Continuity and Differentiability — Class 12 MCQs with Answers

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

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   f is continuous at x=a if:

   A.lim x→a f(x) = f(a)✓ Correct

   B.f(a) = 0

   C.f differentiable

   D.f bounded

   Answer: A. 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. Q2Easy

   d/dx (cos x) =

   A.sin x

   B.−sin x✓ Correct

   C.cos x

   D.−cos x

   Answer: B. −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. Q3Easy

   Derivative of f(g(x)):

   A.g'(f(x))

   B.f(g(x))

   C.f'(g(x)) · g'(x)✓ Correct

   D.f'(x) · g'(x)

   Answer: C. f'(g(x)) · g'(x)

   Explanation: d/dx[f(g(x))] = f'(g(x))·g'(x): differentiate outer f at the inner value g(x), then multiply by g'(x) — swapping the order to g'(f(x)) is the usual mix-up.

4. Q4Easy

   Implicit differentiation is used when:

   A.Functions are simple

   B.y is explicit

   C.Both

   D.y is defined implicitly in terms of x✓ Correct

   Answer: D. y is defined implicitly in terms of x

   Explanation: Implicit differentiation is needed whenever y is tangled up with x in an equation (like x²+y²=1) rather than isolated as y=f(x) explicitly.

5. Q5Easy

   Logarithmic differentiation is useful for:

   A.Simple polynomials

   B.Products, quotients, and powers of functions✓ Correct

   C.Constants

   D.Trigonometric

   Answer: B. Products, quotients, and powers of functions

   Explanation: Logarithmic differentiation turns products, quotients, and powers of functions into sums via ln — essential for forms like x^x where the plain power rule fails.

6. Q6Easy

   Second derivative of f w.r.t. x denoted by:

   A.f''(x) or d²f/dx²✓ Correct

   B.f'(x)

   C.f(x)²

   D.∫ f dx

   Answer: A. f''(x) or d²f/dx²

   Explanation: The second derivative — rate of change of the rate of change — is written f''(x) or d²f/dx², not f(x)² (squaring the function) or ∫f dx (integration).

7. Q7Medium

   Where is f(x) = 1/x discontinuous?

   A.x = 1

   B.x = 0✓ Correct

   C.x = −1

   D.Continuous everywhere

   Answer: B. x = 0

   Explanation: f(x)=1/x is discontinuous at x=0 because division by zero is undefined there — it's continuous at every other real number, including x=1 and x=−1.

8. Q8Medium

   d/dx [f(g(x))] =

   A.f(g(x))

   B.f'(x) · g'(x)

   C.f'(g(x)) · g'(x)✓ Correct

   D.g(f(x))

   Answer: C. f'(g(x)) · g'(x)

   Explanation: d/dx[f(g(x))] = f'(g(x))·g'(x) — the chain rule: differentiate the outer function at g(x), then multiply by the inner function's derivative g'(x).

9. Q9Medium

   d(x³ + 2x² − x + 5)/dx =

   A.3x² + 4x

   B.x² + x − 1

   C.3x² + 4x − 1✓ Correct

   D.x³ + x + 5

   Answer: C. 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.

10. Q10Medium

    For x² + y² = 25, dy/dx =

    A.−y/x

    B.x/y

    C.y/x

    D.−x/y✓ Correct

    Answer: D. −x/y

    Explanation: Differentiate x²+y²=25 implicitly: 2x+2y·y'=0, so y'=−x/y — losing the chain-rule factor y' on the y² term produces the wrong sign or ratio.

11. Q11Medium

    d(x^x)/dx =

    A.x^x

    B.x · x^(x−1)

    C.x^x (ln x + 1)✓ Correct

    D.x^x · ln x

    Answer: C. x^x (ln x + 1)

    Explanation: For y=x^x, take ln y=x ln x, then (1/y)y'=ln x+1, so y'=x^x(ln x+1) — treating x^x like a plain power xⁿ (giving x·xˣ⁻¹) ignores the exponent is also x.

12. Q12Hard

    Mean Value Theorem: there exists c in (a,b) such that:

    A.f'(c) = (f(b) − f(a))/(b − a)✓ Correct

    B.f(c) = 0

    C.f(c) = average

    D.f''(c) = 0

    Answer: A. f'(c) = (f(b) − f(a))/(b − a)

    Explanation: Lagrange's MVT: at some point strictly inside the interval, the derivative equals the average rate of change of f between the endpoints — the function vanishing there, or its second derivative vanishing there, are unrelated conditions MVT doesn't require.

13. Q13Hard

    If x² + y² = 1, dy/dx =

    A.x/y

    B.−x/y✓ Correct

    C.−y/x

    D.0

    Answer: B. −x/y

    Explanation: Differentiate x²+y²=1 implicitly: 2x+2y·(dy/dx)=0, so dy/dx=−x/y. Forgetting the chain-rule factor dy/dx on the y² term gives the wrong sign-free x/y.

14. Q14Hard

    d(sin⁻¹ x)/dx =

    A.cos⁻¹ x

    B.−1/√(1 − x²)

    C.1/(1 + x²)

    D.1/√(1 − x²)✓ Correct

    Answer: D. 1/√(1 − x²)

    Explanation: d(sin⁻¹x)/dx=1/√(1−x²), derived from x=sin y via implicit differentiation — confusing this with tan⁻¹x's derivative 1/(1+x²) is the common mix-up.

Continuity and Differentiability — FAQs

More Class 12 Mathematics chapters

• 01Relations and Functions
• 02Inverse Trigonometric Functions
• 03Matrices
• 04Determinants
• 06Application of Derivatives
• 07Integrals

Class 12 — other subjects

• Class 12 Biology chapters
• Class 12 Chemistry chapters
• Class 12 Computer Science chapters
• Class 12 English chapters
• Class 12 Physics chapters