# 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
The closed curve shown in the figure is described by

$r=1+\cos \theta$, where $r=\sqrt{x^{2}+y^{2}}$ $x=r \cos \theta, y=r \sin \theta$

The magnitude of the line integral of the vector field $F=-y \hat{i}+x \hat{j}$ around the closed curve is ___(Round off to 2 decimal places). A 9.42 B 6.36 C 2.45 D 7.54
GATE EE 2023   Engineering Mathematics
Question 3 Explanation:
\begin{aligned} I & =\int_{0}^{2 \pi} \vec{F} \cdot \overrightarrow{d l} \\ & =\int_{0}^{2 \pi}(-y \hat{i}+x)(d x \hat{i}+d y) \\ & =\int_{0}^{2 \pi}(-y d x+x d y) \end{aligned}
Given : $\quad x=r \cos \theta$ and $y=r \sin \theta$
\begin{aligned} \therefore I&=\int_{0}^{2 \pi}[(-r \sin \theta)(-r \sin \theta) d \theta+(r \cos \theta)(r \cos \theta) d \theta] \\ &=\int_{0}^{2 \pi} r^{2} d \theta \\ &=\int_{0}^{2 \pi}(1+\cos \theta)^{2} d \theta \\ &=\int_{0}^{2 \pi}\left(1+\cos ^{2} \theta+2 \cos \theta\right) d \theta \\ &=\int_{0}^{2 \pi}\left(1+\frac{1+\cos 2 \theta}{2}+2 \cos \theta\right) \mathrm{d} \theta \\ &=3 \pi=9.425 \end{aligned}
 Question 4
The value of the integral $\iint_{R} x y d x d y$ over the region $R$, given in the figure, is ___
(rounded off to the nearest integer). A 0 B 1 C 2 D 3
GATE EC 2023   Engineering Mathematics
Question 4 Explanation: \begin{aligned} I & =\iint_{R} x y d x d y \\ & =\int_{y=0}^{1} \int_{x=-y}^{y} x y d x d y+\int_{y=1}^{2} \int_{x=y-2}^{2-y} x y d x d y \\ & =\int_{0}^{1} y\left(\frac{x^{2}}{2}\right)_{-y}^{y} d y+\int_{1}^{2} y\left(\frac{x^{2}}{2}\right)_{y-2}^{2-y} d y \\ & =0+0=0 \end{aligned}
 Question 5
The smallest perimeter that a rectangle with area of 4 square units can have is ______ units. (Answer in integer)
 A 4 B 6 C 8 D 10
GATE ME 2023   Engineering Mathematics
Question 5 Explanation: Given $\quad B \times L=4$
Perimeter $(P)=2 B+2 L$
perimeter to be smallest possible
$\frac{d P}{d B}=0=\frac{d}{d B}\left(2 B+2 \times \frac{4}{B}\right)=0$
$2-\frac{8}{B^{2}}=0$
$B=\pm 2 \Rightarrow B=2 \quad L=2$
Smallest perimeter $=2(B+L)=2(2+2)=8$

There are 5 questions to complete.

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### 3 thoughts on “Calculus”

1. In question 27, the limit is tending to 1 instead of 0. Please update it accordingly.

2. • 