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Question

If cos2θ+2sin2θ+3cos2θ+4sin2θ+......(200)terms=10025 , where θ is an acute angle , then the value of sinθcosθ is

A
132
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B
1+32
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C
312
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D
0
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Solution

The correct option is A 132
Given,

cos2θ+2sin2θ+3cos2θ+4sin2θ+.......+(200) terms

=(cos2θ+3cos2θ+......100terms)+(2sin2θ+4sin2θ+.......100terms)

=1002[cos2θ+(1001)2cos2θ]+1002[2sin2θ+(1001)2sin2θ] .......... Sn=n2[2a+(n1)d]

=50[2cos2θ+198cos2θ]+50[4sin2θ+198sin2θ]

50[2cos2θ+198cos2θ+4sin2θ+198sin2θ]=10025

198(cos2θ+sin2θ)+2(cos2θ+sin2θ)+2sin2θ=4012

198+2+2sin2θ=4012

2sin2θ=12

sin2θ=14

sinθ=±12

θ=30

given,

sinθcosθ

=sin30cos30

=1232

=132

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