Insertion of GP's between 2 Numbers

Q.

The sum of an infinite GP is 57 and the sum of their cubes is 9747, find the GP.

Q. If three geometric means be inserted between 2 and 32, then the third geometric mean will be

1. 4

2. 16
3. 12
4. 8

Q. If n geometric means between a and b be $G_1,G_2,.......G_n$ and a geometric mean be G, then the true relation is

1. $G_1.G_2.......G_n=G$
2. $G_1.G_2.......G_n=G^{1/n}$
3. $G_1.G_2.......G_n=G^n$
4. $G_1.G_2.......G_n=G^{2/n}$

Q. IQ of a person is given by the formula $IQ=\frac{MA}{CA}\times 100$
where MA is mental age and CA is chronological age. If $80\le IQ\le 140$ for a group 12 years old children, find the range of their mental age.

Q.

If the third term of a G.P. is 4 then the product of its first 5 terms is

1. 

2. 

3. 

4. 

Q. If $(a+ib)(c+id)(e+if)(g+ih)=A+iB$, then show that $(a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2)=A^2+B^2$.

Q. The value of $\alpha$ so that the geometric mean of $x$ and $y$, where $x\ne y$ is $\frac{x^{\alpha +2}+y^{\alpha +2}}{x^{\alpha +1}+y^{\alpha +1}}$, is

1. $-\frac{2}{3}$
2. $-\frac{1}{4}$
3. $-\frac{3}{2}$
4. $\frac{7}{6}$

Q. If m is the A.M. of two distinct real numbers I and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals:

1. $4lm{n}^2$
2. $4l^2{m}^2{n}^2$
3. $4l^{2}mn$
4. $4l{m}^{2}n$

Q. If m is the AM of two distinct real numbers l and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals

1. $4l{m}^{2}n$
2. $l^2{m}^2{n}^2$
3. $4l^{2}mn$
4. $lm{n}^2$

Q. Insert two numbers between 3 and 81 so that the resulting sequence is G.P.

Q. If m is the A.M. of two distinct real numbers I and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals:

1. $4l{m}^{2}n$
2. $4l^2{m}^2{n}^2$
3. $4lm{n}^2$
4. $4l^{2}mn$

Q.

The two numbers between $\frac{1}{16},\frac{1}{6}$ such that first three may be in G.P. and the last three in H.P. are respectively.

1. $\frac{1}{12},\frac{1}{9}$
   

2. $-\frac{1}{4},-1$
   

3. $\frac{-1}{12},\frac{-1}{9}$
   

4. $\frac{1}{4},1$

Q. The two geometric means between the number 1 and 64 are
[Kerala (Engg.) 2002]

1. 1 and 64

2. 2 and 16

3. 4 and 16

4. 8 and 16

Q. If n geometric means between a and b be $G_1,G_2,.......G_n$ and a geometric mean be G, then the true relation is

1. $G_1.G_2.......G_n=G$
2. $G_1.G_2.......G_n=G^{1/n}$
3. $G_1.G_2.......G_n=G^n$
4. $G_1.G_2.......G_n=G^{2/n}$

Q. If $m$ is the arithmetic mean of two distinct real numbers $l$ and $n(l,n>1)$ and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n,$ then $G_1^4+2G_2^4+G_3^4$ equals

1. $4lm{n}^2$
2. $4l^2{m}^2{n}^2$
3. $4l^{2}mn$
4. $4l{m}^{2}n$

Q. Write the negation of the following statements:
(i) Chennai is the capital of Tamil Nadu
(ii) $\sqrt{2}$ is not a complex number
(iii) All triangles are not equilateral triangle
(iv) The number $2$ is greater than $7$
(v) Every natural number is an integer