KVPY Sample Paper KVPY Stream-SX Model Paper-3

If a, b and c be three distinct real numbers in G.P and \[a+b+c=xb,\] then x cannot be:

A) 4

B) \[\,-\,3\]

C) \[-\,2\]

D) 2

Correct Answer: D

Solution :

Let r be the common ratio of the G.P. then \[a+b+c\,=\,xb\]

\[\Rightarrow \] \[a\,(1+r+{{r}^{2}})\,=\,x(ar)\]

\[\Rightarrow \] \[x\,=\,\left( 1+r+\frac{1}{r} \right)\]

where \[\left( r+\frac{1}{r} \right)\,\le \,-\,2\,\text{or}\,\ge \,2\]

\[\Rightarrow \] Range of x is \[(-\alpha ,-\,1]\,\cup \,[3,\alpha )\,\]

\[\therefore \]x can not be equal to 2.