Question

Identify the statement(s) which are always True?

• A. A summable infinite geometric progression with non zero common ratio less than unity in absolute value is a decreasing progression.
• B. An infinitely decreasing geometric progression having the property that its sum is twice the sum of its first $n$ terms $( n >2)$ has a unique common ratio.
• C. $\tan 1$ is greater than $\tan 2$.
• D. The expression $y=\cos ^2 x +\cos ^2( x +\alpha)-2 \cos \alpha \cos x \cos ( x +\alpha)$ is independent of $x$.

Answer 1

Answer: (C), (D)

(A) It is correct only if first term and common ratio are both positive.
(B)$\text { False: } \frac{a}{1-r}=\frac{2 a\left(1-r^n\right)}{1-r} (r \neq 1) $
$1=2\left(1-r^n\right) $
$ 1-r^n=\frac{1}{2} \Rightarrow r=\left(\frac{1}{2}\right)^{\frac{1}{n}}$
is $n$ is odd then $r$ is unique if $n$ is even $= \pm \sqrt[n]{\frac{1}{2}}$
(C) $\tan 1$ is + ve; $\tan 2$ is - ve
(D) True.