JEE Main 2025 7 April Morning Shift
Exam JEE Main
Year 2025
Date 7 April
Shift Morning Shift
Subject Mathematics
Chapter Trigonometric Equations JEE Main 2025 · 7 April · Morning Shift · Mathematics · Trigonometric Equations
If for θ ∈[-(π)/(3), 0] , the points (x, y)=(3 (θ+(π)/(3)), 2...
If for
θ ∈ [ − π 3 , 0 ] \theta \in\left[-\frac{\pi}{3}, 0\right] θ ∈ [ − 3 π , 0 ] , the points
( x , y ) = ( 3 tan ( θ + π 3 ) , 2 tan ( θ + π 6 ) ) (x, y)=\left(3 \tan \left(\theta+\frac{\pi}{3}\right), 2 \tan \left(\theta+\frac{\pi}{6}\right)\right) ( x , y ) = ( 3 tan ( θ + 3 π ) , 2 tan ( θ + 6 π ) ) lie on
x y + α x + β y + γ = 0 x y+\alpha x+\beta y+\gamma=0 x y + αx + β y + γ = 0 , then
α 2 + β 2 + γ 2 \alpha^2+\beta^2+\gamma^2 α 2 + β 2 + γ 2 is equal to :
Step-by-step solution
< b r / > x = 3 ( tan θ + 3 1 − 3 tan θ ) x − 3 tan θ = 3 tan θ + 3 3 tan θ = x − 3 3 3 + 3 x … ( 1 ) 2 ( tan θ + 1 3 1 − tan θ 3 = y ) 2 ( 3 tan θ + 1 ) = y ( 3 − tan θ ) … . < b r / > \begin{aligned}<br/>& x=3\left(\frac{\tan \theta+\sqrt{3}}{1-\sqrt{3} \tan \theta}\right) \\
& x-\sqrt{3} \tan \theta=3 \tan \theta+3 \sqrt{3} \\
& \tan \theta=\frac{x-3 \sqrt{3}}{3+\sqrt{3} x} \ldots(1) \\
& 2\left(\frac{\tan \theta+\frac{1}{\sqrt{3}}}{1-\frac{\tan \theta}{\sqrt{3}}}=y\right) \\
& 2(\sqrt{3} \tan \theta+1)=y(\sqrt{3}-\tan \theta) \ldots .<br/>\end{aligned} < b r / > x = 3 ( 1 − 3 tan θ tan θ + 3 ) x − 3 tan θ = 3 tan θ + 3 3 tan θ = 3 + 3 x x − 3 3 … ( 1 ) 2 ( 1 − 3 t a n θ tan θ + 3 1 = y ) 2 ( 3 tan θ + 1 ) = y ( 3 − tan θ ) … . < b r / > using (1) and (2)
< b r / > 2 ( x − 3 3 3 + x + 1 ) = y ( 3 − ( x − 3 3 ) 3 ( 3 + x ) ) 2 3 ( x − 3 3 + x + 3 ) = y ( 3 ( 3 + x ) − x + 3 3 ) 4 3 x − 12 = y ( 2 x + 6 3 ) x y − 2 3 x + 3 3 y − 6 = 0 ⇒ α = − 2 3 , β = 3 3 , γ = − 6 α 2 + β 2 + γ 2 = 12 + 27 + 36 = 75 < b r / > \begin{aligned}<br/>& 2\left(\frac{x-3 \sqrt{3}}{\sqrt{3}+x}+1\right)=y\left(\sqrt{3}-\frac{(x-3 \sqrt{3})}{\sqrt{3}(\sqrt{3}+x)}\right) \\
& 2 \sqrt{3}(x-3 \sqrt{3}+x+\sqrt{3})=y(3(\sqrt{3}+x)-x+3 \sqrt{3}) \\
& 4 \sqrt{3} x-12=y(2 x+6 \sqrt{3}) \\
& x y-2 \sqrt{3} x+3 \sqrt{3} y-6=0 \\
& \Rightarrow \alpha=-2 \sqrt{3}, \beta=3 \sqrt{3}, \gamma=-6 \\
& \alpha^2+\beta^2+\gamma^2=12+27+36=75<br/>\end{aligned} < b r / > 2 ( 3 + x x − 3 3 + 1 ) = y ( 3 − 3 ( 3 + x ) ( x − 3 3 ) ) 2 3 ( x − 3 3 + x + 3 ) = y ( 3 ( 3 + x ) − x + 3 3 ) 4 3 x − 12 = y ( 2 x + 6 3 ) x y − 2 3 x + 3 3 y − 6 = 0 ⇒ α = − 2 3 , β = 3 3 , γ = − 6 α 2 + β 2 + γ 2 = 12 + 27 + 36 = 75 < b r / >
Practice Trigonometric Equations on Quantrex Academy →
← Previous question The number of solutions of the equation 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3 (5 θ)/(2) in…Next question → If 10 ^4 θ+15 ^4 θ=6 , then the value of 27 cosec^6 θ+8 ^6 θ 16 ^8 θ is:
More from Trigonometric Equations
Match each entry in List-I to the correct entry in List-II and choose the correct option.… JEE Advanced 2026 Passage: Consider the curve C_1 given by y = e^ -x for x ∈ [0, 10π] , and the curve C_2… JEE Advanced 2026 Let S = \ θ ∈ (-2π, 2π) : θ + 1 = 3 θ\ . Then ∑_ θ ∈ S θ is equal to: JEE Main 2026 If S = \ θ ∈ [-π, π] : θ (5θ)/(2) = 7θ (7θ)/(2)\ , then n(S) is equal to _______. JEE Main 2026 Let S = \ x ∈ [-π, π] : x ( x + x) = a, a ∈ ℤ\ . Then n(S) is equal to : JEE Main 2026 The number of elements in the set \ x ∈[0,180^ ∘ ]: (x+100^ ∘ )= (x+50^ ∘ ) x (x-50^ ∘ )\… JEE Main 2026 Number of solutions of 3 2 θ+8 θ+3 3 =0, θ ∈[-3 π, 2 π] is: JEE Main 2026 The number of solutions of the equation 2 θ (θ)/(2)+ (5 θ)/(2)=2 ^3 (5 θ)/(2) in… JEE Main 2025 If 10 ^4 θ+15 ^4 θ=6 , then the value of 27 cosec^6 θ+8 ^6 θ 16 ^8 θ is: JEE Main 2025 The number of solutions of equation (4- 3 ) x -2 3 ^2 x=- 4 1+ 3 , x ∈[-2 π, (5 π)/(2)] is JEE Main 2025 The number of solutions of the equation 2 x+3 x=π, x ∈[-2 π, 2 π]-\ ± (π)/(2), ± (3… JEE Main 2025 If θ ∈[-(7 π)/(6), (4 π)/(3)] , then the number of solutions of 3 cosec^2 θ-2( 3 -1)… JEE Main 2025 If θ ∈[-2 π, 2 π] , then the number of solutions of 2 2 ^2 θ+(2- 6 ) θ- 3 =0 , is equal… JEE Main 2025 The sum of all values of θ ∈[0,2 π] satisfying 2 ^2 θ= 2 θ and 2 ^2 θ=3 θ is JEE Main 2025 Let 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 Let (π)/(2) < x < π be such that x= -5 11 . Then ( (11 x)/(2))( 6 x- 6 x)+( (11 x)/(2))(… JEE Advanced 2024
Full Trigonometric Equations list
Trigonometric Equations JEE Maths PYQs JEE Mathematics PYQs
All JEE Main PYQs
