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If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly 7 solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for…

Q1

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly 7 solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in N\), then \(\sum_{k=1}^n \frac{k}{2^k}\) is equal to :

[JEE Main 2024, 27 Jan (Shift 2)]

a

\(\frac{1}{2^{13}}\left(2^{14}-15\right)\)

b

\(1-\frac{15}{2^{13}}\)

c

\(\frac{1}{2^{15}}\left(2^{14}-14\right)\)

d

\(\frac{1}{2^{14}}\left(2^{15}-15\right)\)

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