Integration by Partial Fractions

Q.

If $x\left(\sqrt{1+y}\right)+y\left(\sqrt{1+x}\right)=0$, then $\frac{dy}{dx}$ is equal to

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

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

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

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

Q.

Let $f:R-\left\{3\right\}\to R-\left\{1\right\}$ be defined by $f\left(x\right)=\frac{\left(x-2\right)}{\left(x-3\right)}$. Let $g:R$ be given as $g\left(x\right)=2x-3$. Then, the sum of all the values of $x$ for which $f^{-1}\left(x\right)+g^{-1}\left(x\right)=\frac{13}{2}$ is equal to?

1. $7$

2. $5$

3. $2$

4. $3$

Q.

How do you find the derivative of $\cot \left(x\right)$ ?

Q.

What is the formula of $\log \left(\frac{1}{x}\right)$?

Q.

If $y=\log x^x$ , then the value of $\frac{dy}{dx}$ is

1. $x^x\left(1+\log x\right)$

2. $\log ex$

3. $\log \frac{e}{x}$

4. $\log \frac{x}{e}$

Q.

What is the derivative of $2^x$?

Q.

A scooter accelerates from rest for time$t_1$ at constant rate ${\alpha }_1$ and then retards at a constant rate ${\alpha }_2$ for time$t_2$ and comes to rest. The correct value of$t_1/t_2$ will be:

1. $({\alpha }_1+{\alpha }_2)/{\alpha }_2$

2. ${\alpha }_2/{\alpha }_1$

3. $({\alpha }_1+{\alpha }_2)/{\alpha }_1$

4. ${\alpha }_1/{\alpha }_2$

Q.

If $0<x,y<\pi$ and $\cos \left(x\right)+\cos \left(y\right)-\cos \left(x+y\right)=\frac{3}{2},$ then $\sin \left(x\right)+\cos \left(y\right)$ is equal to:

1. $\frac{\left(1+\sqrt{3}\right)}{2}$

2. $\frac{\left(1-\sqrt{3}\right)}{2}$

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

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

Q.

The domain for which the functions$f\left(x\right)=2x^2-1$ and $g\left(x\right)=1-3x$ is equal, i.e.$f\left(x\right)=g\left(x\right),$ is

1. $\{0,2\}$

2. $\left\{\frac{1}{2},-2\right\}$

3. $\left\{-\frac{1}{2},2\right\}$

4. $\left\{\frac{1}{2},2\right\}$

Q.

Prove that $[\frac{(3+2i)}{(2-5i)}+\frac{(3-2i)}{(2+5i)}]$is rational.

Q.

If the Boolean expression $(p\Rightarrow q)\iff (q*(~P\left)\right)$is a tautology, then the Boolean expression $p*(~q)$ is equivalent to:

1. $p\Rightarrow ~q$

2. $p\Rightarrow q$

3. $q\Rightarrow p$

4. $~q\Rightarrow p$

Q.

If $\alpha$and $\beta$are the roots of the quadratic equation $x^2+x+1=0$, then the equation, whose roots are ${\alpha }^{19}$and ${\beta }^7$, is

1. $x^2-x+1=0$

2. $x^2-x-1=0$

3. $x^2+x-1=0$

4. $x^2+x+1=0$

Q.

Let $a,b,c\in R.$ If $f\left(x\right)=a{x}^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f\left(x\right)+f\left(y\right)+xy$, $\forall x,y\in R$ then$\sum _{n=1}^{10}f\left(n\right)=$ is equal to

1. $165$

2. $190$

3. $255$

4. $330$

Q.

Solution of $(1+xy)ydx+(1-xy)xdy=0$ is

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

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

3. $\log \left(\frac{x}{y}\right)-\left(\frac{1}{xy}\right)=c^{}$

4. None of these

Q. Let $f\left(x\right)$ be a quadratic function such that $f\left(0\right)=1$ and $\int \frac{f\left(x\right)}{x^2(x+1{)}^3}dx$ is a rational function. Then the value of $f^{\prime }\left(0\right)$ is

Q.

If $x,2x+2,3x+3$ are in G.P., then the fourth term is:

1. $27$

2. $-27$

3. $13.5$

4. $-13.5$

Q.

Let $\left[x\right]$ denote the greatest integer less than or equal to x. If for $n\in N,(1–x+x^3)n=\sum _{j=0}^{3n}{a}_j{x}^j,then\sum _{j=0}^{\left(\frac{3n}{2}\right)}{a}_{2j}+4\sum _{j=0}^{\left(\frac{3n-1}{2}\right)}{a}_{2j}+1=$

1. $1$

2. $n$

3. $2^n-1$

4. 2

Q.

For a suitably chosen real constant a, let a function, $f:R-\{-a\}\to R$ be defined by $f\left(x\right)=\frac{(a-x)}{(a+x)}$ . Further suppose that for any real number $x\ne -aandf\left(x\right)\ne -a,\left(fof\right)\left(x\right)=x$. Then $f\left(\frac{-1}{2}\right)$ is equal to:

Q. The function $f\left(x\right)$, that satisfies the condition $f\left(x\right)=x+\underset{0}{\overset{\pi /2}{\int }}\sin x\cdot \cos yf\left(y\right)dy$, is

1. $x+(\pi +2)\sin x$
2. $x+\frac{2}{3}(\pi -2)\sin x$
3. $x+\frac{\pi }{2}\sin x$
4. $x+(\pi -2)\sin x$

Q.

Solve The Following:

$\frac{3}{2}x+\frac{5}{3}y=7$ and

$9-10y=14$

Q.

If $a_1,a_2,a_3...a_n$are in AP and $a_1=0$, then the value of $\left(\frac{a_3}{a_2}+\frac{a_4}{a_3}+\dots .+\frac{a_n}{a_{n-1}}\right)-a_2\left(\frac{1}{a_2}+\frac{1}{a_2}+....+\frac{1}{a_{n-2}}\right)$ is equal to

Q.

The derivative of $f\left(x\right)=3\left|2+x\right|$at the point $x_0=-3$ is

1. $3$

2. $-3$

3. $0$

4. does not exist

Q.

The value of $c$ in $\left(0,2\right)$ satisfying the mean value theorem for the function $f\left(x\right)=x{(x-1)}^2$, $x\in [0,2]=$?

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

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

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

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

5. $\frac{5}{3}$

Q.

Find the integral of $\left[{\tan }^{-1}x\right]dx$ from $0$ to $102$, where $\left[x\right]$ is the largest integer not exceeding $x$.

Q.

There are $5$ girls and $7$ boys. A team of $3$ boys and $2$ girls is to be formed such that no two specific boys are in the same team. Number of way to do so are

1. $400$

2. $250$

3. $200$

4. $300$

Q. Find the period of real valued function satisfying
f(x) +. f(x+4). =f(x+2) + f(x+6)

Q.

How do you find the derivative of $y=\cos \left(3x\right)$ ?

Q.

The sum of two non–zero numbers is $4$.

The minimum value of the sum of their reciprocals is

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

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

3. $1$

4. None of these.

Q.

For $\left|x\right|<1,y=1+x+x^2+......\infty ,$then $\frac{dy}{dx}-y$ equal to

1. $x/y$

2. $x^2/y^2$

3. $x/y^2$

4. $x{y}^2$

Q.

$Ifa.i=4,then\left(axj\right).(2j-3k)isequalto?$

1. $12$

2. $2$

3. $0$

4. $-12$