JEE Main 2024 9 April Evening Shift
Exam JEE Main
Year 2024
Date 9 April
Shift Evening Shift
Subject Mathematics
Chapter Definite Integration JEE Main 2024 · 9 April · Evening Shift · Mathematics · Definite Integration
Let ∫_0^x 1-(y^ (t))^2 d t=∫_0^x y(t) d t, 0 ≤ x ≤ 3, y ≥ 0, y(0)=0...
Let
∫ 0 x 1 − ( y ′ ( t ) ) 2 d t = ∫ 0 x y ( t ) d t , 0 ≤ x ≤ 3 , y ≥ 0 , y ( 0 ) = 0 \int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0 ∫ 0 x 1 − ( y ′ ( t ) ) 2 d t = ∫ 0 x y ( t ) d t , 0 ≤ x ≤ 3 , y ≥ 0 , y ( 0 ) = 0 . Then at
x = 2 , y ′ ′ + y + 1 x=2, y^{\prime \prime}+y+1 x = 2 , y ′′ + y + 1 is equal to
Options A 1 B 2 C 2 \sqrt{2} 2 D 1 / 2 1 / 2 1/2
Step-by-step solution
1 − ( y ′ ( x ) ) 2 = y ( x ) 1 − ( d y d x ) 2 = y 2 ( d y d x ) 2 = 1 − y 2 \begin{aligned} & \sqrt{1-\left(y^{\prime}(x)\right)^2}=y(x) \\ & 1-\left(\frac{d y}{d x}\right)^2=y^2 \\ & \left(\frac{d y}{d x}\right)^2=1-y^2\end{aligned} 1 − ( y ′ ( x ) ) 2 = y ( x ) 1 − ( d x d y ) 2 = y 2 ( d x d y ) 2 = 1 − y 2 d y 1 − y 2 = d x OR d y 1 − y 2 = − d x ⇒ sin − 1 y = x + c , sin − 1 y = − x + c x = 0 , y = 0 ⇒ c = 0 sin − 1 y = x , as y ≥ 0 sin x = y ⇒ d y d x = cos x d 2 y d x 2 = − sin x ⇒ − sin x + sin x + 1 = 1 \begin{aligned} & \frac{d y}{\sqrt{1-y^2}}=d x \text{\text{ OR } } \frac{d y}{\sqrt{1-y^2}}=-d x \\ & \Rightarrow \sin ^{-1} y=x+c, \sin ^{-1} y=-x+c \\ & x=0, y=0 \Rightarrow c=0 \\ & \sin ^{-1} y=x, \text{ as } y \geq 0 \\ & \sin x=y \\ & \Rightarrow \frac{d y}{d x}=\cos x \\ & \frac{d^2 y}{d x^2}=-\sin x \\ & \Rightarrow-\sin x+\sin x+1=1\end{aligned} 1 − y 2 d y = d x OR 1 − y 2 d y = − d x ⇒ sin − 1 y = x + c , sin − 1 y = − x + c x = 0 , y = 0 ⇒ c = 0 sin − 1 y = x , as y ≥ 0 sin x = y ⇒ d x d y = cos x d x 2 d 2 y = − sin x ⇒ − sin x + sin x + 1 = 1 $
Practice Definite Integration on Quantrex Academy →
← Previous question Let \(f: [ 0, (π)/(2)] arrow[ 0,1]\) be the function defined by \(f(x)= ^2 x\) and let…Next question → The value of the integral ∫_ -1 ^2 _e(x+ x^2+1 ) d x is
More from Definite Integration
The value of the definite integral ∫_ 0 ^ 2 1 3^ (x+3)/(2) \, dx is JEE Advanced 2026 The value of the integral ∫_ 0 ^ 2 x(x^2+x+1) ( x+1 )( x^4+x^2+1 ) \, dx is equal to: JEE Main 2026 If ∫_ π/6 ^ π/4 ( (x-(π)/(3)) (x+(π)/(3))+1)dx = α _e( 3 -1) , then 9α^2 is equal to… JEE Main 2026 Let A = bmatrix 1 & 3 & -1 2 & 1 & α 0 & 1 & -1 bmatrix be a singular matrix. Let f(x) =… JEE Main 2026 Let f: ℝ arrow ℝ be such that f(xy) = f(x)f(y) , for all x, y ∈ ℝ and f(0) ≠ 0 . Let g… JEE Main 2026 The value of the integral ∫ _ -1 ^ 1 ((x^3 + |x| + 1)/(x^2 + 2|x| + 1)) dx is equal to : JEE Main 2026 The value of the integral ∫_ -π/4 ^ π/4 ( 32 ^4 x 1 + e^ x )dx is: JEE Main 2026 Let (2^ 1-a + 2^ 1+a ) , f(a) , (3^a + 3^ -a ) be in A.P. and α be the minimum value of… JEE Main 2026 Let f: [1, ∞) arrow R be a differentiable function defined as f(x) = ∫_1^x f(t)\,dt +… JEE Main 2026 The value of the integral ∫_0^∞ ( _e(x))/(x^2 + 4)\,dx is: JEE Main 2026 The value of the integral ∫_ π/6 ^ π/3 ((4 - ^2 x)/( ^4 x)) dx is: JEE Main 2026 The integral ∫_ 0 ^ 1 ^ -1 (1+x+x^2)dx is equal to: JEE Main 2026 Let f be a twice differentiable function such that f(x)=∫_ 0 ^ x (t-x)dt-∫_ 0 ^ x f(t)… JEE Main 2026 Let ∫_ -2 ^ 2 (| x| + [x x])\,dx = 2(3 - 2) + β , where [·] is the greatest integer… JEE Main 2026 The value of ∫_0^ 20π ( ^4 x + ^4 x) \, dx is equal to: JEE Main 2026 Let [·] denote the greatest integer function. Then the value of ∫_0^3 ( e^x + e^ -x [x]!… JEE Main 2026
Full Definite Integration list
Definite Integration JEE Maths PYQs JEE Mathematics PYQs
All JEE Main PYQs
