CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q242 Marksprincipal_values
CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Question 24 (Inverse Trigonometric Functions)
Evaluate \(\sin[\cot^{-1}(\sqrt2\cos(\tan^{-1}1))]\). OR On {1,2,3}, is R={(1,1),(3,3),(1,2)} symmetric? Find the smallest R₁ making R∪R₁ an equivalence relation.
Alternative 1 2 Marks
Evaluate \(\sin[\cot^{-1}(\sqrt2\cos(\tan^{-1}1))]\).
Answer: 1/√2.
- Use tan⁻¹(1)=π/4, so √2 cos(π/4)=1.
- The principal inverse cotangent of 1 is π/4.
- Its sine is 1/√2.
Alternative 2 2 Marks
On {1,2,3}, is R={(1,1),(3,3),(1,2)} symmetric? Find the smallest R₁ making R∪R₁ an equivalence relation.
Answer: No. R₁={(2,1),(2,2)}.
- Symmetry fails since (1,2) is present and its reverse is absent.
- An equivalence extension must include (2,1) for symmetry and (2,2) for reflexivity.
- Adding only these gives the equivalence classes {1,2} and {3}, proving sufficiency and minimality.