Inequality

Q. If $A=\{x:x^2-5x+6=0\}$ and $B=\{y:y\in Z,3<|y-2|\le 5\}$, then the number of relations from $A$ to $B$ is

Q. Given $n^4<{10}^n$ for a fixed positive integer $n\ge 2$, then $(n+1{)}^4<{10}^{n+1}$

1. True
2. False

Q. If $f:R^{+}\to R^{+}$ be defined as $f\left(x\right)=\{\begin{array}{c}{2}^x,x\in (0,1)\\ 3^x,x\in [1,\infty )\end{array},$
then which among the following options is correct?

1. Range of $f\left(x\right)$ is $(0,\infty )$
2. $f\left(x\right)$ is one-one function
3. Range of $f\left(x\right)$ is $(0,2)\cup (3,\infty )$
4. $f\left(x\right)$ is many one function

Q. The number of integral value(s) of $x$ that satisfy the inequality $({\log }_{0.5}x{)}^2+{\log }_{0.5}x-2\le 0$ is

Q. $Letf:[-\frac{1}{3},3]\to Randg:[-\frac{1}{3},3]\to Rdefinedbyf\left(x\right)=[x^2-4]andg\left(x\right)=|x-2|f\left(x\right)+|3x-5|f\left(x\right)$, where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ for $x\in R,$ then

1. f is discontinuous exactly at eight points in $[-\frac{1}{3},3]$
2. f is discontinuous exactly at nine points in $[-\frac{1}{3},3]$
3. g is discontinuous exactly at ten points in $[-\frac{1}{3},3]$
4. g is discontinuous exactly at nine points in $[-\frac{1}{3},3]$

Q. If $y=sinnx+cosnx$ (x, n real), then $-\sqrt{2}\le y\le \sqrt{2}$

1. True
2. False

Q. Which among the following represents the graph of $y=\sqrt{x+1}$

1. 
2. 
3. 
4. 

Q. Sum of all integral values of $x$ satisfying the inequality $\left(3^{\frac{5}{2}}{\log }_3\right(12-3x\left)\right)-\left(3^{{\log }_{2}x}\right)>32$ is

Q. Which of the following hold good?

1. $a^4+b^4+c^4>abc(a+b+c)$
2. $a^5+b^5+c^5+d^5>abcd(a+b+c+d)$
3. $a^5+b^5+c^5>abc(ab+bc+ca)$
4. $\frac{a^8+b^8+c^8}{a^3{b}^3{c}^3}>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
5. $\frac{b^2+c^2}{b+c}+\frac{c^2+a^2}{c+a}+\frac{a^2+b^2}{a+b}>a+b+c$

Q. Suppose $x$ and $y$ are natural numbers, then the number of ordered pairs $(x,y)$ which satisfy $x+y\le 5$ is

1. $25$
2. $16$
3. $9$
4. $10$

Q. If $x=a+\frac{a}{r}+\frac{a}{r^2}+\cdots \infty ,$ $y=b-\frac{b}{r}+\frac{b}{r^2}-\cdots \infty$ and $z=c+\frac{c}{r^2}+\frac{c}{r^4}+\cdots \infty$ for $|r|>1,$ then the value of $\frac{xy}{z}$ is

1. $\frac{ab}{c}$
2. $\frac{a{b}^2}{c}$
3. $\frac{ac}{b}$
4. $\frac{a^{2}b}{c}$

Q. If a, b, c are the sides of a triangle then which of the following hold good?

1. $a^2+b^2+c^2>ab+bc+ca$
2. $\frac{a^2+b^2+c^2}{ab+bc+ca}$ lies between 1 and 2
3. $a^3+b^3+c^3>3abc$
4. $3(ab+bc+ca)\le (a+b+c{)}^2\le 4(bc+ca+ab)$

