Relation between AM,GM,HM for 2 Numbers

Q. Let the harmonic mean and the geometric mean of two positive numbers be in the ratio $4:5.$ Then the two numbers are in the ratio

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

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

1. False
2. True

Q.

The angles of a triangle are in the ratio $2:3:5$. Find the measure of each of the angles.

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. 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.

If two angles of a triangle are $102°$ and $56°$, what is the measure of the third angle?

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. Let the harmonic mean and the geometric mean of two positive numbers be in the ratio $4:5.$ Then the two numbers are in the ratio

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

Q.

In an $\underline{}$ triangle, each angle has measure $60°$.

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.

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

1. $2\sqrt{ab}>(a+b)$

2. $2\sqrt{ab}<(a+b)$

3. $2\sqrt{ab}$ = $(a+b)$

4. None of these

Q. If $\tan \theta =\sqrt{n}$, where $n\in N\ge 2$, then $\sec 2\theta$ is always

1. a rational number
2. an irrational number
3. a positive integer
4. a negative integer