Calculus


Question 1
Two vectors \left[\begin{array}{llll}2 & 1 & 0 & 3\end{array}\right]^{\top} and \left[\begin{array}{llll}1 & 0 & 1 & 2\end{array}\right]^{\top} belong to the null space of a 4 \times 4 matrix of rank 2 . Which one of the following vectors also belongs to the null space?
A
\left[\begin{array}{llll}1 & 1 & -1 & 1\end{array}\right]^{\top}
B
\left[\begin{array}{llll}2 & 0 & 1 & 2\end{array}\right]^{\top}
C
\left[\begin{array}{llll}0 & -2 & 1 & -1\end{array}\right]^{\top}
D
\left[\begin{array}{llll}3 & 1 & 1 & 2\end{array}\right]^{\top}
GATE CE 2023 SET-2   Engineering Mathematics
Question 1 Explanation: 
Given matrix is 4 \times 4 and rank of matrix is 2 .
Therefore, rank of matrix \neq No. of variables Thus, there two linearly dependent vectors \& two linearly independent vectors are present.
\begin{aligned} & X_{1}=\left[\begin{array}{llll} 2 & 1 & 0 & 3 \end{array}\right]^{\top} \\ & X_{2}=\left[\begin{array}{llll} 1 & 0 & 1 & 2 \end{array}\right]^{\top} \end{aligned}
\therefore \quad X=\mathrm{K}_{1}\left[\begin{array}{l}2 \\ 1 \\ 0 \\ 3\end{array}\right]+\mathrm{K}_{2}\left[\begin{array}{l}1 \\ 0 \\ 1 \\ 2\end{array}\right]
For \mathrm{K}_{1}=1 \text{ and } \mathrm{~K}_{2}=-1
X=\left[\begin{array}{c}1 \\ 1 \\ -1 \\ 1\end{array}\right]=\left[\begin{array}{llll}1 & 1 & -1 & 1\end{array}\right]^{\top}
Question 2
Let \phi be a scalar field, and \mathbf{u} be a vector field. Which of the following identities is true for div(\phi \mathbf{u}) ?
A
\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{div}(\mathbf{u})+\mathbf{u} \cdot \operatorname{grad}(\phi)
B
\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{div}(\mathbf{u})+\mathbf{u} \times \operatorname{grad}(\phi)
C
\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{grad}(\mathbf{u})+\mathbf{u} \cdot \operatorname{grad}(\phi)
D
\operatorname{div}(\phi \mathbf{u})=\phi \operatorname{grad}(\mathbf{u})+\mathbf{u} \times \operatorname{grad}(\phi)
GATE CE 2023 SET-2   Engineering Mathematics
Question 2 Explanation: 
div(\phi \mathrm{u})=\phi div(\mu)+ugrad(\phi)


Question 3
For the function f(x)=e^{x}|\sin x| ; x \in \mathbb{R}, which of the following statements is/are TRUE?
A
The function is continuous at all x
B
The function is differentiable at all x
C
The function is periodic
D
The function is bounded
GATE CE 2023 SET-1   Engineering Mathematics
Question 3 Explanation: 
f(x)=e^{x}|\sin x|

From above graph its clear that for every x \in \mathbb{R}
lim _{h+0} f(x-h)=\lim _{h+0} f(x+h)=f(x)
So, function is always continuous but in the graph there are corner points so function is not differentiable.
Question 4
The following function is defined over the interval [-\mathrm{L}, \mathrm{L}] :

f(x)=p x^{4}+q x^{5}

If it is expressed as a Fourier series,

f(x)=a_{0}+\sum_{n=1}^{\infty}\left\{a_{n} \sin \left(\frac{\pi x}{L}\right)+b_{n} \cos \left(\frac{\pi x}{L}\right)\right\} \text {, }

which options amongst the following are true?
A
a_n,n=1,2,...,\infty depends on p
B
a_n,n=1,2,...,\infty depends on q
C
b_n,n=1,2,...,\infty depends on p
D
b_n,n=1,2,...,\infty depends on q
GATE CE 2023 SET-1   Engineering Mathematics
Question 4 Explanation: 
\begin{aligned} f(x) & =p x^{4}+q x^{5} \\ b_{n} & =\frac{1}{\ell} \int_{-\ell}^{\ell} f(x) \cos \left(\frac{n \pi x}{\ell}\right) d x\\ & =\frac{1}{\ell} \int_{-\ell}^{\ell}\left(p x^{4}+q x^{5}\right) \cos \left(\frac{n \pi x}{\ell}\right) d x \\ & =\frac{1}{\ell} \int_{-\ell}^{\ell} \mathrm{p}\left(\mathrm{x}^{4}\right) \cos \frac{\mathrm{n} \pi \mathrm{x}}{\ell} \mathrm{dx} \\ & \left(\because q x^{5} \frac{\cos n \pi x}{\ell} \text { is odd function }\right) \end{aligned}
So, b_{n} depends on p.
\begin{aligned} a_{n}= & \frac{1}{\ell} \int_{-\ell}^{\ell} f(x) \sin \left(\frac{n \pi x}{\ell}\right) d x \\ a_{n}= & \frac{1}{\ell} \int_{-\ell}^{\ell}\left(p x^{4}+q x^{5}\right) \sin \frac{n \pi x}{\ell} d x \\ a_{n}= & 0+\frac{1}{\ell} \int_{-\ell}^{\ell} q x^{5} \sin \left(\frac{n \pi x}{\ell}\right) d x \\ & \left(\because p x^{4} \frac{\sin n \pi x}{\ell}\right. \text { is odd function) } \end{aligned}
So, a_{n} depends on p.
Question 5
For the integral

\mathrm{I}=\int_{-1}^{1} \frac{1}{\mathrm{x}^{2}} \mathrm{dx}

which of the following statements is TRUE?
A
\quad \mathrm{I}=0
B
\quad \mathrm{I}=2
C
\quad \mathrm{I}=-2
D
The integral does not converge
GATE CE 2023 SET-1   Engineering Mathematics
Question 5 Explanation: 
\begin{aligned} I & =\int_{-1}^{1} \frac{1}{x^{2}} d x \\ & =2 \int_{0}^{1} \frac{1}{x^{2}} d x \quad(\because \quad f(-x)=f(x)) \\ & =2 \lim _{t \rightarrow 0^{+}} \int_{t}^{1} \frac{d x}{x^{2}} \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{-1}{x}\right)_{t}^{1} \\ & =-2 \lim _{t \rightarrow 0^{+}}\left(1-\frac{1}{t}\right) \\ & =2 \lim _{t \rightarrow 0^{+}}\left(\frac{1}{t}-1\right) \\ & =2 \lim _{h \rightarrow 0^{+}}\left(\frac{1}{0+h}-1\right)\\ & =2(\infty-1) \\ & =\infty \;\;\; (Divergent) \end{aligned}




There are 5 questions to complete.

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