KVPY Sample Paper KVPY Stream-SX Model Paper-22

Number of possible ordered pair(s) (a, b) for each of which the equality, \[a\,(\cos x-1)+{{b}^{2}}=\cos \,(ax+{{b}^{2}})-1\] holds true for all \[x\in R\] are:

A) 0

B) 1

C) 2

D) infinite

Correct Answer: C

Solution :

Put \[x=0\] \[{{b}^{2}}=\cos \] \[{{b}^{2}}-1\] or \[\cos {{b}^{2}}=1+{{b}^{2}}\]\[\Rightarrow \]\[b=0\]

Now the equation becomes \[-\,2\,\,a{{\sin }^{2}}\frac{x}{2}=\cos a\,x-1=-\,2{{\sin }^{2}}\frac{ax}{2}\]

or \[a{{\sin }^{2}}\frac{x}{2}={{\sin }^{2}}\frac{ax}{2}\] must be true \[\forall \,\,x\in R\]\[\Rightarrow \]\[a=0\] or \[a=1\]