CSIR NET Maths previous year question papers

Single choice · Continuity and Differentiation

1. A

   f is unbounded.

2. B

   f is increasing.

3. C

   $lim_{x\to\infty}$ f(x) = 2.

4. D

   f is decreasing.

Single choice · Continuity and Differentiation

1. A

   f∘g is uniformly continuous, but g∘f is not uniformly continuous.

2. B

   g∘f is uniformly continuous, but f∘g is not uniformly continuous.

3. C

   Neither f∘g nor g∘f is uniformly continuous.

4. D

   Both f∘g and g∘f are uniformly continuous.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f′(x) ≤ 0 for all x > 0.

2. B

   $lim_{x\to\infty}$ f′(x) = 0.

3. C

   $lim_{x\to\infty}$ x f′(x) need not exist.

4. D

   $lim_{x\to\infty}$ x f′(x) = 0.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   F is non-decreasing on $\mathbb{R}$

2. B

   $\lim(x\to\infty) F(x) = 1$

3. C

   F is left-continuous on $\mathbb{R}$

4. D

   F is right-continuous on $\mathbb{R}$

Single choice · Continuity and Differentiation

1. A

   For every bounded subset $A \subseteq \mathbb{R}, f$⁻$^{1}(A)$ is a bounded subset of $\mathbb{R}$.

2. B

   For every Cauchy sequence $(x_{n})_{n}\ge_{1}$ in $\mathbb{R}, (f(x_{n}))_{n}\ge_{1}$ is a Cauchy sequence in $\mathbb{R}$.

3. C

   There exists $x \in \mathbb{R}$ such that f(x) = x.

4. D

   There exists $x \in \mathbb{R}$ such that f(x) = 0.

Single choice · Continuity and Differentiation

1. A

   f is monotone.

2. B

   f is continuous but not strictly monotone.

3. C

   f is not continuous.

4. D

   f is continuous but not uniformly continuous.

Single choice · Continuity and Differentiation

1. A

   0

2. B

   1

3. C

   Countably infinite

4. D

   Uncountable

Single choice · Continuity and Differentiation

1. A

   If f is continuous on the set S, then f is continuous on the set $\mathbb{R}$ \ S

2. B

   If g is continuous, then f is continuous on the set S

3. C

   If g is identically 0 and f is continuous on the set $\mathbb{R}$ \ S, then f is identically 0

4. D

   If g is identically 0 and f is continuous on the set S, then f is identically 0

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   S is empty.

2. B

   S is non-empty.

3. C

   f must be uniformly continuous.

4. D

   f need not be uniformly continuous.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   lim(x→0) f(x) exists.

2. B

   f is continuous at 0.

3. C

   f is differentiable at 0.

4. D

   lim(x→0) f′(x) does not exist.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   F is bounded.

2. B

   F is continuous on $\mathbb{R}$.

3. C

   F is uniformly continuous on $\mathbb{R}$.

4. D

   F is not uniformly continuous on $\mathbb{R}$.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is strictly increasing

2. B

   f is strictly decreasing

3. C

   f is either strictly increasing or strictly decreasing

4. D

   f is onto

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f need not be surjective

2. B

   f must be surjective if K = [0, 1]

3. C

   f is injective and f⁻$^{1} : f(K) \to K$ is continuous

4. D

   f is injective, but f⁻$^{1} : f(K) \to K$ need not be continuous

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is continuous on $\mathbb{R}$

2. B

   f is differentiable on $\mathbb{R}$

3. C

   f is differentiable only at 0

4. D

   f is not differentiable at 0

Single choice · Continuity and Differentiation

1. A

   f(x) = sin(1/x)

2. B

   f(x) = $e^{-1/x^{2}}$

3. C

   f(x) = eˣ cos(1/x)

4. D

   $f(x) = \cos x \cdot \cos(\pi/x)$

Single choice · Continuity and Differentiation

1. A

   F(x) has exactly one real root.

2. B

   F(x) has exactly four real roots.

3. C

   F(x) has at most two real roots.

4. D

   F(x) has at most one real root.

Single choice · Continuity and Differentiation

1. A

   f never vanishes in [0, 1].

2. B

   f has at most finitely many zeros in [0, 1].

3. C

   f has infinitely many zeros in [0, 1].

4. D

   f(1/2) = 0.

Single choice · Continuity and Differentiation

1. A

   S = ∅

2. B

   S ≠ ∅ and S is a proper subset of $(1, \infty)$

3. C

   $(2, \infty)$ is a proper subset of S

4. D

   S ∩ (0, 1) ≠ ∅

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is necessarily one-one.

2. B

   f need not be one-one.

3. C

   f is necessarily onto.

4. D

   f need not be onto.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f is continuous on $[0, \infty)$.

2. B

   f is uniformly continuous on $[0, \infty)$.

3. C

   g is continuous on $[0, \infty)$.

4. D

   g is uniformly continuous on $[0, \infty)$.

Multi-select · full marks only for the exact set · Continuity and Differentiation

1. A

   f(x) = [x] sin(1/x) for x ≠ 0, f(0) = 0 (where [x] is the integer part) has a discontinuity at 0 which is removable.

2. B

   $f : [0,\infty) \to \mathbb{R}, f(x) = \sin(\log x)$ for x ≠ 0, f(0) = 0, has a discontinuity at 0 which is NOT removable.

3. C

   f(x) = $e^{1/x}$ for x < 0, f(x) = $e^{1/(x+1)}$ for x ≥ 0, has a jump discontinuity at 0.

4. D

   If $f, g : [0,1] \to \mathbb{R}$ are of bounded variation, then fg has at most countably many discontinuities.