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List - IList - II(I)Number of solutions of the equation(P)0ex+ex=tanx  x[0,π2)(II)Number of solutions of the equations(Q)1x+y=2π3 and cosx+cosy=32 is(III)Number of solutions of the equation(R)2cosx+2sinx=1, x[0,2π) is(IV)Number of solutions of the equation(S)Infinite(3sinx+cosx)3sin2xcos2x+2=4 is

Which of the following is only INCORRECT combination?


Your Answer
A
(I)(Q)
Your Answer
B
(II)(P)
Correct Answer
C
(III)(S)
Your Answer
D
(IV)(S)

Solution

The correct option is C (III)(S)
ex+ex=tanx
      
    only one solution

x+y=2π3
cosx+cosy=32cosx+cos(2π3x)=32
cosx+3 sinx=3
sin(x+π6)=6, which is not possible.

cosx+2 sinx=1
2sin2 x2=2sinxsin x2(sin x22 cosx2)=0 
sin(x2)=0, tan(x2)=2
So two solutions

3 sin2xcos2x+2=22 cos(π3+2x)=2.2 sin2(π6+2x) 
3sin2xcos2x+2=2sin(π6+2x)
3 sinx+cosx=2 sin(x+π6)
So sin(x+π6)=y,   (2y)2y=22y=1
sin(π6+x)=1, which has infinite number of solutions.

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