Geometric Mean

Q.

If $9$ AM’s and HM’s are inserted between $2$ and $3$ and if the harmonic mean $H$ is corresponding to arithmetic mean $A$, then $A+\frac{6}{H}$ is equal to

1. $1$

2. $3$

3. $5$

4. $6$

Q. If $4a^{2}b,a{b}^2,p$ are in continued proportion then find the value of p.

Q.

If $G$ is the geometric mean of $x$ and $y$, then $\frac{1}{G^2-x^2}+\frac{1}{G^2-y^2}=$

1. $G^2$

2. $\frac{1}{G^2}$

3. $\frac{2}{G^2}$

4. $3G^2$

Q. The expression $\frac{a^{n+1}+b^{n+1}}{a^n+b^n}is[a\ne b\ne 0]$ is (where a and b are unequal non-zero numbers)

1. H.M. between a and b if n = 0
2. all are correct
3. A.M. between a and b if n = -1
4. G.M. between a and b$=-\frac{1}{2}$

Q. Divide $x^2-9x-10byx+1$. Write down all steps involved in division.

Q. For $n=25,\sum x=125,\sum x^2=650,\sum y=100,\sum y^2=460,\sum xy=508$, correlation coefficient is

1. $0.99$
2. $0.207$
3. $0.66$
4. $0.89$

Q. If n geometric means be inserted between a and b then the $n^{th}$ geometric mean will be

1. $a{\left(\frac{b}{a}\right)}^{\frac{n}{(n-1)}}$
2. $a{\left(\frac{b}{a}\right)}^{\frac{n-1}{\left(n\right)}}$
3. $a{\left(\frac{b}{a}\right)}^{\frac{n}{(n+1)}}$
4. $a{\left(\frac{b}{a}\right)}^{\frac{1}{\left(n\right)}}$

Q.

Which of the following is an empty set

1. 

2. 

3. 

4. 

Q. Let $\alpha ,\beta$ be the roots of $x^2-x+p=0$ and $\gamma ,\delta$ be the roots of $x^2-4x+q=0$. If $\alpha ,\beta ,\gamma ,\delta$ are in G.P., the integral values of p, q are respectively

1. – 6, – 32
2. – 6, 3
3. – 2, 3
4. – 2, – 32

Q. Show that
(1) ${(x+y+z)}^3>27(y+z-x)(z+x-y)(x+y-z)$
(2) $xyz>(y+z-x)(z+x-y)(x+y-z)$.

Q. If one geometric mean $G$ and two arithmetic means $A_1$ and $A_2$ are inserted between two given quantities, then $(2A_1-A_2)(2A_2-A_1)$

1. $G^2$
2. $G$
3. $G^3$
4. $2G$

Q. Find the polynomial equation whose roots are the squares of the roots of $x^3-x^2+8x-6=0$

Q. If $a,b$ and $c$ are real numbers and $a^2+b^2+c^2-ab-bc-ca=0$ then show that $a=b=c$?

Q. If one geometric mean G and two arithmetic means $A_1$ and $A_2$ be inserted between two given quantities, prove that $G^2=(2A_1-A_2)(2A_2-A_1)$.

Q. If $4a^{2}b,a{b}^2,p$ are in continued proportion then find the value of p. [2 Marks]

Q. The geometric mean of two numbers is $81$. One of the numbers is $9$. What is the other number?

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

Q. If a+b =5 and $a^2+b^2=13$, find ab.

Q. If the geometric mean between a and b is $\frac{a^{n+1}+b^{n+1}}{a^n+b^n},$ then the value of n is

1. 1

2. –1/2

3. 1/2
4. 2

Q. If the mean of x and 1/x is M then the mean of $x^3$ and $1/x^3$ is

1. $\frac{M(M^2-3)}{2}$
2. $M(4M^2-3)$
3. $M^3$
4. $M^3+3$

Q. The geometric mean of the series $1,2,4,8,\cdots 2^n$is

1. $2^{n/2}$
2. $2^{n/4}$
3. $2^{2n}$
4. none of these

Q. If ${\log }_{7}x+{\log }_{13}x=1$ and $x={13}^{{\log }_{k}7}$ then $k$ is divisible by

1. 7
2. 13
3. 17
4. 119