Question 1 |
The product of eigenvalues of the matrix P is
P=\begin{bmatrix} 2 & 0 & 1\\ 4 & -3 & 3\\ 0 & 2 & -1 \end{bmatrix}
P=\begin{bmatrix} 2 & 0 & 1\\ 4 & -3 & 3\\ 0 & 2 & -1 \end{bmatrix}
-6 | |
2 | |
6 | |
-2 |
Question 1 Explanation:
Product of Eigen value = determinant value
\begin{array}{l} =2(3-6)+1(8-0) \\ =2(-3)+8=-6+8=2 \end{array}
\begin{array}{l} =2(3-6)+1(8-0) \\ =2(-3)+8=-6+8=2 \end{array}
Question 2 |
The value of lim _{x \to 0}\: \frac{x^{3}-\sin(x)}{x} is
0 | |
3 | |
1 | |
-1 |
Question 2 Explanation:
\begin{aligned} \lim _{x \rightarrow 0} \frac{x^{3}-\sin x}{x} &=\left(\frac{0}{0} \mathrm{form}\right) \\ \lim _{x \rightarrow 0} \frac{3 x^{2}-\cos x}{1} &=0-\cos 0 \\ &=0-1=-1 \end{aligned}
Question 3 |
consider the following partial differential equation for u(x,y) with the constant c\gt 1:
\frac{\partial y}{\partial x}+c \: \frac{\partial u}{\partial x} =0
solution of this equations is
\frac{\partial y}{\partial x}+c \: \frac{\partial u}{\partial x} =0
solution of this equations is
u(x,y)=f(x+cy) | |
u(x,y)=f(x-cy) | |
u(x,y)=f(cx+y) | |
u(x,y)=f(cx-y) |
Question 3 Explanation:
\small \begin{aligned} u &=f(x-c y) \\ \frac{\partial u}{\partial x} &=f^{\prime}(x-c y)(1) \\ \frac{\partial u}{\partial y} &=f^{\prime}(x-c y)(-c) \\ &=-c \cdot f^{\prime}(x-c y)\\ &=-c \cdot \frac{\partial u}{\partial x} \\ \therefore \quad \frac{\partial u}{\partial y} & +c \frac{\partial u}{\partial x} =0 \end{aligned}
Question 4 |
The differential equation \frac{\mathrm{d^{2}}y}{\mathrm{d} x^{2}}+16y=0 for y(x) with the two boundary conditions
\left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=0}=1 and \left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=\frac{\pi}{2}}=-1 has
\left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=0}=1 and \left | \frac{\mathrm{d} y}{\mathrm{d} x} \right |_{x=\frac{\pi}{2}}=-1 has
no solution | |
exactly two solutions | |
exactly one solutions | |
infinitely many solutions |
Question 4 Explanation:
\begin{aligned} \left(d^{2}+16\right) y &=0 \\ A E \text { is } m^{2}+16 &=0 \\ m &=\pm 4 i \end{aligned}
Solution is
y=c_{1} \cos 4 x+c_{2} \sin 4 x
\begin{aligned} y' &=-4 c_{1} \sin 4 x+4 c_{2} \cos 4 x \\ y'(0) &=1 \\ 1 &=4 c_{2} \\ c_{2} &=1 / 4 \\ y'(\pi / 2) &=-1 \\ -1 &=-4 c_{1} \sin 2 \pi+4 c_{2} \cos 2 \pi \\ -1 &=0+4 c_{2} \\ c_{2} &=-1 / 4 \end{aligned}
Therefore the given differential equation has no solution.
Solution is
y=c_{1} \cos 4 x+c_{2} \sin 4 x
\begin{aligned} y' &=-4 c_{1} \sin 4 x+4 c_{2} \cos 4 x \\ y'(0) &=1 \\ 1 &=4 c_{2} \\ c_{2} &=1 / 4 \\ y'(\pi / 2) &=-1 \\ -1 &=-4 c_{1} \sin 2 \pi+4 c_{2} \cos 2 \pi \\ -1 &=0+4 c_{2} \\ c_{2} &=-1 / 4 \end{aligned}
Therefore the given differential equation has no solution.
Question 5 |
A six-face fair dice is rolled a large number of times. The mean value of the outcomes is ________
3.5 | |
3 | |
2.5 | |
1 |
Question 5 Explanation:
\begin{aligned} \text{mean} &=E(x)=\Sigma x \cdot P(x) \\ &=1(1 / 6)+2(1 / 6)+3(1 / 6)+4 \\ &(1 / 6)+5(1 / 6)+6(1 / 6) \\ &=\frac{1}{6}(1+2+3+4+5+6) \\ &=\frac{21}{6}=3.5 \end{aligned}
There are 5 questions to complete.