Karnataka 2nd PUC Math Prep
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Karnataka 2nd PUC Math Prep
Sample questions with model answers
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Show that the relation R on the set \( A = \{ x \in \mathbb{Z} : 0 \leq x \leq 12 \} \) given by R = \{(a, b) : ab - 1 \text{ is a multiple of 4}\} is an equivalence relation.
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Prove reflexivity, symmetry, and transitivity.
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Prove that \( \cos^{-1} \left( \frac{1}{5} \right) + \cos^{-1} \left( \frac{1}{13} \right) = \frac{1}{5} \).
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Use cosine addition formulas.
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Prove that for any square matrix A with real number entries, A is a symmetric matrix and is a skew symmetric matrix.
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Use properties of symmetric and skew symmetric matrices.
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Find \( \frac{dy}{dx} \), if \( \sin y = 1 - \cos x \).
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Use implicit differentiation.
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