# Engineering Mathematics

 Question 1
Which of the following is/are the eigenvector(s) for the matrix given below?
$\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}$
MSQ
 A $\begin{pmatrix} -1\\ 1\\ 0\\ 1 \end{pmatrix}$ B $\begin{pmatrix} 1\\ 0\\ -1\\ 0 \end{pmatrix}$ C $\begin{pmatrix} -1\\ 0\\ 2\\ 2 \end{pmatrix}$ D $\begin{pmatrix} 0\\ 1\\ -3\\ 0 \end{pmatrix}$
GATE CSE 2022      Linear Algebra
Question 1 Explanation:
 Question 2
Consider solving the following system of simultaneous equations using LU decomposition.
\begin{aligned} x_1+x_2-2x_3&=4 \\ x_1+3x_2-x_3&=7 \\ 2x_1+x_2-5x_3&=7 \end{aligned}
where L and U are denoted as
$L= \begin{bmatrix} L_{11} & 0 & 0 \\ L_{21}& L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{bmatrix}, U= \begin{bmatrix} U_{11} & U_{12} & U_{13} \\ 0& U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{bmatrix}$
Which one of the following is the correct combination of values for $L32, U33,$ and $x_1$?
 A $L_{32}=2,U_{33}=-\frac{1}{2},x_1=-1$ B $L_{32}=2,U_{33}=2,x_1=-1$ C $L_{32}=-\frac{1}{2},U_{33}=2,x_1=0$ D $L_{32}=-\frac{1}{2},U_{33}=-\frac{1}{2},x_1=0$
GATE CSE 2022      Linear Algebra
Question 2 Explanation:
 Question 3
The value of the following limit is _____
$\lim_{x \to 0+} \frac{ \sqrt{x}}{1-e^{2 \sqrt{x}}}$
 A -0.5 B 0.5 C 0 D 1
GATE CSE 2022      Calculus
Question 3 Explanation:
 Question 4
Consider the following two statements with respect to the matrices $A_{m \times n},B_{n \times m},C_{n \times n} \text{ and }D_{n \times n},$

Statement 1: $tr(AB) = tr(BA)$
Statement 2: $tr(CD) = tr(DC)$

where$tr()$ represents the trace of a matrix. Which one of the following holds?
 A Statement 1 is correct and Statement 2 is wrong. B Statement 1 is wrong and Statement 2 is correct. C Both Statement 1 and Statement 2 are correct. D Both Statement 1 and Statement 2 are wrong.
GATE CSE 2022      Linear Algebra
Question 4 Explanation:
 Question 5
For two n-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ is defined as follows:

$s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])$

Let $\mathcal{L}$ be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors $P,Q \in \mathcal{L}, s(P,Q)=0$. What is the maximum cardinality possible for the set $\mathcal{L}$?
 A 9 B 10 C 11 D 100
GATE CSE 2021 SET-2      Calculus
Question 5 Explanation:
 Question 6
Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function on the interval [-3,3] and a differentiable function in the interval (-3,3) such that for every $x$ in the interval, $f'(x) \leq 2$. If $f(-3) =7$, then $f(3)$ is at most __________
 A 19 B 32 C 11 D 54
GATE CSE 2021 SET-2      Calculus
Question 6 Explanation:
 Question 7
Suppose that P is a 4x5 matrix such that every solution of the equation Px=0 is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of P is _______
 A 1 B 2 C 3 D 4
GATE CSE 2021 SET-2      Linear Algebra
Question 7 Explanation:
 Question 8
Consider the following matrix.
$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$
The largest eigenvalue of the above matrix is __________.
 A 1 B 3 C 4 D 6
GATE CSE 2021 SET-1      Linear Algebra
Question 8 Explanation:
 Question 9
Consider the following expression.
$\lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}$
The value of the above expression (rounded to 2 decimal places) is ___________.
 A 0.25 B 0.45 C 0.75 D 0.85
GATE CSE 2021 SET-1      Calculus
Question 9 Explanation:
 Question 10
If $x+2 y=30$,then $\left(\frac{2 y}{5}+\frac{x}{3}\right)+\left(\frac{x}{5}+\frac{2 y}{3}\right)$ will be equal to
 A 8 B 16 C 18 D 20
ISRO CSE 2020      Linear Algebra
Question 10 Explanation:

There are 10 questions to complete.

### 3 thoughts on “Engineering Mathematics”

1. Update Q184 as A instead of C