Class 12 Mathematics: Advanced Continuity and Differentiability

1. Q1. For what value of 'k' is the function f(x) = { (sin(k+1)x + sin x) / x , if x < 0 ; c , if x = 0 ; (sqrt(x+bx^2) - sqrt(x)) / (bx^(3/2)) , if x > 0 } continuous at x = 0?
   • k = 1/2
   • k = -1/2
   • k = 0
   • k = 1
2. Q2. If f(x) = |x-1| + |x-2| + |x-3|, then f(x) is not differentiable at:
   • x = 1, 2, 3
   • x = 1, 2 only
   • x = 2, 3 only
   • x = 1, 3 only
3. Q3. Let f(x) = { e^(1/x) / (1 + e^(1/x)) , if x ≠ 0 ; 0 , if x = 0 }. Which of the following statements is true regarding the continuity of f(x) at x = 0?
   • f(x) is continuous at x = 0
   • f(x) has a removable discontinuity at x = 0
   • f(x) has a jump discontinuity at x = 0
   • f(x) is continuous from the right but not from the left at x = 0
4. Q4. If y = cot⁻¹( (√(1+sin x) + √(1-sin x)) / (√(1+sin x) - √(1-sin x)) ), then dy/dx is equal to:
   • 1/2
   • -1/2
   • 1
   • 0
5. Q5. The derivative of tan⁻¹((√(1+x²) - 1)/x) with respect to tan⁻¹x is:
   • 1/2
   • 2
   • 1
   • x

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