CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Q263 Marksprincipal_values
CBSE Class 12 Mathematics 2026 Main Paper 65(B) · Question 26 (Inverse Trigonometric Functions)
Evaluate \(\arcsin(\sin(2\pi/3))+\arctan(\tan(5\pi/6))+\arccos(\cos(13\pi/6))\). OR Show f:ℕ→ℕ, f(n)=n²+n+1 is one-one but not onto.
Alternative 1 3 Marks
Evaluate \(\arcsin(\sin(2\pi/3))+\arctan(\tan(5\pi/6))+\arccos(\cos(13\pi/6))\).
Answer: π/3.
- The principal values are π/3, −π/6 and π/6.
- Their sum is π/3.
- All three angles lie in their respective principal ranges.
Alternative 2 3 Marks
Show f:ℕ→ℕ, f(n)=n²+n+1 is one-one but not onto.
Answer: Injective but not onto.
- For consecutive nonnegative inputs f(n+1)−f(n)=2n+2>0.
- Thus the values strictly increase, proving injectivity.
- The natural number 2 has no preimage, proving non-surjectivity under either usual convention for ℕ.