CSIR NET Maths previous year question papers

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   $3 \partial{}f/\partial{}x_{1}(1, 1) + 3 \partial{}f/\partial{}x_{2}(1, 1) = f(1, 1)$

2. B

   $\partial{}f/\partial{}x_{1}(1, -1) - \partial{}f/\partial{}x_{2}(1, -1) = 3f(1, -1)$

3. C

   $x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 6f(x_{1}, x_{2})$

4. D

   $x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 9f(x_{1}, x_{2})$

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $g:\Omega_\varepsilon\to(-\delta,\delta)$ such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all $(x,y)\in\Omega_\varepsilon$.

2. B

   There exist $\varepsilon>0, \delta>0$ and a continuously differentiable function $h:\Omega_\varepsilon\to(-\delta,\delta)$ such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all $(x,z)\in\Omega_\varepsilon$.

3. C

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $k_{1}(0)=k_{2}(0)=0$ and $F(x,k_{1}(x),k_{2}(x))=-1$ for all $x\in(-\varepsilon,\varepsilon)$.

4. D

   There exist $\varepsilon>0, \delta>0$ and continuously differentiable functions $j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta)$ such that $j_{1}(0)=j_{2}(0)=0$ and $F(j_{1}(z),j_{2}(z),z)=-1$ for all $z\in(-\varepsilon,\varepsilon)$.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   lim(x,y)→(0,0) f(x,y) does not exist.

2. B

   The partial derivatives of f at (0,0) do not exist.

3. C

   lim(x→0) f(x,0) = lim(y→0) f(0,y).

4. D

   f is differentiable at (0,0).

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is differentiable at 0 when $\alpha > 1$.

2. B

   f is differentiable at 0 when $\alpha = 1/2$.

3. C

   f is continuous at 0 when $\alpha = 0$.

4. D

   f is continuous at 0 when $\alpha > 0$.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   The function h_(a,b) is differentiable on $\mathbb{R}$ for each $(a, b) \in \mathbb{R}^{2}$.

2. B

   There exists $(a, b) \in \mathbb{R}^{2}$ such that h_(a,b) is not differentiable at t = 0.

3. C

   The function f is differentiable at the point (0, 0).

4. D

   If the function f is differentiable at (0, 0) then g is identically zero.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is continuous at (0, 0).

2. B

   The partial derivatives $\partial{}f/\partial{}x$ and $\partial{}f/\partial{}y$ exist at (0, 0).

3. C

   f is differentiable at (0, 0).

4. D

   f is not differentiable at (0, 0).

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is differentiable on $\mathbb{R}^{2}$ \ {(0, 0)}.

2. B

   All the directional derivatives of f exist at (0, 0).

3. C

   f is differentiable on $\mathbb{R}^{2}$.

4. D

   f is not continuous at (0, 0).

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   $\partial{}f/\partial{}x(0, 0)$ exists

2. B

   $\partial{}f/\partial{}y(0, 0)$ exists

3. C

   f is not continuous at (0, 0)

4. D

   f is not differentiable at (0, 0)

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   f is injective in a neighbourhood of (0, 0)

2. B

   There exists an open neighbourhood U of (0, 0) in $\mathbb{R}^{2}$ such that $f_{3}$ is a function of $f_{1}$ and $f_{2}$

3. C

   f maps an open neighbourhood of (0, 0) in $\mathbb{R}^{2}$ onto an open subset of $\mathbb{R}^{3}$

4. D

   (0, 0) is an isolated point of f⁻$^{1}(${f(0,0)})

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   There is no continuous real-valued function g defined on any interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

2. B

   There is exactly one continuous real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

3. C

   There is exactly one differentiable real-valued function g defined on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

4. D

   There are two distinct differentiable real-valued functions g on an interval of $\mathbb{R}$ containing 0 such that f(x, g(x)) = 0.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   F is continuous.

2. B

   F is uniformly continuous.

3. C

   F is differentiable.

4. D

   F has partial derivatives of all orders.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   F is injective.

2. B

   f is increasing.

3. C

   For every $(x', y') \in (0,1)^{2}$, there exists a unique $(x, y) \in (0,1)^{2}$ such that F(x, y) = (x′, y′).

4. D

   The total derivative DF(x, y) is invertible for all $(x, y) \in (0,1)^{2}$.

Multi-select · full marks only for the exact set · Functions of Several Variables

1. A

   Let $A_{1} =$ {$x^{2} + y^{2} < 1$}. For each $\gamma \in p(A_{1})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{1})$.

2. B

   Let $A_{2} =$ {$x^{2} + y^{2} \le 1$}. For each $\gamma \in p(A_{2})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{2})$.

3. C

   Let $A_{3} =$ {xy = 0}. For each $\gamma \in p(A_{3})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{3})$.

4. D

   Let $A_{4} =$ {xy = 1}. For each $\gamma \in p(A_{4})$ there exists $\varepsilon > 0$ with $(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{4})$.