Continuity and Differentiability in past papers.

CBSE Board Examination 2026

• Board 2026 Q21Continuity of a piecewise function at $x = 3$⁠, or $\frac{dy}{dx}$ on $\sqrt3(x^2 + y^2) = 4xy$
  Continuity at a point; implicit differentiation2 marks
• Board 2026 Q34If $x = \cos t$⁠, $y = \cos mt$⁠, prove $(1 - x^2)y'' - xy' + m^2y = 0$
  Second derivative of a parametric function5 marks

CBSE Board Examination 2025

• Board 2025 Q5Is $f(x) = |x| + |x - 1|$ continuous and differentiable at $0$ and $1$⁠?
  Continuity and differentiability of |x| + |x − 1|1 mark
• Board 2025 Q20Value of k for continuity at $x = 5$
  Continuity of a piecewise function1 mark
• Board 2025 Q21Differentiate $2^{\cos^2 x}$ w.r.t. $\cos^2 x$⁠, or $\frac{dy}{dx}$ when $\tan^{-1}(x^2 + y^2) = a^2$
  Derivative w.r.t. a function; implicit functions2 marks
• Board 2025 Q32Prove $\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}$⁠, or $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{4}$ for a parametric curve
  Implicit, parametric and second derivatives5 marks

CBSE Board Examination 2024

• Board 2024 Q3Continuity of $f(x) = x^2 + 3$ ($x \ne 0$⁠), $f(0) = 1$
  Continuity of a piecewise function1 mark
• Board 2024 Q13Derivative of $\sin(x^2)$ at $x = \sqrt{\pi}$
  Derivative by the chain rule1 mark
• Board 2024 Q21Is $f(x) = x^2|x|$ differentiable at $x = 0$⁠? or prove $\sqrt x\,\frac{dy}{dx} = \frac{1 + y^4}{4y}$ for $y = \sqrt{\tan\sqrt x}$
  Differentiability at a point; chain rule2 marks
• Board 2024 Q27Prove $\frac{dy}{dx} = \sqrt{\frac{1 - y^2}{1 - x^2}}$ if $\sqrt{1 - x^2} + \sqrt{1 - y^2} = a(x - y)$⁠, or differentiate $(\tan x)^x$
  Implicit and logarithmic differentiation3 marks

CBSE Board Examination 2023

• Board 2023 Q13k for which $3x + 5$ ($x \ge 2$⁠), $kx^2$ ($x < 2$⁠) is continuous
  Continuity of a piecewise function1 mark
• Board 2023 Q24Show that $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = 4y^2$ for $y = \left(x + \sqrt{x^2 - 1}\right)^2$
  Chain rule: show a derivative relation2 marks
• Board 2023 Q38Case study: a roller-coaster path $f(x) = a(x + 9)(x + 1)(x - 3)$
  Second order derivative of a cubic4 marks

CBSE Sample Paper 2026-27

• Sample 2026-27 Q7Derivative of $\cos^{-1}(2x^2 - 1)$ with respect to $\cos^{-1}x$
  Derivative of one function with respect to another1 mark
• Sample 2026-27 Q8Triplet $(a, b, c)$ that makes a piecewise function differentiable at $x = 1$
  Differentiability of a piecewise function1 mark
• Sample 2026-27 Q22If $y = \sin(x + y)$ and $\frac{dy}{dx} = 0$⁠, prove that $y = \pm 1$
  Implicit differentiation2 marks
• Sample 2026-27 Q28$\frac{d^2y}{dx^2}$ for $x = \sin^3t$⁠, $y = \cos^3t$⁠; or $\frac{dy}{dx}$ for $y = x^{\log x} + (\log x)^x$
  Parametric second derivative; log differentiation3 marks

