Linear Algebra - Engineering Mathematics - Mining Engineering Previous Year Questions

1

2007 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2007

If the rank of a matrix A is $r$, the rank of the matrix $\mathbf{A}^{T}$ is

A

$r$, if and only if $\mathbf{A}^{T} = \mathbf{A}$

B

$r$, for all A

C

$p$, where $p \neq r$

D

$r-1$, where $r \geq 1$

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2

2007 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2007

The value of \(A \cdot B\), if \(A + B = \begin{bmatrix} 1 & -1 \\ 3 & 0 \end{bmatrix}\) and \(A - B = \begin{bmatrix} 3 & 1 \\ 1 & 4 \end{bmatrix}\), is

A

\(\begin{bmatrix} 1 & 1 \\ 0 & 3 \end{bmatrix}\)

B

\(\begin{bmatrix} 1 & 1 \\ 0 & 3 \end{bmatrix}\)

C

\(\begin{bmatrix} 1 & 1 \\ 0 & 3 \end{bmatrix}\)

D

\(\begin{bmatrix} 1 & 1 \\ 0 & 3 \end{bmatrix}\)

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3

2008 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2008

Q.1 The trace of the following matrix is \[ \begin{pmatrix} 2 & 2 & 3 \\ 3 & 2 & 3 \\ 4 & 1 & -2 \end{pmatrix} \]

A

6

B

7

C

8

D

9

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4

2008 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2008

The inverse of the following matrix is: \[\begin{pmatrix} 4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}\]

A

\(\begin{pmatrix} 16 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)

B

\(\begin{pmatrix} 0.25 & 0 & 0 \\ 0 & 0.50 & 0 \\ 0 & 0 & 1.00 \end{pmatrix}\)

C

\(\begin{pmatrix} 2 & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\)

D

\(\begin{pmatrix} 16 & 0 & 0 \\ 0 & 4 & 1 \\ 0 & 0 & 1 \end{pmatrix}\)

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5

2008 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2008

The solution of the following system of linear equations is
x+4y+3z=0
3x+5y+2z=0
8x+10y+12z=0

A

(0,0,0)

B

(1,-1,1)

C

(2, -1, -2)

D

(-3,0,1)

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6

2009 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2009

If A is an orthogonal matrix, then

A

\(A^T = A^{-1} \)

B

\(A^T = -A^{-1} \)

C

\(A = A^{-1} \)

D

\(A = -A^{-1} \)

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7

2009 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2009

The sum of the eigenvalues of the matrix \( \begin{bmatrix} 1 & 2 \\ 1 & 0 \end{bmatrix} \) is

A

-3

B

-1

C

1

D

3

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8

2009 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2009

Consider the following linear programming problem:
Maximize
\[ z = 3x + 2y \]
Subject to
\[ 3x + 2y \geq 15 \]
\[ 2x + 3y \leq 6 \]
\[ x \geq 0, y \geq 0 \]
The above linear programming problem has

A

unique optimal solution

B

multiple optimal solutions

C

unbounded solution

D

infeasible solution

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9

2010 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2010

Two determinants of order \( n \) are multiplied. The order of the resultant determinant is

A

\(n \)

B

\(2n \)

C

\(n^2 \)

D

\(n/2 \)

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10

2010 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2010

The feasibility region of an LP problem in variables x and y is given by the following constraints, in addition to the non-negativity constraints:
\(y \le 60\), \(x \le 90\), \(x + y \le 70\).
The number of corner point feasible solutions for this problem are

A

3

B

4

C

5

D

6

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11

2011 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2011

The two vectors are orthonormal, if

A

vector product is zero and norm of each vector is also zero

B

vector product is one and norm of each vector is also one

C

cross product is zero and norm of each vector is one

D

cross product is one and norm of each vector is zero

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12

2011 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2011

Product of the eigen values of the matrix A is
\[ A = \begin{pmatrix} 3 & 2 & 5 \\ 2 & 2 & 1 \\ 1 & 5 & 4 \end{pmatrix} \]

A

6

B

8

C

10

D

35

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13

2012 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2012

The cofactor matrix of \(P = \begin{bmatrix} 3 & 1 & 2 \\ 2 & 3 & 1 \\ 1 & 2 & 3 \end{bmatrix}\) is

A

\(\begin{bmatrix} 21 & -5 & 2 \\ 2 & 21 & -5 \\ -5 & 2 & 21 \end{bmatrix}\)

B

\(\begin{bmatrix} 21 & 2 & -5 \\ 2 & 7 & 15 \\ -5 & 21 & 2 \end{bmatrix}\)

C

\(\begin{bmatrix} -5 & 2 & 21 \\ 15 & 7 & 2 \\ -2 & 21 & -5 \end{bmatrix}\)

D

\(\begin{bmatrix} 15 & 7 & 2 \\ -5 & 2 & 21 \\ -2 & 21 & -5 \end{bmatrix}\)

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14

2013 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2013

If the transpose of a matrix is equal to its inverse, then the matrix is

A

symmetric

B

orthogonal

C

skew symmetric

D

singular

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15

2013 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2013

If the following linear system of equations has non-trivial solutions
\(px + y + z = 0\)
\(2x + y - 2z = 0\)
\(x + 2y - 3z = 0\)
the value of p is

A

1

B

0

C

-1

D

-7

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16

2016 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2016

If \([A][B] = [I]\) then

A

\([B] = [A]^T\)

B

\([A] = [B]^T\)

C

\([B] = [A]^{-1}\)

D

\([B] = [A]\)

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17

2016 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2016

The value of \( x \) in the simultaneous equations is __________

Check answer

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18

2017 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2017

Consider the following linear programming problem:
Maximize   Z = 6X + 10Y
Subject to  X ≤ 4
                Y ≤ 6
                3X + 2Y ≤ 18
                X ≥ 0 , Y ≥ 0
The maximum value of the objective function is ________

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19

2017 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2017

If the rank of the following matrix is less than 3, the values of x are
\[ A = \begin{bmatrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{bmatrix} \]

A

1, −1/2

B

1, 1/2

C

2, −1/4

D

2, −3/4

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20

2018 · Mining Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2018

If \( X = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} \), then \( XX^T \) is

A

\(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \)

B

\(\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \)

C

\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \)

D

\(\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix} \)

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