IIT-JEE 2007 · Paper 1 · Mathematics Q65

MathematicsMatrix match+6Question 65 of 66

In the following $[x]$ denotes the greatest integer less than or equal to $x$ .

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

Column I

A
$x|x|$
B
$\sqrt{|x|}$
C
$x+[x]$
D
$|x-1|+|x+1|$

Column II

P
continuous in $(-1,1)$
Q
differentiable in $(-1,1)$
R
strictly increasing in $(-1,1)$
S
not differentiable at least at one point in $(-1,1)$

Answer and worked solution

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

A, $x|x|$ : equals $\pm x^2$ ; continuous, derivative $2|x|$ exists everywhere, and it is strictly increasing. → P, Q, R.
B, $\sqrt{|x|}$ : continuous, but has a cusp at 0 (not differentiable); decreasing on $(-1,0)$ . → P, S.
C, $x+[x]$ : jumps from $-1$ to $0$ at $x=0$ , so it is discontinuous and not differentiable there; on $(-1,1)$ it is $x-1$ then $x$ , strictly increasing overall. → R, S.
D, $|x-1|+|x+1|$ : equals 2 on $(-1,1)$ , so continuous and differentiable, but constant (not strictly increasing). → P, Q.

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

Topics:Differentiability Continuity Monotonicity

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IIT-JEE 2007 · Paper 1, Mathematics question 65