Question 1 |
If v_1, v_2,..., v_6 are six vectors in \mathbb{R}^4, which one of the following statements is False?
It is not necessary that these vectors span \mathbb{R}^4. | |
These vectors are not linearly independent. | |
Any four of these vectors form a basis for \mathbb{R}^4. | |
If {v_1, v_3,v_5, v_6} spans \mathbb{R}^4, then it forms a basis for \mathbb{R}^4. |
Question 1 Explanation:
v_1, v_2,..., v_6 are six vectors in \mathbb{R}^4.
For a 4-dimensional vector space,
(i) any four linearly independent vectors form a basis (or)
(ii) Any set of four vectors in \mathbb{R}^4 spans \mathbb{R}^4, then it forms a basis.
Therefore, clearly options (A), (B), (D) are true.
Option (C) is FALSE
For a 4-dimensional vector space,
(i) any four linearly independent vectors form a basis (or)
(ii) Any set of four vectors in \mathbb{R}^4 spans \mathbb{R}^4,
Therefore, clearly options (A), (B), (D) are true.
Option (C) is FALSE
Question 2 |
For a vector field \vec{A}, which one of the following is False?
\vec{A} is solenoidal if \bigtriangledown \cdot \vec{A}=0 | |
\bigtriangledown \times \vec{A} is another vector field. | |
\vec{A} is irrotational if \bigtriangledown ^2 \vec{A}=0. | |
\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A} |
Question 2 Explanation:
Divergence and curl operator is performed on a vector field \vec{A}
Curl operation provides a vector orthogonal to the given vector field \vec{A}
\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}
If a vector field is irrortational then \bigtriangledown \times \vec{A}=0
If a vector field is solenoidal then \bigtriangledown \cdot \vec{A}=0
If a field is scalar A, then \bigtriangledown ^2 \vec{A}=0, is a laplacian equation.
Hence option (C) is incorrect
Curl operation provides a vector orthogonal to the given vector field \vec{A}
\bigtriangledown \times(\bigtriangledown \times \vec{A})=\bigtriangledown (\bigtriangledown \cdot \vec{A})-\bigtriangledown ^2 \vec{A}
If a vector field is irrortational then \bigtriangledown \times \vec{A}=0
If a vector field is solenoidal then \bigtriangledown \cdot \vec{A}=0
If a field is scalar A, then \bigtriangledown ^2 \vec{A}=0, is a laplacian equation.
Hence option (C) is incorrect
Question 3 |
The partial derivative of the function
f(x,y,z)=e^{1-x \cos y}+xze^{-1/(1+y^2)}
with respect to x at the point (1,0,e) is
f(x,y,z)=e^{1-x \cos y}+xze^{-1/(1+y^2)}
with respect to x at the point (1,0,e) is
-1 | |
0 | |
1 | |
\frac{1}{e} |
Question 3 Explanation:
\begin{aligned} \text{Given, } f(x,y,z)&=e^{1-x\cos y}+xze^{-1/(1+y^{2})} \\ \frac{\partial f}{\partial x}&=e^{1-x\cos y}(0-\cos y)+ze^{-1/1+y^{2}} \\ \left ( \frac{\partial f }{\partial x} \right )_{(1,0,e)}&=e^{0}(0-1)+e\cdot e^{-1/(1+0)} \\ &=-1+1=0
\end{aligned}
Question 4 |
The general solution of \frac{d^2y}{dx^2}-6\frac{dy}{dx}+9y=0 is
y=C_1e^{3x}+C_2e^{-3x} | |
y=(C_1+C_2x)e^{-3x} | |
y=(C_1+C_2x)e^{3x} | |
y=C_1e^{3x} |
Question 4 Explanation:
Taking \frac{\mathrm{d} }{\mathrm{d} x}=D
Given, D^{2}-6D+9=0
(D-3)^2=0
D=3,3
So, Solution of the given Differential equation
y=(C_{1}+C_{2}x)e^{3x}
Given, D^{2}-6D+9=0
(D-3)^2=0
D=3,3
So, Solution of the given Differential equation
y=(C_{1}+C_{2}x)e^{3x}
Question 5 |
The output y[n] of a discrete-time system for an input x[n] is
y[n]=\begin{matrix} max\\ -\infty \leq k\leq n \end{matrix}\; |x[k]|.
The unit impulse response of the system is
y[n]=\begin{matrix} max\\ -\infty \leq k\leq n \end{matrix}\; |x[k]|.
The unit impulse response of the system is
0 for all n | |
1 for all n | |
unit step signal u[n]. | |
unit impulse signal \delta[n]. |
There are 5 questions to complete.