Relation between AM,GM,HM for 2 Numbers

Q.

What is $p^2+q^2$ formula?

Q.

The sum of two numbers is 6 times their geometric mean. Show that numbers are in the ratio $(3+2\sqrt{2}):(3-2\sqrt{2})$

Q. 53.Solve for real X for following inequalities: 4/(x-1)<3/(x-5)

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. 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. 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 sum of principal arguments of complex numbers 1+i, -1+i3½, -3½-i, 3½-i, i, -3i, 2, -1 is

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 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. 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

Q. The arithmetic mean of $a$ 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 $27{\log }_{b}a$ is

Q.

If $a,bandc$ are$+ve$, then $a^2+b^2+c^2$is

1. $>a+b+c$

2. $>ab+bc+ca$

3. $<a+b+c$

4. $<ab+bc+ca$

Q. 9. Let Iz-3-i|+|z-1-3i|=6-2;find the difference between maximum and minimum value of argument.

Q.

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

1. $A^2$ = GH

2. $H^2$ = AG

3. G = AH

4. $G^2$ = AH

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 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. The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.

Q. The sum of principal arguments of complex numbers 1+i, -1+i3½, -3½-i, 3½-i, i, -3i, 2, -1 is

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. $2$

Q. If $2,b,c$ are in A.P., $b,c,d$ are in G.P., and $c,d,18$ are in H.P. $($where $b,c,d\in R^{+})$, then the value of $b+c+d$ is

Q.

Fill the following boxes with $<\text{or}>$ sign

$5.3\times 7.6\times 1\left(\right)1\times 53\times 7.6$

Q. Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.