(3-2\sqrt{-54})}^{1/2}$ can be:

1. $\sqrt{6}$
2. $-2\sqrt{6}$
3. $6$
4. $-\sqrt{6}$

Q. Let f(x) be a function satisfying f ’(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = $x^2$. Then, the value of the integral ${\int }_0^{1}f\left(x\right)g\left(x\right)dx,$ is

1. $e-\frac{e^2}{2}-\frac{5}{2}$
2. $e-\frac{e^2}{2}-\frac{3}{2}$
3. $e+\frac{e^2}{2}+\frac{5}{2}$
4. $e+\frac{e^2}{2}-\frac{3}{2}$

Q.

The function $x+\left(\frac{1}{x}\right),(x\ne 0)$ is a non-increasing function in the interval

1. $[–1,1]$

2. $[0,1]$

3. $[–1,0)\cup (0,1]$

4. $[–1,2]$

Q.

If $\log \left(a\right)=\frac{1}{2}\log \left(b\right)=\frac{1}{5}\log \left(c\right)$, then $a^4{b}^3{c}^{-2}=$

1. $1$

2. $2$

3. $3$

4. $4$

Q.

Prove that $\cos \left(3x\right)=4{\cos }^3\left(x\right)-3\cos \left(x\right)$

Q. The value of $\underset{x\to 0^{+}}{lim}[x\sin \left(\frac{1}{x}\right)+(\sin x{)}^{1/x}+{\left(\frac{1}{x}\right)}^{\sin x}],$ is

1. $1$
2. $0$
3. $-1$
4. Limit does not exist

Q. Let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral $\underset{0}{\overset{1}{\int }}[-8x^2+6x-1]dx$ is equal to

1. $-1$
2. $\frac{-5}{4}$
3. $\frac{\sqrt{17}-16}{8}$
4. $\frac{\sqrt{17}-13}{8}$

Q.

In a triangle ABC, $\frac{a(a+c-b)}{b(b+c-a)}$ is equal to

1. $\frac{(1–\cos A)}{(1–\cos B)}$

2. $\frac{(1+\cos B)}{(1+\cos A)}$

3. $\frac{(1–\cos A)}{(1+\cos B)}$

4. None of these

Q.

If$I_1={\int }_0^{\frac{\mathrm{\pi }}{4}}{\sin }^{2}xdx$and$I_2={\int }_0^{\frac{\mathrm{\pi }}{4}}{\cos }^{2}xdx$, then

1. $I_1=I_2$

2. $I_1<I_2$

3. $I_1>I_2$

4. $I_2=I_1+\pi /4$

Q.

$\mathrm{If}\mathrm{y}={\sec }^{-1}\left(\frac{\left(2\mathrm{x}\right)}{(1+{\mathrm{x}}^2)}\right)+{\sin }^{-1}\frac{[\mathrm{x}-1]}{[\mathrm{x}+1]},\mathrm{then}\frac{\mathrm{dy}}{\mathrm{dx}}=$

1. $1$

2. $\frac{x+1}{x-1}$

3. does not exist

4. none of these

Q. If f(y) =$e^y$, g(y) = y ; y > 0 and F(t) = ${\int }_0^{t}f(t-y)$ g(y) dy, then

1. $F\left(t\right)=1-e^{-t}(1+t)$
2. $F\left(t\right)=t{e}^t$
3. $F\left(t\right)=t{e}^{-t}$
4. $F\left(t\right)=e^t-(1+t)$