KVPY Sample Paper KVPY Stream-SX Model Paper-1

Let E' denote the complement of an event E. Let E, F, G be pairwise independent events with \[P\,(G)>0\] and \[P\,(E\cap F\cap G)=0.\] Then, \[\frac{P\,(E'\cap F')}{G}\] equals

A) \[P\,(E')+P\,(F')\]

B) \[P\,(E')-P\,(F')\]

C) \[P\,(E')-P\,(F)\]

D) \[P\,(E)-P\,(F')\]

Correct Answer: C

Solution :

[d]

Given,

\[2\sin \alpha +3\cos \beta =3\sqrt{2}\] ... (i)

\[3\sin \beta +2\cos \alpha =1\] ... (ii)

On squaring and adding Eqs. (i) and (ii), we get \[4+9+12\,(\sin \alpha \cos \beta +\sin \beta \cos \alpha )=18+1\]

\[\Rightarrow \]\[12\sin \,(\alpha +\beta )=6\]

\[\Rightarrow \]\[\sin \,(\alpha +\beta )=\frac{1}{2}\]

\[\Rightarrow \]\[\alpha +\beta =150{}^\circ \]

\[\Rightarrow \]\[\alpha +\beta \ne 30{}^\circ \]

\[\therefore \]\[\gamma =180-(\alpha +\beta )\]

\[\Rightarrow \]\[\gamma =30{}^\circ \]