NIMCET Mathematics Previous Year Questions (PYQs) – Page 5 of 95
Study Stuff for MCA Examinations
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Vector Projection Problem
Given: A vector of magnitude 5 makes equal angles with x, y, and z axes.
To Find: Sum of magnitudes of projections on each axis.
Let angle with each axis be \( \alpha \). Then, from direction cosine identity: \[ \cos^2\alpha + \cos^2\alpha + \cos^2\alpha = 1 \Rightarrow 3\cos^2\alpha = 1 \Rightarrow \cos\alpha = \frac{1}{\sqrt{3}} \]
Projection on each axis: \( 5 \cdot \frac{1}{\sqrt{3}} \)
Sum = \( 3 \cdot \frac{5}{\sqrt{3}} = \frac{15}{\sqrt{3}} = \boxed{5\sqrt{3}} \)
✅ Final Answer: \( \boxed{5\sqrt{3}} \)
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Conditional Probability – Bayes' Theorem
Given: One ball is transferred from Bag I to Bag II, and then a ball is drawn from Bag II and is black.
Goal: Find the probability that the transferred ball was red, given that a black ball was drawn.
Using Bayes' theorem: \[ P(R|A) = \frac{P(R \cap A)}{P(A)} = \frac{\frac{3}{10} \cdot \frac{5}{10}}{\frac{3}{10} \cdot \frac{5}{10} + \frac{4}{10} \cdot \frac{6}{10} + \frac{3}{10} \cdot \frac{5}{10}} = \frac{15}{54} = \boxed{\frac{5}{18}} \]
✅ Final Answer: \( \boxed{\frac{5}{18}} \)
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