Monotonically Increasing Functions

Q.

Let $f\left(x\right)=\frac{\sqrt{x^2-\left|x\right|+1}}{\sqrt{9-x^2}}$. Its domain does not contain $D$, where $D=$?

1. $\left(-3,-1\right)$

2. $\left(1,3\right)$

3. $\left(-3,3\right)$

4. $\left[0\right]\cup [1,3)$

Q.

What are the properties of function?

Q. The number of points, at which the function $f\left(x\right)=|2x+1|–3|x+2|+|x^2+x–2|,x\in R$ is not differentiable, is

Q.

The value of $se{c}^{-1}\left(\frac{1}{4}\sum _{k=0}^{10}sec\left(\frac{7\mathrm{\pi }}{12}+\frac{k\mathrm{\pi }}{2}\right)sec\left(\frac{7\mathrm{\pi }}{12}+\frac{\left(k+1\right)\mathrm{\pi }}{2}\right)\right)$ in the interval $[-\frac{\pi }{4},\frac{3\pi }{4}]$ equals:

Q.

If $k\le {\sin }^{-1}x+{\cos }^{-1}x+{\tan }^{-1}x\le K$, then

1. $k=0,K=\mathrm{\pi }$

2. $k=0,K=\frac{\mathrm{\pi }}{2}$

3. $k=\frac{\mathrm{\pi }}{4},K=\frac{3\mathrm{\pi }}{4}$

4. None of these

Q.

If ${\left(\frac{2}{3}\right)}^{x+2}={\left(\frac{3}{2}\right)}^{2-2x}$ , then $x$

Q. The interval in which the function $f\left(x\right)=x^3-6x^2+9x+10$ is increasing in

1. $[1,3]$
2. $(-\infty ,1)\cup (3,\infty )$
3. $(-\infty ,1]\cup [3,\infty )$
4. $(-\infty ,-1]\cup [3,\infty )$

Q.

If $y=y\left(x\right)$ is the solution of the differential equation $\left(\frac{\left(5+e^x\right)}{\left(2+y\right)}\right)\frac{dy}{dx}+e^x=0$ satisfying $y\left(0\right)=1$ then the value of $y\left({\log }_{e}13\right)$ is.

1. $1$

2. $0$

3. $2$

4. $-1$

Q.

What is the formula of $a^2+b^2+c^2$?

Q. Let $f\left(x\right)=[2x^2+1]$ and $g\left(x\right)=\{\begin{array}{c}2x-3,x<0\\ 2x+3,x\ge 0\end{array}$, where $\left[t\right]$ is the greatest integer $\le t$. Then, in the open interval $(-1,1)$, the number of points where fog is discontinuous is equal to

Q.

The interval in which $y=x^2{e}^{-x}$ is increasing with respect to x is

a) $(-\infty ,\infty )$
b) (-2, 0)
c) $(2,\infty )$
d) (0.2)

Q. Constant functions are monotonically increasing functions.

1. T

Q.

Verify that $y^2=4a(x+a)$ is a solution of the differential equation $y\left\{1-{\left(\frac{dy}{dx}\right)}^2\right\}=2x\frac{dy}{dx}$.

Q. If $f\left(x\right)=x{e}^{x(1-x)}$, then f(x) is

1. increasing on $[\frac{-1}{2},1]$
2. decreasing on R
3. increasing on R
4. decreasing on $[\frac{-1}{2},1]$

Q.

The inverse of the function $y={\log }_a\left(x+\sqrt{x^2+1}\right)\left(a\ne 1,a>0\right)$

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

2. not defined for all $x$

3. defined for only positive $x$

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

Q. Let $f:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}-55x,& \text{if}x<-5\\ 2x^3-3x^2-120x,& \text{if}-5\le x\le 4\\ 2x^3-3x^2-36x-336,& \text{if}x>4\end{array}$

Let $A=\{x\in R:f$ is increasing$\}.$ Then $A$ is equal to

1. $(-5,-4)\cup (4,\infty )$
2. $(-\infty ,-5)\cup (4,\infty )$
3. $(-5,\infty )$
4. $(-\infty ,-5)\cup (-4,\infty )$

Q.