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

1. $<26$ Km/h
2. $>20$ Km/h
3. $>22$ Km/h
4. $>26$ Km/h

Q. Let $S_n=1+q+q^2+\cdots +q^n$ and $T_n=1+\left(\frac{q+1}{2}\right)+{\left(\frac{q+1}{2}\right)}^2+\cdots +{\left(\frac{q+1}{2}\right)}^n$ where $q$ is a real number and $q\ne 1$. If ${}^{101}{C}_1{+}^{101}{C}_2\cdot S_1+\cdots +{}^{101}{C}_{101}\cdot S_{100}=\alpha T_{100}$, then $\alpha$ is equal to :

1. $200$
2. $2^{100}$
3. $202$
4. $2^{99}$

Q. The solution set of $x^2+4x+9\ge 0$ is

1. $\varphi \left(\text{empty set}\right)$
2. $R$
3. $[-3,4]$
4. $[0,\infty )$

Q. If $x^2+9y^2+25z^2=xyz(\frac{15}{x}+\frac{5}{y}+\frac{3}{z})$ then,

1. $x,y$ and $z$ are in $H.P.$
2. $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $A.P.$
3. $x,y$ and $z$ are in $G.P.$
4. $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in $G.P.$

Q. If $\frac{5}{x-2}>3,$ then

1. $x\in (2,\frac{11}{3})$
2. $x\in (0,\frac{11}{3})$
3. $x\in (-\infty ,\frac{11}{3})$
4. $x\in (-\infty ,2)\cup (\frac{11}{3},\infty )$

Q. For the given expression $\frac{\sqrt{2x-1}}{x-2}<1,x\in$ :

1. $[\frac{1}{2},2)\cup (5,\infty )$
2. $[\frac{1}{2},1)\cup (5,\infty )$
3. $[0,2)\cup (5,\infty )$
4. $(-\infty ,2)\cup (5,\infty )$

Q. $\text{Let}y=\sqrt{\frac{(x+1)(x-3)}{(x-2)}}$ . If $y$ takes real values, then $x$ can lie in

1. $(2,\infty )$
2. $[-1,3]-\left\{2\right\}$
3. $[-1,2)$
4. $[3,\infty )$

Q. Sam and Alex plays in the same soccer team. Last saturday Alex scored $3$ more goals than Sam, but together they scored less than $9$ goals. The possible number of goals that Alex scored?

1. $2$
2. $3$
3. $4$
4. $6$

Q. If a, b, c are real numbers such that $a^2+b^2+c^2=1$ then $ab+bc+ca>-\frac{1}{2}$

1. True
2. False

Q. Suppose $x$ and $y$ are natural numbers, then the number of ordered pairs $(x,y)$ which satisfy $x+y\le 5$ is

1. $25$
2. $16$
3. $9$
4. $10$

Q. If a, b, c are real numbers such that a + 2b + c = 4 then max. value of ab + bc + ca is

1. 2
2. 4
3. 6
4. 8

Q. If $-4\le 4x+4\le 8$, then maximum value of $x$ is

Q. For real a, b and x $-\sqrt{(a^2+b^2)}\le asinx+bcosx\le \sqrt{(a^2+b^2)}$

1. True
2. False

Q. If $\frac{x-3}{x+4}<0,$ then

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

Q. A farmer wants to buy some horses, and every horse he buys requires $2$ acres of land. If the farmer has $18$ acres of land, write an inequality representing the possible number of horses he can buy. (where $a$ is number of horses)

1. $2a\le 18,a\in R$
2. $2a<18,a\in I$
3. $2a\le 18,a\in W$
4. $2a\le 18,a\in N$

Q. Solve the irrational inequality $\sqrt{4-x^2}+\frac{\sqrt{x^2}}{x}\ge 0$

1. $[-\sqrt{3},0)\cup (0,2]$
2. $[-2,2]$
3. $[2,4]\cup [5,13]$
4. $\varphi$

Q. If $y=3^{x-1}+3^{-x-1}$ (x real), then the least value of y is

1. 2
2. 6
3. $\frac{2}{3}$
4. None of these

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

1. True
2. False