Class CBSE Class 12 Mathematics Differential Equations Q #645
KNOWLEDGE BASED
UNDERSTAND
1 Marks 2024 AISSCE(Board Exam) MCQ SINGLE
The general solution of the differential equation \(x~dy+y~dx=0\) is:
(A) \(xy=c\)
(B) \(x+y=c\)
(C) \(x^{2}+y^{2}=c^{2}\)
(D) \(log~y=log~x+c\)
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Correct Answer: A

AI Tutor Explanation

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Step-by-Step Solution

  1. Given the differential equation: \(x~dy + y~dx = 0\)

  2. Rearrange the equation: \(x~dy = -y~dx\)

  3. Separate the variables: \(\frac{dy}{y} = -\frac{dx}{x}\)

  4. Integrate both sides: \(\int \frac{dy}{y} = -\int \frac{dx}{x}\)

  5. Evaluate the integrals: \(log|y| = -log|x| + c'\), where \(c'\) is the constant of integration.

  6. Rewrite the equation: \(log|y| + log|x| = c'\)

  7. Combine the logarithms: \(log|xy| = c'\)

  8. Remove the logarithm by exponentiating: \(|xy| = e^{c'}\)

  9. Let \(c = e^{c'}\), where \(c\) is another constant. Then \(xy = c\)

Correct Answer: \(xy=c\)

AI Suggestion: Option A

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because it requires the student to recognize and apply a standard method for solving a simple differential equation.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (separation of variables and integration) to solve the differential equation.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of solving differential equations using standard techniques taught in the textbook.

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