Graph of |x|

Q. $\text{If}f\left(x\right)=2x-1,\text{then the number of positive solution(s)}\text{of the equation}|f\left(x\right)|=|f(|x|-1)|\text{is/are}$

1. $1$
2. $4$
3. $2$
4. $8$

Q. Let $f:R\to R$ be a function defined as $f\left(x\right)=\{\begin{array}{ccc}3(1-\frac{|x|}{2})& \text{if}& |x|\le 2\\ 0& \text{if}& |x|>2\end{array}$

Let $g:R\to R$ be given by $g\left(x\right)=f(x+2)-f(x-2).$ If $n$ and $m$ denote the number of points in $R$ where $g$ is not continuous and not differentiable, respectively, then $n+m$ is equal to

Q. Let $f\left(x\right)$ be a differentiable function satisfying the condition $f\left(\frac{x}{y}\right)=\frac{f\left(x\right)}{f\left(y\right)},y\ne 0,f\left(y\right)\ne 0$ for all $x,y\in R$ and $f^{\prime }\left(1\right)=2.$ Then

1. Absolute maximum value of $f\left(x\right)$ over the interval $[-3,2]$ is $9.$
2. Area bounded by the curves $y=f\left(x\right),y=2^x$ and $y-$axis in $1^{\text{st}}$ quadrant is $\frac{9-\ln 256}{\ln 8}$ sq. units.
3. $\underset{x\to 0}{lim}\left[\frac{f\left(x\right)}{x}\right]$ does not exist, where $[.]$ denotes greatest integer function.
4. Area bounded by the curves $y=f\left(x\right),y=2^x$ and $y-$axis in $1^{\text{st}}$ quadrant is $\frac{9+\ln 256}{\ln 8}$ sq. units.

Q. If $f\left(x\right)=|x|$ and graph of $f(x-a)$ is


, then the value of $2a$ is

Q. If $y=f\left(x\right)$ makes $+ve$ intercept of $2$ and $0$ unit on $x$ and $y$ axes respectively and enclosed an area of $\frac{3}{4}$ square unit in the first quadrant, then $\underset{0}{\overset{2}{\int }}x{f}^{\prime }\left(x\right)dx$ is equal to

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

Q. The number of solution(s) of $y=|{\log }_e|x||$ and $y=e^x,$ is

Q. The equation $2{\tan }^{2}x-5\sec x=1$ holds true for exactly eleven distinct values of $x\in [0,\frac{n\pi }{2}],n\in N$. The greatest value of $n$ is

1. $21$
2. $24$
3. $22$
4. $23$

Q. The number of points of intersection of the graphs of $(x-1{)}^2+y^2-9=0$ and $|y|=|\ln (|x|-1)|$ is -

1. $4$ when $x\in (0,\infty )$
2. $2$ when $x\in (2,\infty )$
3. $2$ when $x\in (-2,0)$
4. $3$ when $x<0$

Q. The equation $2|x|-1=3|\sin \pi x|$ has

1. $4$ solutions
2. No solution in $[2,\infty )$
3. $6$ solutions
4. Two solutions in $(0,\frac{1}{2})$

Q. How many integers satisfy the condition $|x|\le 3$? ___

Q. The number of points of discontinuity and non-differentiability of the function $f\left(f\right(x\left)\right)$ in its domain are respectively

1. 2, 2
2. 1, 1
3. 1, 2
4. 2, 1

Q. The equation $2\sin x=|x|+a$ has no solution for $a\in$

1. $(\frac{3\sqrt{3}-\pi }{3},\infty )$
2. $(-\infty ,\frac{3\sqrt{3}+\pi }{3})$
3. $(-\infty ,\frac{3\sqrt{3}-\pi }{3})$
4. $(\frac{3\sqrt{3}+\pi }{3},\infty )$

Q. The equation $2\sin x=|x|+a$ has no solution for $a\in$

1. $(\frac{3\sqrt{3}-\pi }{3},\infty )$
2. $(\frac{3\sqrt{3}+\pi }{3},\infty )$
3. $(-\infty ,\frac{3\sqrt{3}-\pi }{3})$
4. $(-\infty ,\frac{3\sqrt{3}+\pi }{3})$

