Class 12 Maths DPP — Ch 5 — Continuity and Differentiability | 25 July 2026

Assertion (A): If a function is continuous, it need not be differentiable.
Reason (R): The function $|x|$ is continuous at $x = 0$ but not differentiable there.

Choose the correct option:

A. Both A and R are true, and R is the correct explanation of A B. Both A and R are true, but R is NOT the correct explanation of A C. A is true, but R is false D. A is false, but R is true

📝 SolutionContinuity does not guarantee differentiability. $|x|$ is a classic example: continuous at 0 but with a sharp corner, so not differentiable there. R correctly explains A.

The derivative of $\tan x$ is:

A. $\sec^2 x$ B. $\csc^2 x$ C. $-\sec^2 x$ D. $\cos^2 x$

📝 Solution$\frac{d}{dx}(\tan x) = \sec^2 x$.

The product rule states $\frac{d}{dx}(uv) =$?

A. $u'v'$ B. $uv' + u'v$ C. $uv' - u'v$ D. $\frac{u'}{v'}$

📝 SolutionThe product rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx} = uv' + u'v$.

The second derivative of $x^3$ is:

A. $3x^2$ B. $6x$ C. $6$ D. $x^2$

📝 SolutionFirst derivative $= 3x^2$; differentiating again gives $6x$.

A function $f(x) = x^2$ is continuous:

A. only at $x = 0$ B. everywhere (all real x) C. nowhere D. only for positive x

📝 SolutionPolynomial functions like $x^2$ are continuous for all real numbers.