Question 1 |
The Fourier cosine series for an even function f(x) is given by
f(x)=a_{0}+\sum_{n=1}^{\infty}a_{n}\cos(nx)
The value of the coefficient a_{2} for the function f(x)=\cos ^{2}(x) in \left [ 0 , \pi \right ] is
f(x)=a_{0}+\sum_{n=1}^{\infty}a_{n}\cos(nx)
The value of the coefficient a_{2} for the function f(x)=\cos ^{2}(x) in \left [ 0 , \pi \right ] is
-0.5 | |
0 | |
0.5 | |
1 |
Question 1 Explanation:
\begin{aligned} \cos ^{2} x &=\frac{1+\cos 2 x}{2} \\ f(x) &=\frac{1}{2}+\frac{\cos 2 x}{2} \\ f(x) &=\frac{a_{0}}{2}+\sum_{n=1}^{\infty} a_{n} \cdot \cos n x \\ a_{0} &=1 \\ a_{1} &=0 \\ a_{2} &=\frac{1}{2} \end{aligned}
Question 2 |
The divergence of the vector field \overrightarrow{u}=e^{x}\left ( \cos y\hat{i} + \sin y\hat{j} \right) is
0 | |
e^{x}\cos y + e^{x}\sin y | |
2e^{x}\cos y | |
2e^{x}\sin y |
Question 2 Explanation:
\begin{aligned} \vec{u} &=e^{x} \cos y \hat{i}+e^{x} \cdot \sin y \hat{j} \\ \nabla \cdot \vec{u} &=\frac{\partial}{\partial x}\left(u_{1}\right)+\frac{\partial}{\partial y}\left(u_{2}\right) \\ &=\frac{\partial}{\partial x}\left(e^{x} \cdot \cos y\right)+\frac{\partial}{\partial y}\left(e^{x} \cdot \sin y\right) \\ &=e^{x} \cos y+e^{x} \cos y \\ \nabla \cdot \vec{u} &=2 e^{x} \cdot \cos y \end{aligned}
Question 3 |
Consider a function u which depends on position x and time t. The partial differential
equation
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}
is known as the
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}
is known as the
Wave equation | |
Heat equation | |
Laplace's equation | |
Elasticity equation |
Question 3 Explanation:
\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}is known as heat equation
Question 4 |
If y is the solution of the differential equation
y^{3}\frac{d y}{dx}+x^{3}=0,y(0)=1, the value of y\left ( -1 \right ) is
y^{3}\frac{d y}{dx}+x^{3}=0,y(0)=1, the value of y\left ( -1 \right ) is
-2 | |
-1 | |
0 | |
1 |
Question 4 Explanation:
\begin{aligned} y^{3} \frac{d y}{d x} &=-x^{3} \\ y^{3} d y &=-x^{3} d x \\ \int y^{3} d y &=-\int x^{3} d x \\ \frac{y^{4}}{4} &=\frac{-x^{4}}{4}+C \\ \frac{x^{4}+y^{4}}{4} &=C \\ y(0) &=1 \\ \frac{0+1}{4} &=C \\ C &=\frac{1}{4} \\ x^{4}+y^{4} &=1 \\ y^{4} &=1-x^{4} \\ y &=\sqrt[4]{1-x^{4}} \\ \text{When,}\quad x &=-1 \\ y &=0 \end{aligned}
Question 5 |
The minimum axial compressive load, P, required to initiate buckling for a pinned-pinned slender column with bending stiffness EI and length L is
P=\frac{\pi ^{2}EI}{4L^{2}} | |
P=\frac{\pi ^{2}EI}{L^{2}} | |
P=\frac{3\pi ^{2}EI}{4L^{2}} | |
P=\frac{4\pi ^{2}EI}{L^{2}} |
Question 5 Explanation:
For both ends hinged buckling load,
P=\frac{\pi^{2} E I}{L^{2}}
P=\frac{\pi^{2} E I}{L^{2}}
There are 5 questions to complete.