Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced 2015Paper 2
Exam
JEE Advanced
Year
2015
Paper
Paper 2
Subject
Mathematics
Chapter
Inverse Trigonometric Functions

JEE Advanced 2015 · Paper 2 · Mathematics · Inverse Trigonometric Functions

If α = 3 ^ - 1 ((6)/(11))andβ = 3 ^ - 1 ((4)/(9)), where the...

If α=3sin⁡−1(611)andβ=3cos⁡−1(49),\alpha = 3\sin^{- 1} \left(\dfrac{6}{11}\right)and\beta = 3\cos^{- 1} \left(\dfrac{4}{9}\right), where the inverse trigonometric functions take only the principal values, then the correct options(s) is(are)

Options

  1. Acosβ>0cos \beta > 0
  2. Bsinβ<0sin \beta < 0
  3. Ccos⁡(α+β)>0\cos \left(\alpha + \beta\right) > 0
  4. Dcosα<0cos \alpha < 0

Correct answer

B. sinβ<0sin \beta < 0

Step-by-step solution

α=3sin⁡−1611\alpha = 3\sin^{- 1} \dfrac{6}{11}
Since π4>sin⁡−1611>π6\dfrac{\pi}{4} > \sin^{- 1} \dfrac{6}{11} > \dfrac{\pi}{6}
3π4>3sin⁡−1611>π2\dfrac{3\pi}{4} > 3\sin^{- 1} \dfrac{6}{11} > \dfrac{\pi}{2}
cosα<0cos \alpha < 0
cos⁡(3cos⁡−149)=4×(49)3−3×49\cos \left(3\cos^{- 1} \dfrac{4}{9}\right) = 4 \times {\left(\dfrac{4}{9}\right)}^{3} - 3 \times \dfrac{4}{9}
=4×64729−129= \dfrac{4 \times 64}{729} - \dfrac{12}{9}
=256−12×8172<0= \dfrac{256 - 12 \times 81}{72} < 0
5π12>cos⁡−149>π3\dfrac{5\pi}{12} > \cos^{- 1} \dfrac{4}{9} > \dfrac{\pi}{3}
5π4>3cos⁡−149>π\dfrac{5\pi}{4} > 3\cos^{- 1} \dfrac{4}{9} > \pi
sin⁡(3cos−149)<0\sin \left(3\dfrac{co{s}^{- 1}4}{9}\right) < 0
3π4>3sin⁡−1611>π2\dfrac{3\pi}{4} > 3\sin^{- 1} \dfrac{6}{11} > \dfrac{\pi}{2}
5π4>3cos⁡−149>π\dfrac{5\pi}{4} > 3\cos^{- 1} \dfrac{4}{9} > \pi
2π>3sin⁡−1611+3cos⁡−149>3π22\pi > 3\sin^{- 1} \dfrac{6}{11} + 3\cos^{- 1} \dfrac{4}{9} > \dfrac{3\pi}{2}
cos⁡(3sin⁡−1611+3cos⁡−149)>0\cos \left(3\sin^{- 1} \dfrac{6}{11} + 3\cos^{- 1} \dfrac{4}{9}\right) > 0
Practice Inverse Trigonometric Functions on Quantrex Academy →

More from Inverse Trigonometric Functions

Considering only the principal values of the inverse trigonometric functions, the value… JEE Advanced 2026Let α = 3 ^ -1 ((6)/(11)) and β = 3 ^ -1 ((4)/(9)) , where inverse trigonometric… JEE Main 2026If ( ^ -1 (x 2 )) = ( ^ -1 1-x^2 ) , x ∈ (0,1) , then the value of x is : JEE Main 2026Let [·] denote the greatest integer function. If the domain of the function f(x) = ^ -1… JEE Main 2026Let 0 < α < 1 , β = (1)/(3α) and ^ -1 (1-α) + ^ -1 (1-β) = (π)/(4) . Then 6(α + β) is… JEE Main 2026If (π)/(4) + ∑_ p=1 ^ 11 ^ -1 ( 2^ p-1 1 + 2^ 2p-1 ) = α , then α is equal to __________. JEE Main 2026If y = ^ -1 ((3 x - 4 x)/(4 x + 3 x)) + 2 ^ -1 ( x 1+ 1-x^2 ) , then (dy)/(dx) at x = 3 2… JEE Main 2026Considering the principal values of inverse trigonometric functions, the value of the… JEE Main 2026If k= ((π)/(4)+(1)/(2) ^ -1 ((2)/(3)))+ ((1)/(2) ^ -1 ((2)/(3))) , then the number of… JEE Main 2026If the domain of the function f(x)= ^ -1 ( 1 x^ 2 -2 x-2 ) , is (-∞, α] [β, γ] [δ, ∞)… JEE Main 2026The number of solutions of ^ -1 4x + ^ -1 6x = (π)/(6) , where - 1 2 6 < x < 1 2 6 , is… JEE Main 2026Let the maximum value of ( ^ -1 x)^ 2 +( ^ -1 x)^ 2 for x ∈[- 3 2 , 1 2 ] be m n π^ 2… JEE Main 2026If the domain of the function f(x)= ^ -1 ((2 x-5)/(11-3 x))+ ^ -1 (2 x^ 2 -3 x+1) is the… JEE Main 2026The total number of real solutions of the equation θ= ^ -1 (2 θ)-(1)/(2) ^ -1 ((6 θ)/(9+… JEE Advanced 2025The value of ^ -1 ( 1+ ^2(2) -1 (2) )- ^ -1 ( 1+ ^2((1)/(2)) +1 ((1)/(2)) ) is equal to JEE Main 2025The sum of the infinite series ^ -1 ((7)/(4))+ ^ -1 ((19)/(4))+ ^ -1 ((39)/(4))+ ^ -1… JEE Main 2025 Full Inverse Trigonometric Functions list Inverse Trigonometric Functions JEE Maths PYQsJEE Mathematics PYQs All JEE Advanced PYQs