Question 1 |
Given a system of equations
x + 2y + 2z = b_1
5x + y + 3z = b_2
What of the following is true regarding its solutions
x + 2y + 2z = b_1
5x + y + 3z = b_2
What of the following is true regarding its solutions
The system has a unique solution for any given b_1 \; and \; b_2 | |
The system will have infinitely many solutions for any given b_1 \; and \; b_2 | |
Whether or not a solution exists depends on the given b_1 \; and \; b_2 | |
The systems would have no solution for any values of b_1 \; and \; b_2 |
Question 2 |
Let f(x)=xe^{-x}. The maximum value of the function in the interval (0,\infty ) is
e^{-1} | |
e | |
1-e^{-1} | |
1+e^{-1} |
Question 2 Explanation:
\begin{aligned} f(x)&=xe^{-x}\\ f'(x)&=e^{-x}-xe^{-x}=0\\ e^{-x}(1-x)&=0\\ x&=1\\ f''(x)&=-e^{-x}-e^{-x}+xe^{-x}\\ &=e^{-x}(x-2)\\ f''(1)&=e^{-1}(-1)=-e^{-1} \lt 0 \end{aligned}
Hence f(x) has maximum value at x=1
f(1)=1\cdot e^{-1}=e^{-1}
Hence f(x) has maximum value at x=1
f(1)=1\cdot e^{-1}=e^{-1}
Question 3 |
The solution for the differential equation
\frac{d^{2}x}{dt^{2}}=-9x
with initial conditions x(0)=1 and \frac{dx}{dt}|_{t=0}=1, is
\frac{d^{2}x}{dt^{2}}=-9x
with initial conditions x(0)=1 and \frac{dx}{dt}|_{t=0}=1, is
t^{2}+t+1 | |
sin 3t+ \frac{1}{3} cos3t+\frac{2}{3} | |
\frac{1}{3} sin 3t+cos3t | |
cos3t+t |
Question 3 Explanation:
\begin{aligned} \frac{d^2x}{dt^2}&=-9x\\ \frac{d^2x}{dt^2}+9x&=0\\ (D^2+9)x &=0 \end{aligned}
Auxiliary equation is m^2+9=0
\begin{aligned} m&= \pm 3i\\ x&= C_1 \cos 3t+C_2 \sin 3t\;\;...(i)\\ x(0) &=1\;\;i.e.\;x\rightarrow 1\; when t\rightarrow 0 \\ 1&=C_1 \\ \frac{dx}{dt}&=-3C_1 \sin 3t+3C_2 \cos 3t\;\;...(ii) \\ x'(0)&=1\;\; i.e.\;x'\rightarrow 1\; when \; t\rightarrow 0 \\ 1&=3C_2 \\ C_2&=\frac{1}{3}\\ \therefore \;x&=\cos 3t+\frac{1}{3}\sin 3t \end{aligned}
Auxiliary equation is m^2+9=0
\begin{aligned} m&= \pm 3i\\ x&= C_1 \cos 3t+C_2 \sin 3t\;\;...(i)\\ x(0) &=1\;\;i.e.\;x\rightarrow 1\; when t\rightarrow 0 \\ 1&=C_1 \\ \frac{dx}{dt}&=-3C_1 \sin 3t+3C_2 \cos 3t\;\;...(ii) \\ x'(0)&=1\;\; i.e.\;x'\rightarrow 1\; when \; t\rightarrow 0 \\ 1&=3C_2 \\ C_2&=\frac{1}{3}\\ \therefore \;x&=\cos 3t+\frac{1}{3}\sin 3t \end{aligned}
Question 4 |
Let X(s)=\frac{3s+5}{s^{2}+10s+21} be the Laplace Transform of a signal x(t). Then, x(0^+) is
0 | |
3 | |
5 | |
21 |
Question 4 Explanation:
Given, X(s)=\left [ \frac{3s+5}{s^2+10s+21} \right ]
Using initial value theorem,
\begin{aligned} x(0^+) &= \lim_{s \to \infty }[sX(s)]\\ x(0^+) &= \lim_{s \to \infty } \left [ \frac{s(3s+5)}{s^2+10+21} \right ] \\ &= \lim_{s \to \infty }\left [ \frac{3+\frac{5}{s}}{1+\frac{10}{s}+\frac{21}{s^2}} \right ]=3 \end{aligned}
Using initial value theorem,
\begin{aligned} x(0^+) &= \lim_{s \to \infty }[sX(s)]\\ x(0^+) &= \lim_{s \to \infty } \left [ \frac{s(3s+5)}{s^2+10+21} \right ] \\ &= \lim_{s \to \infty }\left [ \frac{3+\frac{5}{s}}{1+\frac{10}{s}+\frac{21}{s^2}} \right ]=3 \end{aligned}
Question 5 |
Let S be the set of points in the complex plane corresponding to the unit circle. (That is, S={z:|z|=1}). Consider the function f(z)=zz* where z* denotes the complex conjugate of z. The f(z) maps S to which one of the following in the complex plane
unit circle | |
horizontal axis line segment from origin to (1, 0) | |
the point (1, 0) | |
the entire horizontal axis |
Question 5 Explanation:
\begin{aligned} z&=x+iy\\ z^*&=x-iy\\ zz^*&=(x+iy)(x-iy)\\ &=x^2+y^2 \end{aligned}
which is equal to (1) always as given
\begin{aligned} |z|&=1\\ zz^*&=x^2+y^2 \end{aligned}
There are 5 questions to complete.