KC Sinha Mathematics Solution Class 11 Chapter 18 Geometric Progressions Exercise 18.5

 Exercise 18.5

Page no 18.31

Question 1

If $a, b, c$ are the $p$ th, $q$ th and $r$ th tems of an A.P. and a G.P. both, prove that $a^{b} b^{c} c^{n}=a^{c} b^{a} e^{b} .$

$\lceil H O T S \mid$

[Hint : $\left.a^{h} b^{c} c^{a}=a^{c} b^{a} c^{h} \Leftrightarrow a^{h-c} b^{c-4} c^{a r-h}=1\right]$

Question 2

An A.P. and a GR. of positive terms have the same first term and the sum of their first, second and third terms are respectively $1, \frac{1}{2}$ and 2 . Show that the sum of their fourth terns is $\frac{19}{2}$

Question 3

If $\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r}$ and $p .4 . r$ are in A.P. show that $\frac{1}{x}, \frac{1}{y}+\frac{1}{z}$ are in A.P

Page no 18.32

Question 4

(i) If $a, b, c$ are in G. and $a^{l / x}=b^{V / y}=c^{l / z}$, prove that $x, y, z$ are in A.P.

(ii) If $a, b, c, d$ be in G.P. and $a^{x}=b^{y}=c^{2}=d^{w}$, prove that $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}, \frac{1}{w}$ are in A.P.

Question 5

If reciprocals of $(y-x), 2(y-a),(y-z)$ are in A.P., prove that $x-a, y-a$, z- a are in G.P.

Hint : Let $\alpha=x-a, \beta=y-a, \gamma=z-a$, then $y-x=\beta-\alpha$ and $y-z=\beta-\gamma \mid$

Question 6

Show that if three quantities be in A.P. and G.P. both, then they are equal.

Question 7

If $a, b, c$ are in A.P. $\frac{1}{p}, \frac{1}{q}, \frac{1}{r}$ are in A.P. and $a p, b q, c r$ are in G.P. show that

$\frac{p}{r}+\frac{r}{p}=\frac{a}{c}+\frac{c}{a}$

Question 8

If $a, b, x$ are in A.P., $a, b, y$ are in G.P. and $\frac{1}{a}, \frac{1}{b}, \frac{1}{z}$ are in H.P. prove thet $4 z(x-y)(y-z)=y(x-z)^{2}$

|HOTS|

Question 9

If $x, 1, z$ are in A.P., $x, 2, z$ are in G.P. show that $\frac{1}{x}+\frac{1}{4}, \frac{1}{z}$ are in A.P.