Real Analysis

JKPSC Assistant Professor (Mathematics) · Mathematics · 69 practice questions with answers and explanations: 40 Easy, 29 Medium, 0 Hard.

Sample questions

Sample Easy question 1

If f(x) = x³ −5x² −4x +1, find f′(0).

  1. 4
  2. −1
  3. −8
  4. −4
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Answer: D. f′(x) is the derivative term by term; at x = 0 it equals −4.

Sample Easy question 2

Find the limit of (x² − 25)/(x − 5) as x tends to 5.

  1. 11
  2. 5
  3. 25
  4. 10
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Answer: D. Factor as (x − 5)(x + 5); the limit is 5 + 5 = 10.

Sample Easy question 3

If F(x) is the integral of t³ dt from 0 to x, then F′(2) equals:

  1. 2
  2. 8
  3. 12
  4. 4
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Answer: B. By the Fundamental Theorem of Calculus, F′(x) = x³, so F′(2) = 8.

Sample Medium question 4

Which of these functions is uniformly continuous on R?

  1. x²
  2. eˣ
  3. sin x
  4. 1/x
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Answer: C. sin x is Lipschitz with constant 1.

Sample Medium question 5

A bounded function on [a, b] with finitely many discontinuities is:

  1. Differentiable
  2. Riemann integrable
  3. Not integrable
  4. Continuous
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Answer: B. Such functions are Riemann integrable.

Sample Medium question 6

The Bolzano–Weierstrass theorem says that every bounded sequence in R has:

  1. A monotone limit equal to zero
  2. A convergent subsequence
  3. A limit
  4. An unbounded subsequence
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Answer: B. It underlies compactness arguments.

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