JEE Advanced 2014 Paper 2 Mathematics Q19

Mathematics

Problem

Match List I with List II and select the correct option. Column I — P: Let y(x)=cos⁡(3cos⁡−1x)y(x)=\cos(3\cos^{-1}x), x∈[−1,1]x\in[-1,1], x≠±3/2x\ne\pm\sqrt3/2. Then 1y(x)((x2−1)d2ydx2+xdydx)\dfrac1{y(x)}\left((x^2-1)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}\right) equals; Q: Let A1,A2,…,AnA_1,A_2,\ldots,A_n (n>2)(n>2) be the vertices of a regular polygon of nn sides with centre at the origin and position vectors a⃗k\vec a_k. If ∣∑k=1n−1(a⃗k×a⃗k+1)∣=∣∑k=1n−1(a⃗k⋅a⃗k+1)∣\left|\sum_{k=1}^{n-1}(\vec a_k\times\vec a_{k+1})\right|=\left|\sum_{k=1}^{n-1}(\vec a_k\cdot\vec a_{k+1})\right|, then the minimum value of nn is; R: If the normal from P(h,1)P(h,1) on the ellipse x2/6+y2/3=1x^2/6+y^2/3=1 is perpendicular to the line x+y=8x+y=8, then the value of hh is; S: Number of positive solutions of tan⁡−1(12x+1)+tan⁡−1(14x+1)=tan⁡−1(2x2)\tan^{-1}\left(\dfrac1{2x+1}\right)+\tan^{-1}\left(\dfrac1{4x+1}\right)=\tan^{-1}\left(\dfrac2{x^2}\right) is Column Ii — 1: 11; 2: 22; 3: 88; 4: 99

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