Insertion of GP's between 2 Numbers

Q.

If m : n is the duplicate ratio of m + x : n + x; then $x^2=mn$.

1. True

2. False

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. 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 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. $n$ arithmetic means are inserted in between $x$ and $2y$ and then between $2x$ and $y.$ In case the $r^{th}$ means in each case be equal, then find the ratio $\frac{x}{y}.$

1. $\frac{r}{n+r+1}$
2. $\frac{n}{n+r+1}$
3. $\frac{n}{n-r+1}$
4. $\frac{r}{n-r+1}$

Q. $n$ arithmetic means are inserted in between $x$ and $2y$ and then between $2x$ and $y$. In case the $r^{th}$
means in each case be equal, then find the ratio $x/y.$

1. $\frac{n}{n-r+1}$
2. $\frac{r}{n+r+1}$
3. $\frac{n}{n+r+1}$
4. $\frac{r}{n-r+1}$

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. Show that the sum of the arithmetic means, between two given quantities, equidistant from the beginning and end is constant.

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

1. 1 and 64

2. 4 and 16

3. 2 and 16

4. 8 and 16