Class CBSE Class 12 Mathematics Derivatives Q #1424
KNOWLEDGE BASED
UNDERSTAND
2 Marks 2025 AISSCE(Board Exam) VSA
If $y=5\cos x-3\sin x$, prove that $\frac{d^{2}y}{dx^{2}}+y=0$.
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Detailed Solution

Step 1: Differentiate the given function with respect to $x$

Given $y = 5\cos x - 3\sin x$. Differentiating with respect to $x$, we get: $$ \frac{dy}{dx} = \frac{d}{dx}(5\cos x - 3\sin x) = -5\sin x - 3\cos x $$

Step 2: Differentiate again with respect to $x$

Differentiating $\frac{dy}{dx}$ with respect to $x$, we get: $$ \frac{d^2y}{dx^2} = \frac{d}{dx}(-5\sin x - 3\cos x) = -5\cos x + 3\sin x $$

Step 3: Substitute $y$ and $\frac{d^2y}{dx^2}$ into the equation $\frac{d^{2}y}{dx^{2}}+y=0$

Now, we need to prove that $\frac{d^2y}{dx^2} + y = 0$. Substituting the values of $\frac{d^2y}{dx^2}$ and $y$, we have: $$ \frac{d^2y}{dx^2} + y = (-5\cos x + 3\sin x) + (5\cos x - 3\sin x) = 0 $$ Thus, $\frac{d^2y}{dx^2} + y = 0$.

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the process of differentiation and then apply it to prove the given equation.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply the procedure of differentiation twice and then substitute the results to verify the given equation.<\/span>
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question is directly based on the textbook concepts of differentiation and proving a given equation.

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