Harmonic Mean

Q.

If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P., then the common ratio of the G.P. is

1. 3

2. 

3. 2

4. 

Q.

If one geometric mean G and two arithemtic means p and q be inserted between two numbers, then $G^2$ is equal to:

1. (2p-q) (2q-p)

2. (4p-q) (4q-p)

3. (4p+q) (4q+p)

4. (3p-q)(3q-p)

Q. If cos (x – y), cos x and cos (x+ y) are in HP, then cos x sec$\left(\frac{y}{2}\right)$ is equal to

1. None of these
2. 
3. 
4. 

Q. If $H_1,H_2,\dots \dots ,H_{20}$ be 20 harmonic means between $2$ and $3$, then the value of $\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}$ is

1. $20$
2. $21$
3. $40$
4. $41$

Q.

If $H$ is the harmonic mean between $P$ and $Q$, then the value of $\frac{H}{P}+\frac{H}{Q}$ is

1. $2$

2. $\frac{PQ}{P+Q}$

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

4. $\frac{P+Q}{PQ}$

Q.

If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are $A\pm \sqrt{(A+G)(A-G)}.$

Q.

5 harmonic means are inserted between 5 and 10. The common difference of corresponding A.P is.

1. 

2. 

3. 

4. 

Q. If $x>1,y>1,z>1$ are in GP, then $\frac{1}{1+lnx},\frac{1}{1+lny},\frac{1}{1+lnz}$ are in

1. AP
2. GP
3. HP
4. None of these

Q. If H is the harmonic mean between p and q, then the value of $\frac{H}{p}+\frac{H}{q}$ is
[MNR 1990; UPSEAT 2000, 01]

1. 2
2. $\frac{pq}{p+q}$
3. $\frac{p+q}{pq}$
4. None of these

Q. If the harmonic mean between a and b be H, then the value of $\frac{1}{H-a}+\frac{1}{H-b}$ is

1. ab
2. a+b
3. $\frac{1}{a}+\frac{1}{b}$
4. $\frac{1}{a}-\frac{1}{b}$

Q.

If, $\frac{1}{\mathrm{x}}+\frac{1}{\mathrm{y}}=\mathrm{k}$ and $\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{y}}=\mathrm{k}$, then the value of $\mathrm{y}$ _____

Q. If the A.M. between a and b is m times their H.M. then a:b =

1. $\sqrt{m}+\sqrt{m-1}:\sqrt{m}-\sqrt{m-1}$
2. $\sqrt{m}-\sqrt{m-1}:\sqrt{m+1}+\sqrt{m-1}$
3. $\sqrt{m}+\sqrt{m+1}:\sqrt{m}-\sqrt{m+1}$
4. None of these

Q. If $9$ arithmetic means and harmonic means are inserted between $2$ and $3$, then the value of $A+\frac{6}{H}$ is
(where $A$ is any of the A.M.'s and H the corresponding H.M.)

Q. If $\frac{2}{3},H_1,H_2,H_3,H_4,\frac{2}{13}$ is an H.P., then the value of $\frac{H_1{H}_4}{H_2{H}_3}$ is

1. $\frac{14}{27}$
2. $\frac{21}{39}$
3. $\frac{55}{42}$
4. $\frac{63}{55}$

Q. The harmonic mean of $\frac{a}{1-ab}$ and $\frac{a}{1+ab}$ is
[MP PET 1996; Pb. CET 2001]

1. a
2. $\frac{a}{1-a^2{b}^2}$
3. $\frac{a}{\sqrt{1-a^2{b}^2}}$
4. $\frac{1}{1-a^2{b}^2}$

Q.

The roots of $\left(x-a\right)\left(x-a-1\right)+\left(x-a-1\right)\left(x-a-2\right)+\left(x-a\right)\left(x-a-2\right)=0$, $a\in R$ are always:

1. imaginary

2. real and distinct

3. equal

4. rational and equal

Q.

If $x+y+z=1$, then $1-3{\mathrm{x}}^2-3{\mathrm{y}}^2-3{\mathrm{z}}^2+2{\mathrm{x}}^3+2{\mathrm{y}}^3+2{\mathrm{z}}^3$ is equal to:

1. $6xyz$

2. $3xyz$

3. $2xyz$

4. $xyz$

Q. If the A.M. between a and b is m times their H.M. then a:b =

1. None of these
2. $\sqrt{m}+\sqrt{m-1}:\sqrt{m}-\sqrt{m-1}$
3. $\sqrt{m}-\sqrt{m-1}:\sqrt{m+1}+\sqrt{m-1}$
4. $\sqrt{m}+\sqrt{m+1}:\sqrt{m}-\sqrt{m+1}$