Relation between AM,GM,HM for 2 Numbers

Q.

If a and b are two different positive real numbers, then which of the following relations is true

1. 

2. 

3. None of these

4. 

Q.

If the aritmetic, geometric and harmonic menas between two positive real numbers be A, G and H, then

1. 

2. 

3. 

4. 

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

Q.

If the harmonic mean between two positive numbers is to their geometric mean as 12:13. The numbers are in the ratio

1. 12:13
   

2. $\frac{1}{12}:\frac{1}{13}$
   

3. 4:9
   

4. $\frac{1}{4}:\frac{1}{9}$

Q. If x, y, z are distinct positive integers then value of expression (x + y) (y + z) (z + x) is

1. is equal to 8xyz
2. is greater than 8xyz
3. is smaller than 8xyz
4. is equal to 4xyz

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. The arithmetic mean, geometric mean and harmonic mean of three distinct natural number can be equal.

1. False
2. True

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. The least value of $6{\tan }^2\theta +54{\cot }^2\theta +18$ is
$\text{I.}$ $54$ when $A.M.\ge G.M.$ is applicable for $6{\tan }^2\theta ,54{\cot }^2\theta ,18$
$\text{II.}$ $54$ when $A.M.\ge G.M.$ is applicable for $6{\tan }^2\theta ,54{\cot }^2\theta ;18$ is added further
$\text{III.}$ $78$ when ${\tan }^2\theta ={\cot }^2\theta$

1. $\text{I}$ is correct
2. $\text{I}$ and $\text{II}$ are correct
3. $\text{III}$ is correct
4. $\text{I}$ is false, $\text{II}$ is correct

Q. Let $f:R\to R$ be such that for all $x\in R,(2^{1+x}+2^{1-x}),f\left(x\right)$ and $(3^x+3^{-x})$ are in A.P., then the minimum value of $f\left(x\right)$ is:

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