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Question

Let the first term a of an infinite G.P. is the value of x, where the function f(x)=7+2xloge255x152x has the greatest value and the common ratio r is equal to limx0x0t2x2tan(π+x) dt. Also, let S be the sum of infinite terms of G.P.

List IList II (A)a(P)4(B)1r(Q)3(C)S(R)2(D)arS(S)1(T)5

Which of the following is the only CORRECT combination?

A
(A)(R),(B)(Q)
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B
(A)(P),(B)(R)
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C
(A)(S),(B)(P)
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D
(A)(Q),(B)(T)
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Solution

The correct option is A (A)(R),(B)(Q)
f(x)=7+2xln255x152x
f(x)=2ln255x1ln5+52xln5
=ln5(45x5+255x)
=ln555x(52x205x125)
=ln55x+1(5x25)(5x+5)

For maximum value of f(x), we equate f(x) to 0
(5x25)(5x+5)=0
5x25=0
x=2
At x=2, f(x) changes sign from +ve to ve.
Hence, x=2 is point of maximum.
a=2

Now, r=limx0x0t2x2tan(π+x) dt
r=limx0x0t2 dtx2tanx
r=limx0x23x2=13 (Using L Hospital rule)

S=a1r=2113=3
aSr=233=1

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