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 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 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 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)

wheretr() 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 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 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 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 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 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 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


There are 10 questions to complete.

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