A particle starts moving with initial velocity $3 m / s$ along x -axis from origin. Its acceleration is varying with x in parabolic nature as shown in figure. At $x = 1 m$ tangent to the graph makes an angle $4 5^{\circ}$ with positive $x$ -axis as shown in diagram. Then at $x = 3 m$ :-

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d x d a ​ = tan 4 5 ∘ = 1 d x d a ​ = 2 k x At x = 1 : 2 k ( 1 ) = 1 ⇒ k = 2 1 ​ a = 2 1 ​ x 2 x = 3 : a = 2 1 ​ ( 3 ) 2 = 2 9 ​ = 4.5 m / s 2 Velocity using a = v d x d v ​ v d v = 2 1 ​ x 2 d x x = 0 to x = 3 : ∫ 3 v ​ v d v = 2 1 ​ ∫ 0 3 ​ x 2 d x 2 1 ​ ( v 2 − 9 ) = 2 1 ​ ⋅ 3 27 ​ v 2 − 9 = 9 ⇒ v 2 = 18 v = 3 2 ​ m / s

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JEE Advanced 2026