Harmonic Mean

Q.

The coefficient of x in the equation $x^2$ + px +q =0 was taken as 17 in place of 13, its roots were found to be -2 and -15, the roots of the original equation are

1. – 3, 10

2. 3, 10

3. None of these

4. -5, -18

Q.

The harmonic mean of the roots of the equation $(5+\sqrt{2}){\mathrm{x}}^2-(4+\sqrt{5})\mathrm{x}+8+2\sqrt{5}=0$ is

Q.

If the roots of the equation $(a^2-bc)x^2+2(b^2-ac)x+c^2-ab=0$ are equal, then

1. b = 0

2. a³ + b³ + c³ = 3abc

3. a³ + b³ + c³ = abc

4. a = 0

Q. $Ify=e^{-x}cosxand{y}_4+ky=0,where{y}_4=\frac{d^{4}y}{d{x}^4},thenk$=

1. 4
2. -4
3. 2
4. -2

Q. If $A$ be the $A.M.$ and $H$ be the $H.M.$ of two numbers $a$ and $b,$ then the value of $\left(\frac{a-A}{a-H}\right)\cdot \left(\frac{b-A}{b-H}\right)$ is

1. $\frac{H}{A}$
2. $H$
3. $\frac{A}{H}$
4. $A$

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 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. $\sqrt{m}+\sqrt{m+1}:\sqrt{m}-\sqrt{m+1}$

Q. If $b-c,2b-x$ and $b-a$ are in $H.P.$ then $a-\frac{x}{2},b-\frac{x}{2}$ and $c-\frac{x}{2}$are in

1. $A.P.$
2. $G.P.$
3. $H.P.$
4. None

Q. If $A$ be the $A.M.$ and $H$ be the $H.M.$ of two numbers $a$ and $b,$ then the value of $\left(\frac{a-A}{a-H}\right)\cdot \left(\frac{b-A}{b-H}\right)$ is

1. $\frac{H}{A}$
2. $H$
3. $\frac{A}{H}$
4. $A$

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. $\frac{63}{55}$
4. $\frac{55}{42}$

Q.

If $a,b,c$are in $HP$, then which of the following is true.

1. $\frac{1}{(b-a)}+\frac{1}{(b-c)}=\frac{1}{b}$

2. $\frac{ac}{(a+c)}=b$

3. $\frac{(b+a)}{(b-a)}+\frac{(b+c)}{(b-c)}=1$

4. None of these

Q. If $x^a=x^{b/2}{z}^{b/2}=z^c$, then $a,b,c$ are in

1. $A.P.$
2. $G.P.$
3. $H.P.$
4. None

Q. If $\cos (x-y),\cos x,\cos (x+y)$ are in H.P., where $y\ne 2n\pi ,n\in Z$, then the value of $\left[\cos x\sec \frac{y}{2}\right]$ is/are

(where [.] denotes greatest integer function)

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

Q. Let $A=a^{2}b+a{b}^2-a^{2}c-a{c}^2,$ $B=b^{2}c+b{c}^2-a^{2}b-a{b}^2$ and $C=a^{2}c+a{c}^2-b^{2}c-b{c}^2$, where $a>b>c>0$. If the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are in

1. A.P.
2. G.P.
3. H.P.
4. A.G.P.

Q. If roots of the equation $x^3+3p{x}^2+3qx+r=0,p,q,r\ne 0$ are in H.P., then which of the following is correct?

1. $pqr=2q^3+3r^2$
2. $pqr=2q^3+r^2$
3. $3pqr=q^3+2r^2$
4. $3pqr=2q^3+r^2$

Q. If $\cos (x-y),\cos x,\cos (x+y)$ are in H.P., where $y\ne 2n\pi ,n\in Z$, then the value of $\left[\cos x\sec \frac{y}{2}\right]$ is/are

(where [.] denotes greatest integer function)

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

Q.

Using properties of proportion, solve each of the following for $x$:

(i) $\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}=\frac{a}{b}$.

Q.

If $4^{25}+5^{15}$ is divided by $10$, then the remainder is :

1. $4$

2. $5$

3. $9$

4. $0$

Q. Which of the following statements is/are correct if $a,b\&c$ are three distinct natural numbers in H.P.?

1. $ac\ge b$
2. $\frac{b+c-a}{a},\frac{c+a-b}{b},\frac{a+b-c}{c}$ are in A.P.
3. $\frac{(c-1)}{2c}+\frac{(a-1)}{2a}=1-\frac{1}{b}$
4. $a+c>b$

Q. If the H.M. of two distinct numbers $a$ and $b$ is $\frac{a^{n-1}+b^{n-1}}{a^{n-2}+b^{n-2}}$, then the value of $n$ is

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

Q. If $a_1,a_2,a_3,a_4,a_5$ are the roots of the equation $6x^5-41x^4+97x^3-97x^2+41x-6=0,$ such that $|a_1|\le |a_2|\le |a_3|\le |a_4|\le |a_5|,$ then which of the following is/are correct?

1. the equation has three real roots and two imaginary roots.
2. $a_3,a_4,a_5$ are in A.P.
3. $a_1,a_2,a_3$ are in G.P.
4. $a_1,a_2,a_3$ are in H.P.

Q.

If$H_1,H_2,....H_{20}$be 20 harmonic means between 2 and 3, then$\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}=$

1. 20

2. 21

3. 40

4. 38

Q. The harmonic mean of the roots of equation $(5+\sqrt{2})x^2-(4+\sqrt{5})x+8+2\sqrt{5}=0$ is

1. $2$
2. $6$
3. $4$
4. $noneofthese$

Q. If $a,b,c$ are in $H.P.,$ then $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in

1. $A.P.$
2. $G.P.$
3. $H.P.$
4. None

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. HP
3. GP
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 $H_1,H_2,\dots ,H_{20}$ 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 $x^a=x^{b/2}{z}^{b/2}=z^c$, then $a,b,c$ are in

1. $A.P.$
2. $G.P.$
3. $H.P.$
4. None

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. HP
3. GP
4. None of these