Q. The equation $2\sin x=|x|+a$ has no solution for $a\in$

1. $(\frac{3\sqrt{3}-\pi }{3},\infty )$
2. $(\frac{3\sqrt{3}+\pi }{3},\infty )$
3. $(-\infty ,\frac{3\sqrt{3}-\pi }{3})$
4. $(-\infty ,\frac{3\sqrt{3}+\pi }{3})$

Q. If the number of apples that grow on a tree per day can be modelled by the equation $y=3^x$, where $x$ is the number of days. There is a container in which the apples are stored and consumed all on the same day. If the capacity of the container is $\mathrm{4,\; 000}$, calculate on which day will the tree produce too many apples for the container to support?

1. Day $7$
2. Day $6$
3. Day $9$
4. Day $8$

Q. Let $f\left(x\right)=\{\begin{array}{cc}2+\sqrt{1-x^2},& \left|x\right|\le 1\\ 2e^{{(1-x)}^2},& \left|x\right|>1\end{array}$
The points where $f\left(x\right)$ is not differentiable is/are :

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

Q. How many integers satisfy the condition $|x|\le -5$ ___

Q. $\text{If}f\left(x\right)=2x-1,\text{then the number of positive solution(s)}\text{of the equation}|f\left(x\right)|=|f(|x|-1)|\text{is/are}$

1. $1$
2. $2$
3. $4$
4. $8$

Q. The solution set of the equation $lo{g}_2(3-x)+lo{g}_2(1-x)=3$ is?

1. $\left\{5\right\}$
2. $\varphi$
3. $\{-1,5\}$
4. $\{-1\}$

Q. If $I=\frac{2}{\pi }\underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{4}}{\int }}\frac{dx}{(1+e^{\sin x})(2-\cos 2x)},$
then $27I^2$ equals to

Q. The number of solution of ${\log }_e|x|=\sqrt{2-x^2}$ is

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

Q. The equation $2|x|-1=3|\sin \pi x|$ has

1. $4$ solutions
2. No solution in $[2,\infty )$
3. $6$ solutions
4. Two solutions in $(0,\frac{1}{2})$

Q. The number of real solutions of the equation $3^{-|x|}-2^{|x|}=0$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. The function $f\left(x\right)=min\{|x|,|x-2|,2-|x-1|\}$ is non-differentiable at $x$ equals

1. $-0.5$
2. $0$
3. $2$
4. $2.5$

Q. Solve for x the equation $\left|\begin{array}{ccc}{a}^2& a& 1\\ sin(n+1)x& sinnx& sin(n-1)x\\ cos(n+1)x& cosnx& cos(n-1)x\end{array}\right|=0$

1. $x=\frac{r\pi }{2},r\in I$
2. $x=r\pi ,r\in I$
3. $x=\frac{(2r+1)\pi }{2},r\in I$
4. $x=\frac{r\pi }{4},r\in I$

Q. The equation $2|x|-1=3|\sin \pi x|$ has

1. $4$ solutions
2. No solution in $[2,\infty )$
3. $6$ solutions
4. Two solutions in $(0,\frac{1}{2})$

Q. The function $f\left(x\right)=min\{|x|,|x-2|,2-|x-1|\}$ is non-differentiable at $x$ equals

1. $-0.5$
2. $0$
3. $2$
4. $2.5$

Q. Consider the following statements
$S_1:$ If $x^3+y^2=5(xy-1),$ then $\frac{dy}{dx}$ at $(1,3)$ is equal to $12$
$S_2:$ If $x^e=e^y,$ then $\frac{dy}{dx}$ at $x=1$ is equal to $e$
$S_3:$ If $f\left(x\right)$ is a linear polynomial, then $f^{\prime }\left(\sin x\right)+f^{\prime \prime }\left(\sin x\right)$ is constant.
Which of the following is/are correct about the truth value of above statements

1. $S_1$ is false and $S_3$ is true
2. $S_2$ and $S_3$ both are true
3. $S_1$ and $S_3$ both are true
4. $S_1$ is true and $S_2$ is false