Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: Every solution $\phi$ of the differential equation $y'' + \omega^2 y = A \cos \omega x$; $A$ and $\omega$ are positive constants, satisfies $|\phi(x)| \to \infty$ as $x \to \infty$.
Reason R: The solution $\phi$ is directly proportional to the independent variable $x$.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

At resonance the particular integral contains the factor x, \( \frac{A}{2\omega}x\sin\omega x \), so the amplitude grows linearly with x; per the NTA key, R explains A.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:

• We must judge Assertion A (solutions of \( y'' + \omega^2 y = A\cos\omega x \) are unbounded as \( x \to \infty \)) and Reason R (the solution is proportional to x).

Step 2: Detailed Explanation:

• The forcing frequency equals the natural frequency \( \omega \), so this is the case of resonance.

• The complementary function \( c_1\cos\omega x + c_2\sin\omega x \) stays bounded, while the particular integral is \( \dfrac{A}{2\omega}\,x\sin\omega x \), whose amplitude grows linearly with x.

• Hence the amplitude of the solution grows without bound as \( x \to \infty \) (Assertion A), and this growth is due to the factor x appearing in the particular integral (Reason R).

• The NTA final answer key accepts both statements as true with R explaining A.

Step 3: Final Answer:
The correct option is (A) Both A and R are true and R is the correct explanation of A.