CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q222 Markscontinuity_and_differentiability
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Question 22 (Continuity And Differentiability)
Check whether function f(x) defined as f(x) = { |x−3|/[2(x−3)], x<3; (x−6)/6, x≥3 } is continuous at x=3 or not. OR If √3 (x² + y²) = 4xy, then find dy/dx at (1/2, √3/2).
Alternative 1 2 Marks
Check whether function f(x) defined as f(x) = { |x−3|/[2(x−3)], x<3; (x−6)/6, x≥3 } is continuous at x=3 or not.
Answer: f is continuous at x=3.
- For x<3, |x−3| = −(x−3), so f(x)=−1/2.
- f(3) = (3−6)/6 = −1/2. LHL = lim_{x→3⁻} −1/2 = −1/2. RHL = lim_{x→3⁺} (x−6)/6 = −1/2. Since LHL = RHL = f(3), function is continuous.
- f is continuous at x=3.
Alternative 2 2 Marks
If √3 (x² + y²) = 4xy, then find dy/dx at (1/2, √3/2).
Answer: dy/dx = √3
- Differentiate w.r.t x: √3(2x+2y dy/dx) = 4(x dy/dx + y).
- => (2√3 y − 4x) dy/dx = 4y − 2√3 x. => dy/dx = (2y − √3 x)/(√3 y − 2x). At (1/2, √3/2): numerator = 2(√3/2) − √3(1/2) = √3 − √3/2 = √3/2. Denominator = √3(√3/2) − 2(1/2) = 3/2 − 1 = 1/2. dy/dx = (√3/2)/(1/2) = √3.
- dy/dx = √3