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GATE Calculus Questions with Answers & Explanations

20 Calculus questions from the Engineering Mathematics section of GATE, each with the correct answer and a worked explanation. Try each question first, then tap Show answer to check. When you're ready, take the timed chapter-wise test to see your all-India rank.

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Question 1

If f(x) = 1x³ − 1x² + 8, what is f′(4)?

  1. A48
  2. B56
  3. C43
  4. D40
Show answer
Answer: (D) 40 f′(x) = 3x² + -2x, so f′(4) = 40.
Question 2

What is the value of lim(x→9) (x² − 81)/(x − 9)?

  1. A9
  2. B18
  3. C19
  4. D0
Show answer
Answer: (B) 18 (x² − 9²)/(x − 9) = x + 9 → 18.
Question 3

What is the value of lim(x→8) (x² − 64)/(x − 8)?

  1. A16
  2. B0
  3. C64
  4. D17
Show answer
Answer: (A) 16 (x² − 8²)/(x − 8) = x + 8 → 16.
Question 4

What is the value of ∫₀^2 (3x² + 4x) dx?

  1. A16
  2. B19
  3. C15
  4. D17
Show answer
Answer: (A) 16 ∫ = [3x³/3 + 4x²/2] from 0 to 2 = 16.
Question 5

If f(x) = 5x³ + 4x², what is f′(2)?

  1. A91
  2. B76
  3. C60
  4. D56
Show answer
Answer: (B) 76 f′(x) = 15x² + 8x, so f′(2) = 76.
Question 6

What is the value of lim(x→4) (x² − 16)/(x − 4)?

  1. A4
  2. B8
  3. C16
  4. D9
Show answer
Answer: (B) 8 (x² − 4²)/(x − 4) = x + 4 → 8.
Question 7

What is the value of ∫₀^1 (3x² + 2x) dx?

  1. A6
  2. B4
  3. C2
  4. D3
Show answer
Answer: (C) 2 ∫ = [3x³/3 + 2x²/2] from 0 to 1 = 2.
Question 8

If f(x) = 5x³ + 1x² − 1, what is f′(3)?

  1. A140
  2. B156
  3. C141
  4. D135
Show answer
Answer: (C) 141 f′(x) = 15x² + 2x, so f′(3) = 141.
Question 9

What is the value of lim(x→8) (x² − 64)/(x − 8)?

  1. A17
  2. B64
  3. C16
  4. D0
Show answer
Answer: (C) 16 (x² − 8²)/(x − 8) = x + 8 → 16.
Question 10

What is the value of ∫₀^3 (3x² + 2x) dx?

  1. A36
  2. B33
  3. C38
  4. D32
Show answer
Answer: (A) 36 ∫ = [3x³/3 + 2x²/2] from 0 to 3 = 36.

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Question 11

If f(x) = 5x³ + 3x² − 1, what is f′(4)?

  1. A264
  2. B240
  3. C367
  4. D263
Show answer
Answer: (A) 264 f′(x) = 15x² + 6x, so f′(4) = 264.
Question 12

What is the value of lim(x→3) (x² − 9)/(x − 3)?

  1. A7
  2. B0
  3. C6
  4. D9
Show answer
Answer: (C) 6 (x² − 3²)/(x − 3) = x + 3 → 6.
Question 13

What is the value of ∫₀^1 (6x² + 2x) dx?

  1. A1
  2. B4
  3. C7
  4. D3
Show answer
Answer: (D) 3 ∫ = [6x³/3 + 2x²/2] from 0 to 1 = 3.
Question 14

If f(x) = 1x³ + 4x² − 7, what is f′(2)?

  1. A31
  2. B21
  3. C28
  4. D17
Show answer
Answer: (C) 28 f′(x) = 3x² + 8x, so f′(2) = 28.
Question 15

What is the value of lim(x→6) (x² − 36)/(x − 6)?

  1. A13
  2. B6
  3. C12
  4. D0
Show answer
Answer: (C) 12 (x² − 6²)/(x − 6) = x + 6 → 12.
Question 16

What is the value of ∫₀^2 (3x² + 4x) dx?

  1. A12
  2. B14
  3. C16
  4. D17
Show answer
Answer: (C) 16 ∫ = [3x³/3 + 4x²/2] from 0 to 2 = 16.
Question 17

If f(x) = 2x³ − 4x² − 9, what is f′(3)?

  1. A30
  2. B9
  3. C21
  4. D54
Show answer
Answer: (A) 30 f′(x) = 6x² + -8x, so f′(3) = 30.
Question 18

What is the value of lim(x→2) (x² − 4)/(x − 2)?

  1. A0
  2. B2
  3. C5
  4. D4
Show answer
Answer: (D) 4 (x² − 2²)/(x − 2) = x + 2 → 4.
Question 19

What is the value of ∫₀^2 (6x² + 2x) dx?

  1. A19
  2. B18
  3. C20
  4. D23
Show answer
Answer: (C) 20 ∫ = [6x³/3 + 2x²/2] from 0 to 2 = 20.
Question 20

If f(x) = 1x³ + 6x² + 2, what is f′(1)?

  1. A18
  2. B15
  3. C17
  4. D3
Show answer
Answer: (B) 15 f′(x) = 3x² + 12x, so f′(1) = 15.

These are RankerClub practice questions written in the GATE pattern. They are not official SSC/IBPS/RRB papers.

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