Question 1 |
The value of x for which the matrix A=\begin{bmatrix} 3 & 2&4 \\ 9& 7&13 \\ -6&-4 & -9+x \end{bmatrix}
has zero as an eigenvalue is ________
has zero as an eigenvalue is ________
0 | |
1 | |
2 | |
3 |
Question 1 Explanation:
A has an eigen value is zero
\begin{aligned} \therefore \quad|A|&=0 \\ \left| \begin{array}{lll} 3 & 2 & 4\\9 & 7 & 13 \\ -6 & -4 & -9+x\end{array}\right|&=0 \\ 3(-63+7 x+52)-2(-81+9 x+78)&+4(-36+42)=0 \\ 3(7 x-11)-2(9 x-3)+4(6)&=0 \\ 21 x-33-18 x+6+24&=0 \\ 3 x-3&=0 \\ x&=1 \end{aligned}
\begin{aligned} \therefore \quad|A|&=0 \\ \left| \begin{array}{lll} 3 & 2 & 4\\9 & 7 & 13 \\ -6 & -4 & -9+x\end{array}\right|&=0 \\ 3(-63+7 x+52)-2(-81+9 x+78)&+4(-36+42)=0 \\ 3(7 x-11)-2(9 x-3)+4(6)&=0 \\ 21 x-33-18 x+6+24&=0 \\ 3 x-3&=0 \\ x&=1 \end{aligned}
Question 2 |
Consider the complex valued function f(z)=2z^{3}+b|z|^{3} where z is a complex variable. The value of b for which the function f(z) is analytic is ________
1 | |
2 | |
3 | |
0 |
Question 2 Explanation:
f(z)=2 z^{3}+b_{1}|z|^{3}
Given that f(z) is analytic.
which is possible only when b=0
since \left|z^{3}\right| is differentiable at the origin but not analytic.
2 z^{3} is analytic everywhere
\begin{aligned} \therefore \quad f(z)&=2 z^{3}+b\left|z^{3}\right| \text{ is analytic}\\ \text{only when }\quad b&=0 \end{aligned}
Given that f(z) is analytic.
which is possible only when b=0
since \left|z^{3}\right| is differentiable at the origin but not analytic.
2 z^{3} is analytic everywhere
\begin{aligned} \therefore \quad f(z)&=2 z^{3}+b\left|z^{3}\right| \text{ is analytic}\\ \text{only when }\quad b&=0 \end{aligned}
Question 3 |
As x varies from -1 to +3, which one of the following describes the behaviour of the function f(z)=x^{3}-3x^{2}+1 ?
f(x) increases monotonically. | |
f(x) increases, then decreases and increases again. | |
f(x) decreases, then increases and decreases again. | |
f(x) increases and then decreases. |
Question 3 Explanation:
\begin{array}{l} f(x)=x^{3}-3 x^{2}+1 \\ f^{\prime}(x)=3 x^{2}-6 x \\ f(x)=0 \end{array}
\begin{aligned} 3 x^{2}-6 x &=0 \\ 3 x(x-2) &=0 \\ x &=0,2 \\ f^{\prime \prime}(x) &=6 x-6 \\ \text{At}\quad x &=0 \quad f^{\prime \prime}(0)=-6 \text { maxima } \\ x &=2 \quad f^{\prime \prime}(2)=6 \text { minima } \end{aligned}
\begin{aligned} 3 x^{2}-6 x &=0 \\ 3 x(x-2) &=0 \\ x &=0,2 \\ f^{\prime \prime}(x) &=6 x-6 \\ \text{At}\quad x &=0 \quad f^{\prime \prime}(0)=-6 \text { maxima } \\ x &=2 \quad f^{\prime \prime}(2)=6 \text { minima } \end{aligned}
Question 4 |
How many distinct values of x satisfy the equation sin(x)=x/2, where x is in radians?
1 | |
2 | |
3 | |
4 or more |
Question 4 Explanation:
Hence, 3 solutions.
Question 5 |
Consider the time-varying vector I=\hat{x}15cos(\omega t)+\hat{y}5sin(\omega t) in Cartesian coordinates, where \omega \gt 0 is a constant. When the vector magnitude |I| is at its minimum value, the angle \theta that I makes with the x axis (in degrees, such that 0 \leq \theta \leq 180) is ________
80 | |
90 | |
100 | |
110 |
Question 5 Explanation:
\begin{aligned} I &=\hat{x} 15 \cos \omega t+\hat{y} 5 \sin \omega t \\ \| I &=\sqrt{\left(15 \cos (t)^{2}+(5 \sin \omega t)^{2}\right.} \\ &=\sqrt{225 \cos ^{2} \omega t+25 \sin ^{2} \omega t} \\ &=\sqrt{25+200 \cos ^{2} \omega t} \end{aligned}
|I| is minimum when \cos ^{2}\omega t=0
text{or}\quad \theta=\omega t=90^{\circ}
|I| is minimum when \cos ^{2}\omega t=0
text{or}\quad \theta=\omega t=90^{\circ}
There are 5 questions to complete.