CBSE Sample Paper 2025-26

• Sample 2025-26 Q7$k$ for continuity of $kx + 1$ and $\cos x$ at $x = \pi$
  Continuity of a piecewise function1 mark
• Sample 2025-26 Q8$f'(1)$ for $f(x) = x\tan^{-1}x$
  Derivatives: product rule with tan⁻¹x1 mark
• Sample 2025-26 Q22If $y = \log\tan\left(\frac{\pi}{4} + \frac{x}{2}\right)$⁠, prove $\frac{dy}{dx} - \sec x = 0$
  Derivative of a composite function (chain rule)2 marks
• Sample 2025-26 Q24$f(x + y) = f(x)f(y)$⁠, $f(5) = 2$⁠, $f'(0) = 3$⁠: find $f'(5)$
  Derivative by definition: f(x + y) = f(x)f(y)2 marks
• Sample 2025-26 Q26$x^y = e^{x - y}$⁠: prove $\frac{dy}{dx} = \frac{\log x}{(\log(xe))^2}$⁠, or $\frac{d^2y}{dx^2}$ for the cycloid
  Logarithmic and parametric differentiation3 marks

CBSE Sample Paper 2024-25

• Sample 2024-25 Q17The greatest integer function $f(x) = [x]$ at $x = 2.5$
  Continuity and differentiability of [x]1 mark
• Sample 2024-25 Q19Where $|x| + |x - 1|$ is not differentiable
  Differentiability of |x| + |x − 1|1 mark
• Sample 2024-25 Q23Derivative of $\tan^{-1}x$ with respect to $\log x$⁠, or of $(\cos x)^x$
  Derivative w.r.t. a function; logarithmic form2 marks
• Sample 2024-25 Q34$f''(x)$ for $f(x) = |x|^3$⁠, or $\frac{[1 + (y')^2]^{3/2}}{y''}$ on a circle is constant
  Second derivative of |x|³; parametric form5 marks

CBSE Sample Paper 2023-24

• Sample 2023-24 Q4$k$ for which $\frac{kx}{|x|}$ ($x \lt 0$⁠), $3$ ($x \ge 0$⁠) is continuous at 0
  Continuity of a piecewise function at a point1 mark
• Sample 2023-24 Q17Where $f(x) = x + |x|$ is differentiable
  Differentiability of x + |x|1 mark
• Sample 2023-24 Q31If $(a + bx)e^{\frac{y}{x}} = x$⁠, prove $x\frac{d^2y}{dx^2} = \left(\frac{a}{a + bx}\right)^2$
  Second-order derivative: prove a relation3 marks

CBSE Sample Paper 2022-23

• Sample 2022-23 Q4$k$ so that $\frac{1 - \cos 4x}{8x^2}$ is continuous at $0$
  Continuity at a point: finding k1 mark
• Sample 2022-23 Q16$(1 - x^2)y_2$ when $y = \sin^{-1}x$
  Second order derivative of sin⁻¹x1 mark
• Sample 2022-23 Q24Prove $\frac{dy}{dx} = -\sqrt{\frac{1 - y^2}{1 - x^2}}$ if $y\sqrt{1 - x^2} + x\sqrt{1 - y^2} = 1$
  Derivative using an inverse trig substitution2 marks

CBSE Sample Paper 2020-21

• Sample 2020-21 Q21Values of $k$ that make $\frac{1 - \cos kx}{x\sin x}$ continuous at $x = 0$
  Continuity at a point: finding k2 marks
• Sample 2020-21 Q30$\frac{dy}{dx}$ for $y = e^{x\sin^2 x} + (\sin x)^x$
  Logarithmic differentiation of a sum3 marks
• Sample 2020-21 Q31$[x]$ is not differentiable at $x = 1$⁠, or $\frac{d^2y}{dx^2}$ for $x = a\sec\theta$⁠, $y = b\tan\theta$
  Differentiability; parametric second derivative3 marks

Questions that also use Continuity and Differentiability

Their main topic is in another chapter.

• Board 2024 Q36Case study: a speed camera and the rate of change of the angle of elevation
  Rates of change: angle of elevation4 marks
• Sample 2026-27 Q10Where the derivative of $y = x^4$ exceeds $y$
  Comparing a derivative with the function1 mark
• Sample 2023-24 Q22Interval where $f(x) = xe^x$ is increasing
  Increasing functions: sign of the derivative2 marks
• Sample 2022-23 Q36Case study: body temperature during an intestinal illness
  Increasing/decreasing; local and absolute extrema4 marks