JEE Advanced 2015 Paper 1 Mathematics Q20

Mathematics

Problem

Match each problem in Column I with every possible value in Column II. Column I — A: In a triangle △XYZ\triangle XYZ, let a,b,ca,b,c be the lengths of the sides opposite to the angles X,Y,ZX,Y,Z, respectively. If 2(a2−b2)=c22(a^2-b^2)=c^2 and λ=sin⁡(X−Y)sin⁡Z\lambda=\dfrac{\sin(X-Y)}{\sin Z}, then possible values of nn for which cos⁡(nπλ)=0\cos(n\pi\lambda)=0 is (are); B: In a triangle △XYZ\triangle XYZ, let a,b,ca,b,c be the lengths of the sides opposite to the angles X,Y,ZX,Y,Z, respectively. If 1+cos⁡2X−2cos⁡2Y=2sin⁡Xsin⁡Y1+\cos2X-2\cos2Y=2\sin X\sin Y, then possible value(s) of ab\dfrac ab is (are); C: In R2\mathbb R^2, let 3i^+j^\sqrt3\hat i+\hat j, i^+3j^\hat i+\sqrt3\hat j and βi^+(1−β)j^\beta\hat i+(1-\beta)\hat j be the position vectors of X,Y,ZX,Y,Z with respect to the origin OO, respectively. If the distance of ZZ from the bisector of the acute angle of OX→\overrightarrow{OX} with OY→\overrightarrow{OY} is 32\dfrac3{\sqrt2}, then possible value(s) of ∣β∣|\beta| is (are); D: Suppose that F(α)F(\alpha) denotes the area of the region bounded by x=0x=0, x=2x=2, y2=4xy^2=4x and y=∣αx−1∣+∣αx−2∣+αxy=|\alpha x-1|+|\alpha x-2|+\alpha x, where α∈{0,1}\alpha\in\{0,1\}. Then the value(s) of F(α)+832F(\alpha)+\dfrac83\sqrt2, when α=0\alpha=0 and α=1\alpha=1, is (are) Column Ii — P: 1; Q: 2; R: 3; S: 5; T: 6

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