If ((π)/(12) - x) , \ tan(π)/(12), ((π)/(12) + x) in order are...

\tan

\left(\frac{\pi}{12} -

x\right)

\$tan\frac{\pi}{12},

\tan

\left(\frac{\pi}{12} + x\right)

in order are three consecutive terms of a G.P. then sum of all the solutions in

[0, 314]

k\pi

. The value of

k

is:

Options

A $4950$
B $5050$
C $2525$
D $5010$

Correct answer

B. $5050$

Step-by-step solution

Since they are in G.P., we have:

\tan^2\frac{\pi}{12} = \tan\left(\frac{\pi}{12} - x\right)\tan\left(\frac{\pi}{12} + x\right)\tan^2\frac{\pi}{12} = \frac{\tan^2\frac{\pi}{12} - \tan^2 x}{1 - \tan^2\frac{\pi}{12}\tan^2 x}

Let

t = \tan^2\frac{\pi}{12}

and

y = \tan^2 x

t = \frac{t - y}{1 - ty} \implies t - t^2y = t - y \implies y(1 - t^2) = 0

. So

\tan^2 x (1 - \tan^4\frac{\pi}{12}) = 0

. Since

\tan\frac{\pi}{12} \ne 1

, we must have

\tan^2 x = 0 \implies \tan x = 0

. The solutions are

x = n\pi

, where

n \in \mathbb{Z}

. In the interval

[0, 314]

, since

\pi \approx 3.14159

100\pi \approx 314.159

, so the solutions are

x = 0, \pi, 2\pi, \dots, 99\pi

. The sum is

0 + \pi + 2\pi + \dots + 99\pi = \frac{99 \times 100}{2} \pi = 4950\pi

. Hence,

k = 4950

Practice Sequences and Series on Quantrex Academy →

More from Sequences and Series

Let α = 3+4+8+9+13+14+… upto 40 terms. If ( β)^ (α)/(1020) is a root of the equation… JEE Main 2026 The sum 1 + (1)/(2)(1^2 + 2^2) + (1)/(3)(1^2 + 2^2 + 3^2) + … upto 10 terms is equal to : JEE Main 2026 The value of 1^3 - 2^3 + 3^3 - … + 15^3 is: JEE Main 2026 The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a… JEE Main 2026 For the functions f(θ) = α ^2θ + β ^2θ , and g(θ) = α ^2θ + β ^2θ , α > β > 0 , let _ 0 <… JEE Main 2026 If the sum of the first 10 terms of the series (1)/(1 + 1^4 × 4) + (2)/(1 + 2^4 × 4) +… JEE Main 2026 Let A_1, A_2, A_3, …, A_ 39 be 39 arithmetic means between the numbers 59 and 159 . Then… JEE Main 2026 Let the sum of the first n terms of an A.P. be 3n^2 + 5n . Then the sum of squares of the… JEE Main 2026 ∑_ n=1 ^ 10 ( (528)/(n(n+1)(n+2)) ) is equal to: JEE Main 2026 Let α=(1)/(4)+(1)/(8)+(1)/(16)+…∞ and β=(1)/(3)+(1)/(9)+(1)/(27)+…∞ . Then the value of… JEE Main 2026 The first term of an A.P. of 30 non-negative terms is (10)/(3) . If the sum of this A.P.… JEE Main 2026 Let a_1, a_2, a_3, … be an A.P. and g_1 = a_1, g_2, g_3, … be an increasing G.P. If a_1 =… JEE Main 2026 The sum (1^3)/(1) + (1^3 + 2^3)/(1 + 3) + (1^3 + 2^3 + 3^3)/(1 + 3 + 5) + ⋯ up to 8… JEE Main 2026 Let α, α + 2, α ∈ ℤ , be the roots of the quadratic equation x(x+2) + (x+1)(x+3) +… JEE Main 2026 Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common… JEE Main 2026 If ∑_ k=1 ^ n a_k = 6n^3 , then ∑_ k=1 ^ 6 ( a_ k+1 - a_k 36 )^2 is equal to _______. JEE Main 2026 Full Sequences and Series list Sequences and Series JEE Maths PYQs JEE Mathematics PYQs All Black Book Advanced Problems in Mathematics PYQs