Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202623 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2026 · 23 January · Morning Shift · Mathematics · Trigonometric Ratios & Identities

Let α and β respectively be the maximum and the minimum values of...

Let α\alpha and β\beta respectively be the maximum and the minimum values of the function f(θ)=4(sin⁡4(7π2−θ)+sin⁡4(11π+θ))−2(sin⁡6(3π2−θ)+sin⁡6(9π−θ)),θ∈Rf(\theta)=4\left(\sin ^{4}\left(\frac{7 \pi}{2}-\theta\right)+\sin ^{4}(11 \pi+\theta)\right)-2\left(\sin ^{6}\left(\frac{3 \pi}{2}-\theta\right)+\sin ^{6}(9 \pi-\theta)\right), \theta \in \mathbf{R}. Then α+2β\alpha+2 \beta is equal to :

Options

  1. A4
  2. B6
  3. C5
  4. D3

Correct answer

C. 5

Step-by-step solution

f(θ)=4{sin⁡4(7π2−θ)+sin⁡4(11π+θ)}−2{sin⁡2(3π2−θ)+sin⁡2(9π−θ)}f(\theta) = 4\left\{\sin^4\left(\dfrac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right\} - 2\left\{\sin^2\left(\dfrac{3\pi}{2} - \theta\right) + \sin^2(9\pi - \theta)\right\}
f(θ)=4(cos⁡4(θ)+sin⁡4(θ))−2(cos⁡6θ+sin⁡6θ)f(\theta) = 4(\cos^4(\theta) + \sin^4(\theta)) - 2(\cos^6\theta + \sin^6\theta)
f(θ)=4(1−2sin⁡2θcos⁡2θ)−2(1−3sin⁡2θcos⁡2θ)f(\theta) = 4(1 - 2\sin^2\theta\cos^2\theta) - 2(1 - 3\sin^2\theta\cos^2\theta)
f(θ)=2−2sin⁡2θcos⁡2θf(\theta) = 2 - 2\sin^2\theta\cos^2\theta
f(θ)=2−sin⁡2(2θ)2f(\theta) = 2 - \dfrac{\sin^2(2\theta)}{2}
\alpha = f(\theta)_\max = 2
\beta = f(\theta)_\min = \dfrac{3}{2}
⇒α+2β=5\Rightarrow \alpha + 2\beta = 5
Practice Trigonometric Ratios & Identities on Quantrex Academy →

More from Trigonometric Ratios & Identities

Let α = (1 - 2 ((π)/(11)))(1 - 2 ((3π)/(11)))(1 - 2 ((9π)/(11)))(1 - 2 ((27π)/(11)))(1… JEE Advanced 2026If A = ( 3^∘)/( 9^∘) + ( 9^∘)/( 27^∘) + ( 27^∘)/( 81^∘) and B = 81^∘ - 3^∘ , then (B)/(A)… JEE Main 2026Let P = \ θ ∈ [0, 4π] : ^2θ ≠ 1\ and S = \ a ∈ ℤ : 2( ^8θ - ^8θ) 2θ = a^2, θ ∈ P\ . Then… JEE Main 2026If ((π)/(18)) ((5π)/(18)) ((7π)/(18)) = K , then the value of ((10Kπ)/(3)) is : JEE Main 2026If (A-B) A + ^ 2 C ^ 2 ~A =1, ~A, ~B, C ∈(0, (π)/(2)) , then JEE Main 2026If x=(5)/(12) for some x ∈(π, (3 π)/(2)) , then 7 x( (13 x)/(2)+ (13 x)/(2))+ 7 x( (13… JEE Main 2026The value of 3 cosec 20^ ∘ - 20^ ∘ 20^ ∘ 40^ ∘ 60^ ∘ 80^ ∘ is equal to JEE Main 2026The least value of ( ^ 2 θ-6 θ θ+3 ^ 2 θ+2) is JEE Main 2026Let (π)/(2) ((15 θ)/(2))( 8 θ+ 8 θ)+ ((15 θ)/(2))( 8 θ- 8 θ) is equal to JEE Main 2026Let (α+β)=-(1)/(10) and (α-β)=(3)/(8) , where 0<α<(π)/(3) and 0<β<(π)/(4) . If 2 α= 3(1-r… JEE Main 2026If ^ 2 48^ ∘ - ^ 2 12^ ∘ ^ 2 24^ ∘ - ^ 2 6^ ∘ = α+β 5 2 , where α, β ∈ ℕ , then α+β is… JEE Main 2026The value of cosec 10^ ∘ - 3 10^ ∘ is equal to : JEE Main 2026Let α= 1 60^ ∘ 61^ ∘ + 1 62^ ∘ 63^ ∘ +…+ 1 118^ ∘ 119^ ∘ Then the value of ( cosec 1^ ∘ α… JEE Advanced 2025If x+ ^2 x=1, x ∈(0, (π)/(2)) , then ( ^ 12 x+ ^ 12 x)+3( ^ 10 x+ ^ 10 x+ ^8 x+ ^8 x)+(… JEE Main 2025If ∑_ r=1 ^ 13 \ (1)/( ((π)/(4)+(r-1) (π)/(6)) ((π)/(4)+(r π)/(6)))\ =a 3 +b, a, b ∈ Z… JEE Main 2025The value of ( 70^ ∘ ) ( 10^ ∘ 70^ ∘ -1) is JEE Main 2025 Full Trigonometric Ratios & Identities list Trigonometric Ratios & Identities JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs