Relation between AM,GM,HM for 2 Numbers

Q. The relation R defined in A = defined in A = {1, 2, 3} by aRb, if $|a^2-b^2|\le 5$. Which of the following is false?

1. R = {(1, 1), (2, 2), (3, 3), (2, 1) (1, 2), (2, 3) (3, 2)}
2. Domain of R = {1, 2, 3}
3. Range of R = {5}
   
4. $R^{-1}=R$

Q. If one geometric mean $G$ and two arithmetic means $A_1$ and $A_2$ are inserted between two distinct positive numbers, then $\left(\frac{2A_1-A_2}{G}\right)\left(\frac{2A_2-A_1}{G}\right)$ is equal to

1. $0$
2. $1$
3. $-1.5$
4. $-2.5$

Q. If $\sum _{r=1}^{50}{\tan }^{-1}\frac{1}{2r^2}=p,$ then the value of $\tan p$ is

1. $100$
2. $\frac{50}{51}$
3. $\frac{101}{102}$
4. $\frac{51}{50}$

Q. The value of the integral $\int \frac{x^3+x+1}{x^2-1}dx$ is
(where $C$ is an arbitrary constant)

1. $\frac{x^2}{2}+\ln |x+1|+\frac{3}{2}\ln |x-1|+C$
2. $\frac{x^2}{2}+\frac{1}{2}\ln |x+1|+\frac{3}{2}\ln |x-1|+C$
3. $\frac{x^2}{2}-\ln |x+1|+\frac{2}{3}\ln |x-1|+C$
4. $\frac{x^2}{2}+\ln |x-1|+\frac{3}{2}\ln |x+1|+C$

Q. Let $x,y$ be positive real numbers and $m,n$ positive integers. The maximum value of the expression $\frac{x^m{y}^n}{(1+x^{2m})(1+y^{2n})}$ is :

1. $1$
2. $\frac{m+n}{6mn}$
3. $\frac{1}{4}$
4. $\frac{1}{2}$

Q. a, g, h are arithmetic mean, geometric mean and harmonic mean between two positive numbers x and y respectively. Then identify the correct statement among the following

1. No such relation exists between a, g and h
2. q is the geometric mean between a and h
3. A is the arithmetic mean between g and h
4. h is the harmonic mean between a and g

Q. The minimum value of $(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$ for $a>0,b>0$ and $c>0$ is

Q. If $a,b,c>0,a{b}^2{c}^3=64$ and $a+b+c=k,$ then

1. maximum value of $k=3\sqrt{3}$
2. at maximum value of $k,a:b:c=3:2:1$
3. at minimum value of $k,a:b:c=1:2:3$
4. minimum value of $k=6\times {\left(\frac{16}{27}\right)}^{1/6}$

Q.

The value of $c$ from the Lagranges mean value theorem for which $f\left(x\right)=\sqrt{25-x^2}$ is $[1,5]$ is

1. $5$

2. $1$

3. $\sqrt{15}$

4. None of these

Q. The greatest value of $x^2{y}^3$ where $x>0,y>0$ and $3x+4y=5$ is

1. $\frac{3}{16}$
2. $\frac{3}{8}$
3. $\frac{6}{5}$
4. $\frac{9}{5}$

Q. If $S=\sum _{r=1}^{90}{\text{tan}}^{-1}\left(\frac{2r}{2+r^2+r^4}\right),$ then $8190\left(\cot S\right)$ is equal to

1. $8192$
2. $8190$
3. $9108$
4. $8109$

Q. The sum of the present ages of a father and son is 53 years. Four years ago, the fathers age was four times the age of the son . Find their present ages.

Q. Let $x_n,y_n,z_n,w_n$ denote $n^{\text{th}}$ term of four different arithmetic progressions with positive terms. If $x_4+y_4+z_4+w_4=8$ and $x_{10}+y_{10}+z_{10}+w_{10}=20,$ then maximum possible value of $x_{20}\cdot y_{20}\cdot z_{20}\cdot w_{20}$ is

1. ${10}^4$
2. ${10}^8$
3. ${10}^{10}$
4. ${10}^{20}$

Q.

write the following as decimals

$\frac{12}{5}$

Q.

