Linear Algebra - Engineering Mathematics - Textile Engineering & Fibre Science Previous Year Questions

1

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2007

The eigen values of the matrix $\begin{bmatrix} -3 & 2 \\ 1 & -2 \end{bmatrix}$ are $\lambda_1 = -1$ and $\lambda_2 = -4$. An eigen vector of the given matrix is

A

$\begin{bmatrix} 2 \\ 1 \end{bmatrix}$

B

$\begin{bmatrix} 1 \\ -1 \end{bmatrix}$

C

$\begin{bmatrix} 1 \\ 1 \end{bmatrix}$

D

$\begin{bmatrix} -2 \\ -1 \end{bmatrix}$

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2

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2008

The inverse of the matrix \(\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\) is

A

\(\begin{pmatrix} -\cos \theta & -\sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}\)

B

\(\begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}\)

C

\(\begin{pmatrix} \sin \theta & \cos \theta \\ -\sin \theta & -\cos \theta \end{pmatrix}\)

D

\(\begin{pmatrix} -\cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\)

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3

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2008

The value of the determinant \(\begin{vmatrix} 3 & 1 & 6 \\ 1 & 2 & 2 \\ -3 & 3 & -6 \end{vmatrix}\) is

A

-1

B

0

C

1

D

2

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4

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2009

If the determinant and trace of a \(2 \times 2\) matrix \(M\) are \(-1\) and \(0\) respectively, then the determinant of \(M + M^{-1}\) is

A

-4

B

-2

C

-1

D

0

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5

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2010

The matrix \begin{bmatrix} 1 & 3 & 2 \\ 2 & 3 & 5 \\ 1 & 4 & P \end{bmatrix} has one eigen value equal to 3. The sum of two other eigen values is

A

P

B

P+4

C

P-1

D

P+1

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6

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2010

Cofactor of ‘a’ in the following determinant is \begin{vmatrix} 3 & -4 & -3 \\ 2 & 7 & a \\ 5 & -9 & 2 \end{vmatrix}

A

3

B

5

C

7

D

9

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7

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2010

A student is answering a multiple choice examination with 65 questions with a marking scheme as follows: i) 1 mark for each correct answer, ii) \(-\frac{1}{4}\) for a wrong answer, iii) \(-\frac{1}{8}\) for a question that has not been attempted. If the student gets 37 marks in the test then the least possible number of questions the student has NOT answered is:

A

6

B

5

C

7

D

4

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8

2011 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2011

X and Y are two matrices such that XY and X+Y are both defined. The CORRECT statement from amongst the following is

A

X and Y are square matrices of the same order

B

X is a square matrix whereas Y is a rectangular matrix

C

X and Y are diagonal matrices of different order

D

X and Y are rectangular matrices

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9

2011 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2011

The number of solutions for the following system of inequalities is
\(X_1 \geq 0\)
\(X_2 \geq 0\)
\(X_1+X_2 \leq 10\)
\(2X_1+2X_2 \geq 22\)

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10

2011 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2011

Three sisters (R, S, and T) received a total of 24 toys during Christmas. The toys were initially divided among them in a certain proportion. Subsequently, R gave some toys to S which doubled the share of S. Then S in turn gave some of her toys to T, which doubled T’s share. Next, some of T’s toys were given to R, which doubled the number of toys that R currently had. As a result of all such exchanges, the three sisters were left with equal number of toys. How many toys did R have originally?

A

8

B

9

C

11

D

12

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11

2013 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2013

The inverse of the matrix \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\) is

A

\(\begin{bmatrix} \sin\theta & \cos\theta \\ \cos\theta & -\sin\theta \end{bmatrix}\)

B

\(\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\)

C

\(\begin{bmatrix} -\sin\theta & \cos\theta \\ \cos\theta & \sin\theta \end{bmatrix}\)

D

\(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)

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12

2013 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2013

A number is as much greater than 75 as it is smaller than 117. The number is:

A

91

B

93

C

89

D

96

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13

2013 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2013

X and Y are two positive real numbers such that 2X + Y ≤ 6 and X + 2Y ≤ 8. For which of the following values of (X, Y) the function f(X, Y) = 3X + 6Y will give maximum value?

A

(4/3, 10/3)

B

(8/3, 20/3)

C

(8/3, 10/3)

D

(4/3, 20/3)

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14

2014 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2014

Of the two eigen values of the matrix \[ \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \]

A

One is positive, one is negative

B

Both are positive

C

Both are negative

D

Both form a complex conjugate

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15

2015 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2015

Q.11 – Q.35 carry one mark each.

If 3 and 4 are two eigenvalues of A = [3 a b] [c 2 d] [e f 4] for some real numbers a, b, c, d, e, and f, then the third eigenvalue of A is ________.

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16

2016 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2016

Let $A = \begin{pmatrix} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{pmatrix}$. The determinant of $A^{-1}$ is equal to

A

$\frac{1}{2}$

B

$\frac{4}{3}$

C

$\frac{3}{4}$

D

2

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17

2016 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2016

The eigen values and eigen vectors of \(\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}\) are

A

\(\pm 5\) and \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) respectively.

B

\(\pm 3\) and \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) respectively.

C

\(\pm 4\) and \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) respectively.

D

\(\pm 5\) and \(\begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \end{pmatrix}\) respectively.

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18

2017 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2017

If the scalar projection of the vector \( \vec{a} = 3\hat{i} + \beta\hat{k} \) on the vector \( \vec{b} = 2\hat{i} - 2\hat{j} + \hat{k} \) is 5, then a value of \( \beta \) is equal to

A

21

B

9

C

-1

D

-6

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19

2017 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2017

The system of linear equations \( \begin{cases} \alpha x - 2y = 4 \\ 2x + 4y = \beta \end{cases} \) has

A

No solution for α = 1, β = -8

B

Unique solution for α = -1, β = 8

C

Infinitely many solutions for α = -1, β = -8

D

Infinitely many solutions for α = 1, β = 8

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20

2018 · Textile Engineering & Fibre Science · Engineering Mathematics · Linear Algebra

Textile Engineering & Fibre Science (TF) 2018

Let \( A = \begin{bmatrix} a & b \\ 2 & -b \end{bmatrix} \) and \( X = \begin{bmatrix} -1 \\ 1 \end{bmatrix} \). If \( AX = \begin{bmatrix} -3 \\ 1 \end{bmatrix} \), then \(|A|\) is equal to

A

2

B

-2

C

-6

D

6

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