JEE Advanced 2022 · Paper 2 · Mathematics Q14

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MathematicsMultiple correct+4 / -2Question 14 of 54

For $x \in \mathbb{R}$ , let the function $y(x)$ be the solution of the differential equation
$\frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12}x\right), \qquad y(0) = 0.$
Then, which of the following statements is/are TRUE?

y(x)

is an increasing function

y(x)

is a decreasing function

There exists a real number

\beta

such that the line

y = \beta

intersects the curve

y = y(x)

at infinitely many points

y(x)

is a periodic function

Answer and worked solution

Official answer: C

Solve. With integrating factor $e^{12x}$ and $\omega=\frac\pi{12}$ :
$y=\frac{12\cos\omega x+\omega\sin\omega x}{144+\omega^2}-\frac{12}{144+\omega^2}e^{-12x}.$

Behaviour.

The first part oscillates forever, so $y$ is neither increasing nor decreasing. (A) and (B) are false.
The decaying exponential keeps $y$ from being periodic. (D) is false.
As $x\to\infty$ , $y$ approaches a sinusoid. Any $\beta$ strictly inside its range is crossed in every period, so the line $y=\beta$ meets the curve infinitely often. (C) is true.

(C).

Topics:Linear first-order equations Monotonicity

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JEE Advanced 2022 · Paper 2, Mathematics question 14