One angle of an isosceles triangle is $110°$, find the remaining two angles.

Q. If $x,y,z>0$ and $x+y+z=1,$ then the least value of $\frac{2x}{1-x}+\frac{2y}{1-y}+\frac{2z}{1-z}$ is

Q. Let a, b and c be positive real numbers such that $a+b+c=6$ Then range of $a{b}^2{c}^3$ is

1. (0, ∞)
2. (0, 1)
3. (0, 108]
4. (6, 108]

Q. The arithmetic mean of two numbers $a$ and $b$ $(a<b)$ is $6.$ If the geometric mean $G$ and harmonic mean $H$ of the two numbers satisfy the relation $G^2+3H=48,$ then the value of ${\log }_{b}a$ is

1. $1$
2. $\frac{1}{2}$
3. $\frac{2}{3}$
4. $6$

Q. If $a+b+c=3$ and $a>0,b>0,c>0$, then the greatest value of $a^2{b}^3{c}^2$ is

1. $\frac{3^{10}\cdot 2^4}{7^7}$
2. $\frac{3^{10}\cdot 2^7}{7^7}$
3. $\frac{3^7\cdot 2^4}{7^7}$
4. $\frac{3^7\cdot 2^4}{7^4}$

Q. Let $A_1,A_2,A_3,...,A_{11}$ be $11$ arithmetic means and $H_1,H_2,H_3,...,H_{11}$ be $11$ harmonic means and $G_1,G_2,G_3,...,G_{11}$ be $11$ geometric means between $1$ and $9$, and the value of $\prod _{k=1}^5{A}_k\cdot G_{12-2k}\cdot H_{12-k}$ is $N$. Then which of the following is\are correct?

1. Number of positive divisors of $N$ are $16$
2. Number of positive divisors of $N$ are $15$
3. If $N$ is divided by $10$, then the remainder is $7$
4. If $N$ is divided by $10$, then the remainder is $3$

Q. If the A.M. between two numbers exceeds their G.M. by 12 and the G.M. exceeds their H.M. by 36/5, then the greater of the two numbers is .

1. 72
2. 54
3. 18
4. 48

Q. a1, a2, a3, a4, a5 are the first five terms of an ap such that a1+a3+a5= -12 & a1a2a3=8 find the first term and common difference

Q.

Find The Cube Root Of $27000$ By Prime Factorization Method

Q.

If the A.M. of two positive numbers a and b(a > b) is twice their geometric mean.

Prove that : $a:b=(2+\sqrt{3}):(2-\sqrt{3})$.

Q. If $\tan ({\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{7}+{\tan }^{-1}\frac{1}{13}+\cdots +{\tan }^{-1}\frac{1}{307})$ $=\frac{p}{q},$ where $p,q\in N,$ then the value of $\left[\frac{q}{p}\right]$ is (where $\left[y\right]$ represents greatest integer function of $y$.)

Q. The area enclosed by the simultaneous inequations $x-2y^2\ge 0$ and $1-x-\left|y\right|\ge 0$ is

1. $\frac{7}{12}$ sq. units
2. $\frac{5}{12}$ sq. units
3. $8$ sq. units
4. $\frac{1}{4}$ sq. units

Q.

If three numbers are in GP, then their logarithm will be in

1. A.P

2. G.P

3. H.P

4. None of these

Q. Let $x,y$ be positive real numbers and $m,n$ positive integers. The maximum value of the expression $\frac{x^m{y}^n}{(1+x^{2m})(1+y^{2n})}$ is :

1. $\frac{1}{4}$
2. $1$
3. $\frac{m+n}{6mn}$
4. $\frac{1}{2}$

Q. Write the function in the simplest form:
${\tan }^{-1}\frac{1}{\sqrt{x^2-1}},\left|x\right|>1$

Q. If $a,b$ and $c$ are distinct positive real numbers and $a^2+b^2+c^2=1$, then $ab+bc+ca$ is-

1. Equal to $1$
2. Less than $1$
3. Greater than $1$
4. Any real number