IIT-JEE 2009 · Paper 2 · Mathematics Q30

MathematicsMatrix match+8Question 30 of 57

Match the statements/expressions given in Column I with the values given in Column II.

Column I

A
The number of solutions of the equation $xe^{\sin x}-\cos x=0$ in the interval $\left(0,\frac\pi2\right)$
B
Value(s) of $k$ for which the planes $kx+4y+z=0$ , $4x+ky+2z=0$ and $2x+2y+z=0$ intersect in a straight line
C
Value(s) of $k$ for which $|x-1|+|x-2|+|x+1|+|x+2|=4k$ has integer solution(s)
D
If $y'=y+1$ and $y(0)=1$ , then value(s) of $y(\ln2)$

Column II

Answer and worked solution

Official answer: A → P; B → Q, S; C → Q, R, S, T; D → R

A: $xe^{\sin x}$ rises from 0 while $\cos x$ falls from 1 on $\left(0,\frac\pi2\right)$ , so they cross exactly once. → P.
B: three planes through the origin share a line when the determinant vanishes: $k(k-4)-4(4-4)+(8-2k)=k^2-6k+8=0$ , so $k=2,4$ . → Q, S.
C: $f(x)=|x-1|+|x-2|+|x+1|+|x+2|$ equals 6 for $|x|\le1$ and $4|x|$ for $|x|\ge2$ . At integers it takes the values 6, 8, 12, 16, 20, …, so $4k$ can be 8, 12, 16, 20: $k=2,3,4,5$ . → Q, R, S, T.
D: $y'=y+1$ with $y(0)=1$ gives $y=2e^x-1$ , so $y(\ln2)=3$ . → R.

Answer: A → P; B → Q, S; C → Q, R, S, T; D → R

IIT-JEE 2009 · Paper 2, Mathematics question 30