Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202225 JuneMorning Shift
Exam
JEE Main
Year
2022
Date
25 June
Shift
Morning Shift
Subject
Mathematics
Chapter
Trigonometric Equations

JEE Main 2022 · 25 June · Morning Shift · Mathematics · Trigonometric Equations

The number of values of x in the interval ((π)/(4), (7π)/(4)) for...

The number of values of xx in the interval (π4,7π4)\left(\dfrac{\pi}{4}, \dfrac{7\pi}{4}\right) for which 14cosec2x−2sin⁡2x=21−4cos⁡2x14{cosec}^{2}x - 2\sin^{2}x = 21 - 4\cos^{2}x holds, is ______.The number of values of x in the interval ((π)/(4), (7π)/(4)) for which 14 cosec ^ 2 x - 2 ^ 2 x = 21 - 4 ^ 2 x holds,

Correct answer

0

Step-by-step solution

Given x∈(π4,7π4)x \in \left(\dfrac{\pi}{4}, \dfrac{7\pi}{4}\right)

14cosec2x−2sin⁡2x=21−4cos⁡2x14{cosec}^{2}x - 2\sin^{2}x = 21 - 4\cos^{2}x

=21−4(1−sin⁡2x)= 21 - 4\left(1 - \sin^{2}x\right)

=17+4sin⁡2x= 17 + 4\sin^{2}x

14sin⁡2x−6sin⁡2x=17\dfrac{14}{\sin^{2}x} - 6\sin^{2}x = 17

Let sin⁡2x=p\sin^{2}x = p

14p−6p=17⇒14−6p2=17p\dfrac{14}{p} - 6p = 17 ⇒ 14 - 6{p}^{2} = 17p

6p2+17p−14=06{p}^{2} + 17p - 14 = 0

p=−72,23⇒sin⁡2x=23p = - \dfrac{7}{2}, \dfrac{2}{3} ⇒ \sin^{2}x = \dfrac{2}{3}

⇒sin⁡x=±23⇒ \sin x = \pm \sqrt{\dfrac{2}{3}}

The number of values of x in the interval ((π)/(4), (7π)/(4)) for which 14 cosec ^ 2 x - 2 ^ 2 x = 21 - 4 ^ 2 x holds,

∴∴ There are 44 solutions

Practice Trigonometric Equations on Quantrex Academy →

More from Trigonometric Equations

Match each entry in List-I to the correct entry in List-II and choose the correct option.… JEE Advanced 2026Passage: Consider the curve C_1 given by y = e^ -x for x ∈ [0, 10π] , and the curve C_2… JEE Advanced 2026Let S = \ θ ∈ (-2π, 2π) : θ + 1 = 3 θ\ . Then ∑_ θ ∈ S θ is equal to: JEE Main 2026If S = \ θ ∈ [-π, π] : θ (5θ)/(2) = 7θ (7θ)/(2)\ , then n(S) is equal to _______. JEE Main 2026Let S = \ x ∈ [-π, π] : x ( x + x) = a, a ∈ ℤ\ . Then n(S) is equal to : JEE Main 2026The number of elements in the set \ x ∈[0,180^ ∘ ]: (x+100^ ∘ )= (x+50^ ∘ ) x (x-50^ ∘ )\… JEE Main 2026Number of solutions of 3 2 θ+8 θ+3 3 =0, θ ∈[-3 π, 2 π] is: JEE Main 2026The number of solutions of the equation 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3 (5 θ)/(2) in… JEE Main 2025If for θ ∈[-(π)/(3), 0] , the points (x, y)=(3 (θ+(π)/(3)), 2 (θ+(π)/(6))) lie on x y+α… JEE Main 2025If 10 ^4 θ+15 ^4 θ=6 , then the value of 27 cosec^6 θ+8 ^6 θ 16 ^8 θ is: JEE Main 2025The number of solutions of equation (4- 3 ) x -2 3 ^2 x=- 4 1+ 3 , x ∈[-2 π, (5 π)/(2)] is JEE Main 2025The number of solutions of the equation 2 x+3 x=π, x ∈[-2 π, 2 π]-\ ± (π)/(2), ± (3… JEE Main 2025If θ ∈[-(7 π)/(6), (4 π)/(3)] , then the number of solutions of 3 cosec^2 θ-2( 3 -1)… JEE Main 2025If θ ∈[-2 π, 2 π] , then the number of solutions of 2 2 ^2 θ+(2- 6 ) θ- 3 =0 , is equal… JEE Main 2025The sum of all values of θ ∈[0,2 π] satisfying 2 ^2 θ= 2 θ and 2 ^2 θ=3 θ is JEE Main 2025Let f: ℝ arrow ℝ be a function defined by f(x)=\ array cc x^2 ((π)/(x^2)), & if x ≠ 0 0… JEE Advanced 2024 Full Trigonometric Equations list Trigonometric Equations JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs