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)$\text{ False: }\frac{a}{1-r}=\frac{2a\left(1-r^n\right)}{1-r}(r\ne 1)$
$1=2\left(1-r^n\right)$
$1-r^n=\frac{1}{2}\Rightarrow r={\left(\frac{1}{2}\right)}^{\frac{1}{n}}$
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.