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Question

If log3x(45)=log4x(403), then the integral part of log3x3 is equivalent to

A
Maximum value of (sin82x+cos164x+2)
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B
Number of solutions of 4sinx=x in x[0,2π]
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C
Number of real roots of the equation 2x98+5x97+5x96+...+5x2+5x+3=0
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D
Value of (a1003a9824a99) if an=αnβn, where n1 and α,β are the roots of x272x3=0
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Solution

The correct option is D Value of (a1003a9824a99) if an=αnβn, where n1 and α,β are the roots of x272x3=0
log3x(45)=log4x(403)=k
45=(3x)k, 403=(4x)k
45403=(3x)k(4x)k983=(34)k
k=log3/4(34)3/2=32

Now, log3x45=32
(3x)3/2=45x3=75
log3x3=log3(75)
27<75<81
log327<log375<log381
3<log375<4
Integral part =3

Now, checking options :
sin82x+cos164x1
sin82x+cos164x+23

4sinx=x

Only 2 solutions in x[0,2π]

p(x)=2x98+5x97+5x96+...+5x2+5x+3
=2x(x97+x96+.....+x+1) +3(x97+x96+.....+x+1)
=(2x+3)(x97+x96+.....+x+1)
=(2x+3)(x+1)(x96+x94+...+x2+1)1 x R
So, real roots of p(x)=0 are 1 and 32 i.e. only two real roots.

(α1003α98)(β1003β98)24(a99)
=α98(72α)β98(72β)24(a99)=7224=3

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