If $y=\sqrt{(1-x)(1+x)}$, then

1. $(1-x^2)\frac{dy}{dx}-xy=0$

2. $(1-x^2)\frac{dy}{dx}+xy=0$

3. $(1-x^2)\frac{dy}{dx}-2xy=0$

4. $(1-x^2)\frac{dy}{dx}+2xy=0$

Q.

The function$f\left(x\right)=x+\cos x$ is

1. Always increasing

2. Always decreasing

3. Increasing for a certain range of $x$.

4. None of the above

Q.

If by dropping a stone in a quiet lake a wave moves in a circle at a speed of $3.5cm/sec$, then the rate of increase of the enclosed region when the radius of the circular wave is $10cm$, is $(\pi =\frac{22}{7})$

1. $220sq.cm/sec$

2. $110sq.cm/sec$

3. $35sq.cm/sec$

4. $350sq.cm/sec$

Q.

The equation ${\left(x-3\right)}^9+{\left(x-3^2\right)}^9+.....+{\left(x-3^9\right)}^9=0$ has

1. All the roots real

2. One real and eight imaginary roots

3. Real roots necessay $x=3,3^2,....,3^9$

4. None of these

Q.

Show that: $\mathrm{secA}(1-\mathrm{sinA})(\mathrm{secA}+\mathrm{tanA})=1$

Q.

The least value of the function $f\left(x\right)=ax+\frac{b}{x}\left(a>0,b>0,x>0\right)$ is

1. $\sqrt{ab}$

2. $\frac{2\sqrt{a}}{b}$

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

4. $2\sqrt{ab}$

Q.

If $\alpha ={\sin }^{-1}\left(\frac{\surd 3}{2}\right)+{\sin }^{-1}\left(\frac{1}{3}\right)$ and $\beta ={\cos }^{-1}\left(\frac{\surd 3}{2}\right)+{\cos }^{-1}\left(\frac{1}{3}\right)$, then

1. $\alpha >\beta$

2. $\alpha =\beta$

3. $\alpha <\beta$

4. $\alpha +\beta =2\pi$

Q.

If $f\left(x\right)=\frac{x}{\sin x}$ and $g\left(x\right)=\frac{x}{\tan x}$, where $0<x\le 1$, then in the interval

1. both $f\left(x\right)$ and $g\left(x\right)$ are increasing functions

2. both $f\left(x\right)$ and $g\left(x\right)$ are decreasing functions

3. $f\left(x\right)$ is a decreasing function and $g\left(x\right)$ is an increasing function

4. $f\left(x\right)$ is increasing function

Q. If $\alpha ,\beta$ are the roots of the equation $3x^2+5x-7=0$, then the equation whose roots are $\frac{1}{3\alpha +5},\frac{1}{3\beta +5}$ is

1. $147x^2+35x-5=0$
2. $147x^2+35x-14=0$
3. $21x^2+5x-1=0$
4. $21x^2+5x-3=0$

Q. The entries in a $2X2$ determinant $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|$ are integers chosen randomly and independently, and for each entry, the probability that the entry is odd is $p$. If the probability that value of determinant is even is$\frac{1}{2}$, then find the value of $p$

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

Q.

If $f\left(x\right)=\left[x-2\right]$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, then $f\left(2.5\right)$ is equal to

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

2. $0$

3. $1$

4. Does not exist.

Q. If $f\left(x\right)=30-2x-x^3$, then the number of positive integral value(s) of $x$ satisfying $f\left(f\right(f\left(x\right)\left)\right)>f\left(f\right(-x\left)\right)$ is
(correct answer + 1, wrong answer - 0.25)

1. $3$
2. $2$
3. $1$
4. $4$

Q.

How many real solutions does the equation $x^7+14x^5+16x^3+30x-560=0$ have?

1. $5$

2. $7$

3. $1$

4. $3$

Q.

If A + B + C = 2 S, then sin (S - A) + sin (S - B) + sin (S - C) - sin S =

1. 4 sin A sin B sin C

2. 4 cos A cos B cos C

3. 4 cos cos cos

4. 4 sin . sin . sin