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GATE calculus: 5 practice questions with solutions

By the PrepMatic team · Updated 9 Oct 2026 · 4 min read

Calculus is in the maths syllabus of every engineering paper that has an engineering-mathematics block, and in DA it appears as a "Calculus and Optimization" section. The questions test whether you can use a handful of tools quickly: series expansions for limits, derivative tests for extrema, the mean value theorem and standard integrals. The five original questions below cover each of those.

What GATE tests in calculus

The 2027 syllabi differ by branch, so match this list against your own paper on the official syllabus page:

• CS: limits, continuity and differentiability, maxima and minima, the mean value theorem, integration.
• DA: single-variable calculus and optimisation.
• EC: the above plus multiple integrals, line and surface integrals.
• EE: calculus including Fourier series and vector calculus.
• ME, CE and RA: calculus alongside differential equations, linear algebra and numerical methods.

In practice, expect limits (often via Taylor series or L'Hôpital's rule), maxima and minima on an interval, definite integrals, and partial derivatives for functions of two variables. The paper also has MSQ and NAT questions; all five below are MCQ.

Practice questions

Question 1

The value of lim (x → 0) of (sin x − x cos x) / x³ is:

• A. 1/6
• B. 1/2
• C. 2/3
• D. 1/3

Question 2

Let f(x) = x³ − 6x² + 9x + 2 on the closed interval [0, 4]. The difference between the maximum and the minimum value of f on this interval is:

• A. 2
• B. 4
• C. 6
• D. 8

Question 3

For f(x) = x³ on [0, 2], the value of c in (0, 2) that satisfies the mean value theorem is:

• A. 1
• B. 4/3
• C. 2/√3
• D. √2

Question 4

The value of the integral of sin⁵x dx from 0 to π/2 is:

• A. 1/3
• B. 2/5
• C. 1/2
• D. 8/15

Question 5

The function f(x, y) = x³ + y³ − 3xy has:

• A. a local minimum at (1, 1) and a saddle point at (0, 0)
• B. a local maximum at (0, 0) and a saddle point at (1, 1)
• C. local minima at both (0, 0) and (1, 1)
• D. a local maximum at (1, 1) and a local minimum at (0, 0)

Answers and explanations

1. D. Expand: sin x = x − x³/6 + …, and x cos x = x − x³/2 + …. The difference is x³(1/2 − 1/6) = x³/3, so the limit is 1/3. L'Hôpital's rule also works but needs three differentiations.
2. B. f′(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3), so the critical points are x = 1 and x = 3. Evaluate at the critical points and the endpoints: f(0) = 2, f(1) = 6, f(3) = 2, f(4) = 6. The maximum is 6 and the minimum is 2, so the difference is 4. On a closed interval, never skip the endpoints.
3. C. The mean slope is (f(2) − f(0)) / (2 − 0) = 8/2 = 4. Set f′(c) = 3c² = 4, so c² = 4/3 and c = 2/√3, which is about 1.155 and lies inside (0, 2). The negative root is outside the interval.
4. D. Use Wallis's formula for odd powers: the integral of sinⁿx from 0 to π/2 is (n − 1)(n − 3)…2 / [n(n − 2)…3] when n is odd. For n = 5 that is (4 × 2) / (5 × 3) = 8/15. For even n the result carries a factor of π/2.
5. A. Setting f_x = 3x² − 3y = 0 and f_y = 3y² − 3x = 0 gives y = x² and x = y², so x⁴ = x and x = 0 or 1. The critical points are (0, 0) and (1, 1). Here r = f_xx = 6x, t = f_yy = 6y and s = f_xy = −3. At (0, 0), rt − s² = −9 < 0, a saddle. At (1, 1), rt − s² = 36 − 9 = 27 > 0 and r = 6 > 0, so it is a local minimum.

How to prepare calculus

• Know the Taylor series of eˣ, sin x, cos x, ln(1 + x) and (1 + x)ⁿ up to the x³ term. Most limits at GATE level go faster by substitution than by repeated L'Hôpital.
• For extrema, always build a table: critical points plus endpoints, with the value at each. Question 2 is a typical trap, because the largest value occurs at an endpoint as well as at a critical point.
• Memorise the standard integral results: Wallis's formula, the integral of 1/(1 + x²) and the symmetric-limit rules (odd functions give zero, even functions double the half-range value).
• Practise the two-variable second-derivative test until it takes a minute: find the critical points, compute r, s, t, then test rt − s².
• Know Rolle's theorem, the mean value theorem and the intermediate value theorem well enough to say what must be true of a function. For EC and EE, add multiple, line and surface integrals and Fourier series.

See the GATE engineering mathematics guide for which calculus topics your branch lists and how the 13-mark maths block works. Then move on to linear algebra and differential equations.

Take the full topic test on PrepMatic

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