CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Integration

1. A

   2/e

2. B

   4/e

3. C

   $log_{e}2 - 1$

4. D

   $2 log_{e}2 - 1$

Multi-select · full marks only for the exact set · Integration

1. A

   The composition f ∘ g is Riemann integrable.

2. B

   If g(x) ≠ 0 for each x ∈ [a, b], then f/g is Riemann integrable.

3. C

   The positive square root $\sqrt{f^{2} + g^{2}}$ is Riemann integrable.

4. D

   The composition f ∘ g is Riemann integrable, if both f and g are continuous.

Multi-select · full marks only for the exact set · Integration

1. A

   If f∈F is a continuous function and f(n)=0 for all positive integers n, then $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$.

2. B

   If f∈F is uniformly continuous and $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$, then $\lim(x\to\infty)f(x)=0$.

3. C

   There exists a function f∈F such that $\lim(x\to\infty)\int_{0}$ˣf(t)dt exists in $\mathbb{R}$ but $\lim(x\to\infty)f(x)\ne0$.

4. D

   If f∈F is Lebesgue measurable and $\lim(x\to\infty)\int_{0}$ˣf(t)dt=0, then f=0 almost everywhere on $[0,\infty)$.

Multi-select · full marks only for the exact set · Integration

1. A

   f is a uniformly continuous function.

2. B

   f is a bounded function.

3. C

   There exists $x\in(0,\infty)$ such that f(x)=0.

4. D

   The derivative of f is continuous.

Multi-select · full marks only for the exact set · Integration

1. A

   $L_\alpha$⁺ and $L_\alpha$⁻ exist for every $\alpha\in(0,1)$.

2. B

   If f is surjective, then f is continuous.

3. C

   If f is Riemann integrable, then f is continuous.

4. D

   If the left and right hand limits exist at $\alpha,\beta\in(0,1), \alpha\ne\beta$, then $U_\alpha\cap{}U_\beta=\emptyset$.

Single choice · Integration

1. A

   A = 0

2. B

   A = 1

3. C

   A = sin(1)/2

4. D

   A = sin(1/4)

Multi-select · full marks only for the exact set · Integration

1. A

   F is Riemann integrable.

2. B

   If F(x) = 0 for all x ∈ [0, 1], then f(x) = 0 for all x ∈ [0, 1].

3. C

   F is uniformly continuous on [0, 1].

4. D

   F is differentiable on (0, 1).

Multi-select · full marks only for the exact set · Integration

1. A

   f(0) = 0

2. B

   f′(0) exists and f′(0) = 0.

3. C

   f″(0) exists and f″(0) = g(0).

4. D

   f″(0) exists but f″(0) ≠ g(0).

Multi-select · full marks only for the exact set · Integration

1. A

   f is Riemann integrable on [0, 1].

2. B

   The set of discontinuities of f cannot contain a non-empty open set.

3. C

   f is Lebesgue measurable function.

4. D

   f is Borel measurable function.

Multi-select · full marks only for the exact set · Integration

1. A

   The integral I is convergent

2. B

   The integral I is not convergent

3. C

   The integral $I_{a}$ converges for a = 1/2 but not for a = 0

4. D

   The integral $I_{a}$ converges for all a ≥ 0

Multi-select · full marks only for the exact set · Integration

1. A

   There exists $\lambda$ such that $I_\lambda$ converges, but $K_\lambda$ does not converge.

2. B

   There exists $\lambda$ such that $K_\lambda$ converges, but $I_\lambda$ does not converge.

3. C

   There exists $\lambda$ such that $I_\lambda, K_\lambda$ both converge.

4. D

   There exists $\lambda$ such that neither $I_\lambda$ nor $K_\lambda$ converges.

Multi-select · full marks only for the exact set · Integration

1. A

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0.

2. B

   $\int$₋$_{1}^{1} f(x)p(x)$ dx = 0 for all real polynomials p(x).

3. C

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 odd.

4. D

   $\int$₋$_{1}^{1} f(x)x^{n}$ dx = 0 for all n ≥ 0 even.