Inequality

Q. Let $f:R\to R$ be a function defined by:
$f\left(x\right)=\{\begin{array}{ccc}\begin{array}{c}\text{max}\{t^3-3t\}\\ t\le x\end{array}& ;& x\le 2\\ x^2+2x-6& ;& 2<x<3\\ [x-3]+9& ;& 3\le x\le 5\\ 2x+1& ;& x>5\end{array}$
where $\left[t\right]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I={\int }_{-2}^{2}f\left(x\right)dx$. Then the ordered pair $(m,I)$ is equal to :

Q. The number of integral value(s) of $x,$ satisfying the inequality $2|x+2|-|x+5|\le 4$ is

1. $0$
2. $5$
3. $10$
4. $7$

Q.

Let ${\mathrm{I}}_{\mathrm{n}}={\int }_1^{\mathrm{e}}{\mathrm{x}}^{19}{\left(\log \right|\mathrm{x}\left|\right)}^{\mathrm{n}}\mathrm{dx}$, where $\mathrm{n}\in \mathrm{N}$. If $20\left({\mathrm{I}}_{10}\right)={\mathrm{\alpha I}}_9+{\mathrm{\beta I}}_8$, for natural numbers $\mathrm{\alpha }\mathrm{and}\mathrm{\beta }$, then $\mathrm{\alpha }–\mathrm{\beta }$ equal to ………………… .

Q.

Let $S$ be the set of all complex numbers $z$ satisfying $|z^2+z+1|=1$.

Then which of the following statements is/are TRUE?

1. $\left|z+\frac{1}{2}\right|<\frac{1}{2},\forall z\in S$

2. $\left|z\right|<2,\forall z\in S$

3. $\left|z+\frac{1}{2}\right|\ge \frac{1}{2},\forall z\in S$

4. The $S$ has exactly four elements

Q.

${\int }_0^{\pi /2}\sqrt{1-sin2x}dx\left(a\right)2\sqrt{2}\left(b\right)2(\sqrt{2}+1)\left(c\right)2\left(d\right)2(\sqrt{2}-1)$

Q. Find the integral of $\int {(\sqrt{x}-\frac{1}{\sqrt{x}})}^{2}dx$

Q.

Solve the following linear in equations in R.

Solve:-4x >30, when

(i) $xϵR$ (ii) $xϵZ$ (iii) $xϵN$

Q. Joe enters a race where he has to cycle and run, he cycles a distance of $25\text{km}$, and then runs for $20\text{km}$. His average running speed is half of his average cycling speed.
Joe completes the race in less than $2.5\text{hrs}$ hrs, then the average speed of Joe throughout the race is

1. $<26\text{Km/h}$
2. $<22\text{Km/h}$
3. $>22\text{Km/h}$
4. $>26\text{Km/h}$

Q.

If $f:(0,1)\to R$is defined by $f\left(x\right)=[b-x]/[1-bx]$, where $b$ is a constant such that $0<b<1$. Then, the value of $f\left(b\right)$ is

1. $f^{(-1)}$ is not invertible on $(0,1)$

2. $f\ne f^{(-1)}on(0,1)$ and $f^{}\left(b\right)=1/f^{}\left(0\right)$

3. $f=f^{(-1)}on(0,1)and{f}^{}\left(b\right)=1$

4. $f^{(-1)}$is differentiable on $(0,1)$

Q. Number of integral values of $x$ satisfying the inequality ${\left(\frac{3}{4}\right)}^{6x+10-x^2}<\frac{27}{64}$ is

1. $6$
2. $7$
3. $8$
4. infinite

Q.

State the following is an equation (with a variable). Give a reason for your answer. Identify the variable from the equation with a variable.

$17=\mathrm{x}+17$

Q.

Given that x, y and b are real numbers and x<y, b>0, then

1. $\frac{x}{b}<\frac{y}{b}$

2. $\frac{x}{b}\le \frac{y}{b}$

3. $\frac{x}{b}>\frac{y}{b}$

4. $\frac{x}{b}\ge \frac{y}{b}$

Q.

Evaluate : $\sqrt{-8-6\mathrm{i}}$

1. $1\pm 3\mathrm{i}$

2. $\pm (1-3\mathrm{i})$

3. $\pm (1+3\mathrm{i})$

4. $\pm (3-\mathrm{i})$

Q.

Write the solution set of the inequation $\frac{x^2}{x-2}>0.$

Q.

The sum of $1^3$ + $2^3$ + $3^3$ + $4^3$ + ........... + ${15}^3$ is

Q. The number of $5-$digit numbers whose sum of digits is $45$ is

Q.

Is the sum of $2$ binomials always a binomial?

1. True
2. False

Q.

With the help of two examples each , show that

subtraction is not commutative for integers.

Q.

Let $R$ be a relation on $N$ defined by$R=\left\{(x,y):x+2y=8\right\}$. The domain of $R$ is

1. $\left\{2,4,8\right\}$

2. $\left\{2,4,6,8\right\}$

3. $\left\{2,4,6\right\}$

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

Q. In any triangle the semi-perimeter is greater than each of its sides.

1. True
2. False

Q. The integral value(s) of
$x$ satisfying $\sqrt{-x^2+10x-16}<x-2$ is/are

1. $5$
2. $6$
3. $7$
4. $8$

Q. If $f\left(x\right)=x^3-x^2-2x,$ then possible number of $2\times 2$ matrices that can be formed by taking the elements which are roots of the equation $f\left(|x|\right)=0$ is

Q.

The number of solutions of the equation $a^{f\left(x\right)}+g\left(x\right)=0$, where $a>0$, $g\left(x\right)\ne 0$ and has minimum value $\frac{1}{2}$ is

1. $1$

2. $2$

3. $0$

4. $\infty$

Q.

The solution set of $f\left(x\right)>g\left(x\right)$ where $f\left(x\right)=\frac{1}{2}·5^{2x+1}$ and $g\left(x\right)=5^x+4x\log 5$ is

1. $2$

2. $\left(0,\infty \right)$

3. $\sin a$

4. None of these

Q. The number of integers in solution set of $\sqrt{x}-3\le \frac{2}{\sqrt{x}-2}$ is

Q.

Let I be any interval disjoint from (−1, 1). Prove that the function f given by

is strictly increasing on I.

Q. If $(\frac{x}{3}+1,y-\frac{2}{3})=(\frac{5}{3},\frac{1}{3})$, find the values of $x$ and $y$.

Q. The integral value(s) of $x$ which satisfies the inequality $4x+5\le 2x+17$ is/are

1. $7$
2. $6$
3. $5$
4. $-4$

Q.

Let a, b and c be real numbers such that a+ 2b + c = 4 then the maximum value of ab + bc + ca is

1. 1

2. 2

3. 3

4. 4

Q. The number of integral values of $x$ for which the expression
$\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ holds true is