Composite Function

Q. Find the value of d(root x + 1/root x)^2/dx

Q. Let $f\left(x\right)=x^2,x\in R.$ For any $A\subseteq R$, define $g\left(A\right)=\{x\in R:f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is not true ?

1. $f\left(g\right(S\left)\right)=S$
2. $g\left(f\right(S\left)\right)\ne S$
3. $f\left(g\right(S\left)\right)\ne f\left(S\right)$
4. $g\left(f\right(S\left)\right)=g\left(S\right)$

Q.

Solve $x^2-2ax+(a^2-b^2)=0$.

Q.

Integration of $x\log x$.

Q.

If $x=\sqrt{7}+\sqrt{3}$ and $xy=4$, then $x^4+y^4=$

1. $400$

2. $368$

3. $200$

4. None of these

Q. For $x\in R-\{0,1\},$ let $f_1\left(x\right)=\frac{1}{x}$, $f_2\left(x\right)=1-x$ and $f_3\left(x\right)=\frac{1}{1-x}$ be three given functions. If a function, $J\left(x\right)$ satisfies $\left(f_{2}oJo{f}_1\right)\left(x\right)=f_3\left(x\right)$, then $J\left(x\right)$ is equal to :

1. $f_1\left(x\right)$
2. $f_2\left(x\right)$
3. $f_3\left(x\right)$
4. $\frac{1}{x}\cdot f_3\left(x\right)$

Q.

The value of the function $(x–1){(x–2)}^2$ at its maxima is

1. $1$

2. $2$

3. $0$

4. $\frac{4}{27}$

Q.

If $f\left(x\right)=\cos \left(\log x\right)$ then $f\left(x\right)·f\left(y\right)-\frac{1}{2}\left[f\left(\frac{x}{y}\right)+f\left(xy\right)\right]$ has the value

1. $-1$

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

3. $-2$

4. $0$

Q. If x=sin³t/√(cos2t) and y=cos³t/√(cos2t) then prove that dy/dx =0 when t=π/6

Q.

If $\alpha$and $\beta$ are the roots of the equation $x^2-6x+a=0$ and satisfy the relation $3\alpha +2\beta =16$, then the value of $a$ is

1. $-8$

2. $8$

3. $16$

4. $9$

5. None of these

Q.

The function $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{\lambda }\left|\sin \left(x\right)\right|+{\mathrm{\lambda }}^2\left|\cos \left(x\right)\right|+\mathrm{g}\left(\mathrm{\lambda }\right)$ has period equal to $\frac{\mathrm{\pi }}{2}$ if $\mathrm{\lambda }$ is

1. $2$

2. $1$

3. $3$

4. $4$

Q.

If $f\left(x\right)=cos\left(logx\right)$, then value of $f\left(x\right)$
$f\left(4\right)-\frac{1}{2}\{f\left(\frac{x}{4}\right)+f(4x\left)\right\}$ is

1. $1$

2. $\pm 1$

3. $-1$

4. $0$

Q.

It is given that at x=1, the function $x^4-62x^2+ax+9$ attains its maximum value, on the interval [0, 2]. Find the value of a.

Q.

If $2a+3b+6c=0$, then at least one root of the equation $a{x}^2+bx+c=0$ lies in the interval

1. $(0,1)$

2. $(1,2)$

3. $(2,3)$

4. $(1,3)$

Q.

The range of the function $f\left(x\right)=|x-1|+|x-2|,-1\le x\le 3$, is

1. $[2,5]$

2. $[1,5]$

3. $[3,5]$

4. None of these

Q. Let $f,g:R\to R$ be two functions defined as $f\left(x\right)=|x|+x$ and $g\left(x\right)=|x|-x,\forall x\in R$. Then $(f\circ g)\left(x\right)$ for $x<0$ is

1. $0$
2. $4x$
3. $-4x$
4. $2x$

Q.

Let $f:R\to R,g:R\to R$ two functions such that $f\left(x\right)=2x-3,g\left(x\right)=x^3+5$. The function ${\left(fog\right)}^{-1}\left(x\right)$ is equal to:

1. ${\left(\frac{x+7}{2}\right)}^{\frac{1}{3}}$

2. ${\left(x-\frac{7}{2}\right)}^{\frac{1}{3}}$

3. ${\left(\frac{x-2}{7}\right)}^{\frac{1}{3}}$

4. ${\left(\frac{x-7}{2}\right)}^{\frac{1}{3}}$

Q.

The number of solutions of $\left|\right[x]-2x|=4$, where$\left[x\right]$ denotes the greatest integer$\le x$, is

1. Infinite

2. $4$

3. $3$

4. $2$

Q.

