Continuity and Differentiability — Class 12 MCQs with Answers

Continuity and Differentiability — important questions & MCQs with answers (Class 12 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   f is continuous at x=a if:

   Alim x→a f(x) = f(a)Bf(a) = 0Cf differentiableDf bounded
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   Answer: lim x→a f(x) = f(a)

   Explanation: Continuity at x=a needs limₓ→ₐ f(x) to exist AND equal the value f actually takes at that point — that value being 0, or f merely being bounded, doesn't guarantee the limit matches it.

2. Q2 of 6Easy

   d/dx (cos x) =

   Asin xB−sin xCcos xD−cos x
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   Answer: −sin x

   Explanation: d/dx(cos x) = −sin x — the minus sign is easy to drop; pair sin→cos and cos→−sin together, since confusing the two is the classic slip.

3. Q3 of 6Medium

   d(x³ + 2x² − x + 5)/dx =

   A3x² + 4xBx² + x − 1C3x² + 4x − 1Dx³ + x + 5
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   Answer: 3x² + 4x − 1

   Explanation: Differentiate term by term: d(x³)=3x², d(2x²)=4x, d(−x)=−1, d(5)=0, giving 3x²+4x−1 — dropping the −1 from −x or missing the constant's zero is the usual slip.

4. Q4 of 6Medium

   Assertion (A): If y = log(sin x), 0 < x < π, then dy/dx = 1/sin x. Reason (R): For x > 0, d/dx (log x) = 1/x.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: A is false but R is true

   Explanation: A is false: by the chain rule, dy/dx = (1/sin x)·cos x = cot x. R is true: it is the standard derivative of the natural logarithm.

5. Q5 of 6Hard

   The value of the derivative of x^x at x = 1 is:

   AeB2C0D1
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   Answer: 1

   Explanation: Logarithmic differentiation gives x^x (1 + log x), which at x = 1 equals 1.

6. Q6 of 6Hard

   Assertion (A): If y = sin⁻¹x, −1 < x < 1, then (1 − x²)·d²y/dx² − x·dy/dx = 0. Reason (R): d/dx (tan⁻¹x) = 1/(1 + x²) for every real x.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: A is true: dy/dx = 1/√(1 − x²), so √(1 − x²)·dy/dx = 1; differentiating again and multiplying by √(1 − x²) gives (1 − x²)y″ − xy′ = 0. R is true, but it is about tan⁻¹x and plays no part in A.

Continuity and Differentiability notes: key concepts (Class 12 Mathematics)

This chapter formalises continuity as limit equalling value and shows every differentiable function is continuous. It then extends differentiation: chain, product and quotient rules, derivatives of exponential, log and inverse-trig functions, implicit, logarithmic and parametric methods, second derivatives and the MVT.

Continuity at a point

f is continuous at a if the left limit, right limit and f(a) all agree: lim(x→a) f(x) = f(a). Polynomials are continuous everywhere.

Points of discontinuity

The greatest-integer function [x] is discontinuous at every integer, where its left and right limits differ. f(x) = 1/x is not defined at 0, so 0 is not in its domain: NCERT counts 1/x as continuous.

Differentiability

f is differentiable at a if the left-hand and right-hand derivatives are equal; |x| fails this at x = 0 (a corner).

Differentiable ⇒ continuous

Differentiability forces continuity, but the converse is false — a continuous function can still have a corner.

Chain rule

The derivative of a composite: d/dx f(g(x)) = f′(g(x))·g′(x); differentiate outer, keep inner, times derivative of inner.

Product and quotient rules

(uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′)/v²; used to differentiate products and ratios of functions.

Standard derivatives

d(xⁿ) = n x^(n−1), d(eˣ) = eˣ, d(ln x) = 1/x, d(sin x) = cos x, d(cos x) = −sin x, d(tan x) = sec²x.

Inverse-trig derivatives

d(sin⁻¹x)/dx = 1/√(1−x²), d(cos⁻¹x)/dx = −1/√(1−x²), d(tan⁻¹x)/dx = 1/(1+x²).

Implicit differentiation

When y is tangled with x (e.g. x² + y² = 25), differentiate both sides w.r.t. x treating y as y(x), then solve for dy/dx.

Logarithmic differentiation

Take ln of both sides when the variable is in the base AND exponent, such as xˣ; then d(xˣ)/dx = xˣ(1 + ln x).

Parametric differentiation

If x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt); divide the two derivatives, never cancel dt.

Second-order derivative

d²y/dx² is the derivative of dy/dx; e.g. for y = sin x, d²y/dx² + y = 0.

Rolle's & Mean Value Theorem

If f is continuous on [a, b], differentiable on (a, b) with f(a) = f(b), some c has f′(c) = 0; MVT drops that equality.