Question

Let [.] denotes the greatest integer function.

List-I

(P) If P(x) = [2 cos x], x ∈ [-π, π], then P(x)

(Q) If Q(x) = [2 sin x], x ∈ [-π, π], then Q(x)

(R) If R(x) = [2 tan x/2], x ∈ [$\frac{-π}{2}$, $\frac{π}{2}$], then R(x)

(S) If S(x) = $\begin{bmatrix} 3 cosec \frac{x}{3} \end{bmatrix}$, x∈ [$\frac{π}{2}$, 2π], then S(x)

List-II

(1) is discontinuous at exactly 7 points

(2) is discontinuous at exactly 4 points

(3) is non differentiable at some points

(4) is continuous at infinitely many values

• A. is discontinuous at exactly 7 points
• B. is discontinuous at exactly 4 points
• C. is non differentiable at some points
• D. is continuous at infinitely many values

Answer 1

Answer: (C), (D)

P -> (1), Q -> (3), R -> (2), S -> (4)

Answer 2

1. P(x) = [2 cos x], x ∈ [-π, π]: Discontinuities occur when $2 \cos x$ is an integer. These points in $(-π, π)$ are $-2π/3, -π/2, -π/3, 0, π/3, π/2, 2π/3$. There are 7 such points. Thus, P matches (1).

2. Q(x) = [2 sin x], x ∈ [-π, π]: Discontinuities occur when $2 \sin x$ is an integer. These points in $(-π, π)$ are $-5π/6, -π/2, -π/6, 0, π/6, π/2, 5π/6$. There are 7 such points. If the question implies a unique match, and P is (1), then Q must map to (3) or (4). Since $Q(x)$ is a step function, it is non-differentiable at its discontinuity points. Thus, Q matches (3).

3. R(x) = [2 tan x/2], x ∈ [$\frac{-π}{2}$, $\frac{π}{2}$]: Discontinuities occur when $2 \tan x/2$ is an integer. These points are $-2\arctan(1/2), 0, 2\arctan(1/2)$. Additionally, $x = π/2$ is a point of discontinuity. Thus, there are 4 points of discontinuity. R matches (2).

4. S(x) = [3 cosec x/3], x ∈ [$\frac{π}{2}$, 2π]: Discontinuities occur when $3 \csc x/3$ is an integer. These points are $π/2, 3\arcsin(3/5), 3\arcsin(3/4), 3(π-\arcsin(3/4)), 3(π-\arcsin(3/5))$. There are 5 points of discontinuity. Since $S(x)$ is a step function, it is continuous at infinitely many values (between the discontinuity points). Thus, S matches (4).