IIT-JEE 2008 · Paper 2 · Mathematics Q21

MathematicsMatrix match+6Question 21 of 66

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I

A
The minimum value of $\dfrac{x^2+2x+4}{x+2}$ is
B
Let $A$ and $B$ be $3\times3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $(A+B)(A-B)=(A-B)(A+B)$ . If $(AB)^t=(-1)^kAB$ , where $(AB)^t$ is the transpose of the matrix $AB$ , then the possible values of $k$ are
C
Let $a=\log_3\log_32$ . An integer $k$ satisfying $1<2^{(-k+3^{-a})}<2$ , must be less than
D
If $\sin\theta=\cos\varphi$ , then the possible values of $\frac1\pi\left(\theta\pm\varphi-\frac\pi2\right)$ are

Column II

Answer and worked solution

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

A. $\dfrac{x^2+2x+4}{x+2}=(x+2)+\dfrac4{x+2}-2\ge4-2=2$ for $x>-2$ (AM-GM), equality at $x=0$ . The minimum (local) value is 2. → R.
B. $(A+B)(A-B)=(A-B)(A+B)$ gives $AB=BA$ . Then $(AB)^t=B^tA^t=-BA=-AB$ , so $(-1)^k=-1$ : $k$ odd, e.g. 1 or 3. → Q, S.
C. $3^{-a}=\dfrac1{\log_32}=\log_23\approx1.585$ . The condition $0<-k+\log_23<1$ gives $0.585<k<1.585$ , so $k=1$ , which is less than 2 and less than 3. → R, S.
D. $\sin\theta=\cos\varphi$ gives $\theta=2n\pi+\frac\pi2\mp\varphi$ , so one of $\frac1\pi\left(\theta\pm\varphi-\frac\pi2\right)$ equals $2n$ : even integers, e.g. 0 or 2. → P, R.

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

Topics:Maxima and minima Special matrices Logarithms Trigonometric equations

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IIT-JEE 2008 · Paper 2, Mathematics question 21