The function $f\left(x\right)=2x^3+3x^2-12x+1$ decreases in the interval

1. $(2,3)$

2. $(1,2)$

3. $(-2,1)$

4. $(-3,-2)$

Q.

Let $f\left(x\right)=\left(1+b^2\right)x^2+2bx+1$ and let $m\left(b\right)$ be the minimum value of$f\left(x\right)$. As $b$ varies the range of $m\left(b\right)$ is

1. $\left[0,1\right]$

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

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

4. $(0,1]$

Q. Let $A$ is a $3\times 3$ matrix and $B$ is its adjoint matrix. If $|B|=64,$ then $|A|=$

1. $\pm 2$
2. $\pm 4$
3. $\pm 8$
4. $\pm 12$

Q.

Universal set, $U=\left\{x:x^5-6x^4+11x^3-6x^2=0\right\}$, $A=\left\{x:x^2-5x+6=0\right\}$ and $B=\left\{x:x^2-3x+2=0\right\}$ .Then, $\left(A\cap B\right)$ is equal to

1. $\left\{1,3\right\}$

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

3. $\left\{0,1,3\right\}$

4. $\left\{0,1,2,3\right\}$

Q.

Prove that sin2x+ sin2(x+π/3)+sin2(x-π/3)=3/2

Q.

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined by $\mathrm{f}\left(\mathrm{x}\right)=\left[2\mathrm{x}\right]-2\left[\mathrm{x}\right],$$\forall \mathrm{x}\in \mathrm{R},$where $\left[\mathrm{x}\right]$ is the greatest integer not exceeding $x$, then the range of $f$ is

1. $\left\{\mathrm{x}\in \mathrm{R}:0\le \mathrm{x}\le 1\right\}$

2. $\left\{0,1\right\}$

3. $\left\{\mathrm{x}\in \mathrm{R}:\mathrm{x}>0\right\}$

4. $\left\{\mathrm{x}\in \mathrm{R}:\mathrm{x}\le 0\right\}$

Q. Let $f\left(x\right)=\frac{1}{x},g\left(x\right)=\frac{1}{4x^2-1}$ and $h\left(x\right)=\frac{5x}{x+2}$ be three functions and $k\left(x\right)=h\left(g\right(f\left(x\right)\left)\right).$ If domain and range of $k\left(x\right)$ are $R-\{a_1,a_2,a_3,\dots ,a_n\}$ and $R-A$ respectively, where $R$ is the set of real numbers, then

1. $n+\sum _{i=1}^n{a}_i=5$
2. $n+\sum _{i=1}^n{a}_i=10$
3. Number of integers in set $A$ is $5$
4. Number of integers in set $A$ is $7$

Q. Let $A$ and $B$ be two sets such that $n\left(A\right)=6$ and $n\left(B\right)=3.$ If $x$ denotes the number of onto functions from $A$ to $B$ and $y$ denotes the number of one-one functions from $B$ to $A,$ then the value of $x-y$ is equal to

Q. If $y^x-x^y=1,$ then the value of $\frac{dy}{dx}$ at $x=1$ is

1. $2(1-\ln 2)$
2. $2(1+\ln 2)$
3. $2-\ln 2$
4. $2+\ln 2$

Q. Let $f:A\to B$ be a real function, where $A=\{x_1,x_2,...,x_6\}$ and $B=\{y_1,y_2...,y_{10}\}$ given by $f\left(x\right)=y.$ Then the number of functions from $A$ to $B$ such that $f\left(x_1\right)<f\left(x_2\right)<f\left(x_3\right)<f\left(x_4\right)<f\left(x_5\right)<f\left(x_6\right)$ is

Q. The general solution of $\cos 3x+8{\cos }^{3}x=0$ is .

1. $x\in \left\{\right(2n+1\left)\frac{\pi }{2}\right\}\cup \{n\pi +\frac{\pi }{3}\},n\in Z$
2. $x\in \left\{\right(2n+1\left)\frac{\pi }{2}\right\}\cup \{n\pi \pm \frac{\pi }{3}\},n\in Z$
3. $\mathrm{Not}\mathrm{possible}\mathrm{to}\mathrm{determine}.$

Q. The range of the function $f\left(x\right)=(1+{\sec }^{-1}x)(1+{\cos }^{-1}x)$ is

1. $\{2,{(1+\pi )}^2\}$
2. $(-\infty ,0]\cup [4,\infty )$
3. $\{1,{(1+\pi )}^2\}$
4. $[1,{(1+\pi )}^2]$