CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Q212 Markscontinuity
CBSE Class 12 Mathematics 2026 Main Paper 65/4/3 · Question 21 (Continuity And Differentiability)
Show that f(x)=cos x/(π/2−x) for x≠π/2 and f(π/2)=1 is continuous at π/2. OR Test differentiability at 2 for f(x)=x−1 when x<2 and 2x−3 when x≥2.
Alternative 1 2 Marks
Show that f(x)=cos x/(π/2−x) for x≠π/2 and f(π/2)=1 is continuous at π/2.
Answer: Continuous.
- Set h=x−π/2, so the quotient is sin h/h.
- Its limit as h→0 is 1.
- This agrees with f(π/2), proving continuity.
Alternative 2 2 Marks
Test differentiability at 2 for f(x)=x−1 when x<2 and 2x−3 when x≥2.
Answer: Not differentiable.
- Both branches tend to f(2)=1.
- The left slope is 1 and the right slope is 2.
- Since the one-sided derivatives differ, f′(2) does not exist.