Figure from the original question paper

(D) T is neither one-one nor onto.

Maths 2 Quiz 2 PYQ Jan 2023: 2 Apr 2023 · IITM BS

Question 1

+2 marksOne correct option

A
T is onto but not one-one
B
T is one-one but not onto.
C
T is both one-one and onto
D
T is neither one-one nor onto.

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Correct answer

A
T is onto but not one-one

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Question 2

+2 marksOne or more correct options

Let A be an n × n orthogonal matrix. Choose the correct option(s).

Select all that apply.

A
A is invertible.
B
det(A) = ± 1.
C
det(A) may be zero.
D
Nullity of A may be 1.

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Correct answers

A
A is invertible.
B
det(A) = ± 1.

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Question 3

+3 marksOne or more correct options

Which of the following options is/are true?

Select all that apply.

A
If the rows of a 3×4 matrix A are linearly independent, then AA^(T) is an invertible matrix.
B
If the columns of a 4 × 3 matrix A are linearly independent, then A^(T)A is an invertible matrix.
C
If the rows of a 3 × 4 matrix A are linearly independent, then A^(T)A is an invertible matrix.
D
If the columns of a 4 × 3 matrix A are linearly independent, then AA^(T) is an invertible matrix.

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Correct answers

A
If the rows of a 3×4 matrix A are linearly independent, then AA^(T) is an invertible matrix.
B
If the columns of a 4 × 3 matrix A are linearly independent, then A^(T)A is an invertible matrix.

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Question 4

+3 marksOne or more correct options

An inner product on a vector space $V$ is a function $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ satisfying the following conditions:

Condition 1: $\langle v, v \rangle > 0$ for all $v \in V \setminus \{0\}$ ; $\langle v, v \rangle = 0$ if and only if $v = 0$ .

Condition 2: $\langle v_1 + v_2, v_3 \rangle = \langle v_1, v_3 \rangle + \langle v_2, v_3 \rangle, \forall v_1, v_2, v_3 \in V$ .

Condition 3: $\langle v_1, v_2 \rangle = \langle v_2, v_1 \rangle, \forall v_1, v_2 \in V$ .

Condition 4: $\langle cv_1, v_2 \rangle = c\langle v_1, v_2 \rangle, \forall v_1, v_2 \in V$ .

Let $V = \mathbb{R}^2$ and consider the function defined as:

$\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$

$\langle (x_1, x_2), (y_1, y_2) \rangle = x_1y_1 - x_2y_1 + x_2y_2.$

Which of the following is/are satisfied by the above function?

Select all that apply.

A
Condition 1 is satisfied.
B
Condition 2 is satisfied.
C
Condition 3 is satisfied.
D
Condition 4 is satisfied.

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Correct answers

A
Condition 1 is satisfied.
B
Condition 2 is satisfied.
D
Condition 4 is satisfied.

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Question 5

+2 marksNumerical answer

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Correct answer: 1

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Question 6

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

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Correct answer: 3

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Question 7

+1 markNumerical answer

Based on the above data, answer the given subquestions.

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Correct answer: 2

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Question 8

+2 marksNumerical answer

Based on the above data, answer the given subquestions.

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Correct answer: 2

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Question 9

+2 marksOne or more correct options

Based on the above data, answer the given subquestions.

Select all that apply.

A
A is equivalent to B.
B
A is not equivalent to B.
C
There exist two invertible matrices C and D such that BD = CA.
D
There are no matrices C and D such that BD = CA.

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Correct answers

A
A is equivalent to B.
C
There exist two invertible matrices C and D such that BD = CA.

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Question 10

+1 markNumerical answer

Consider the system of linear equations $AX = b$ , where $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{pmatrix}$ , $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $b = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ .

Let $L$ denote the set of all solutions of the above system. Clearly, $L$ forms an affine space. Let $W$ denote the subspace corresponding to $L$ . Answer the given sub questions.

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Correct answer: 1

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Question 11

+2 marksNumerical answer

Consider the system of linear equations $AX = b$ , where $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{pmatrix}$ , $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $b = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ .

Let $L$ denote the set of all solutions of the above system. Clearly, $L$ forms an affine space. Let $W$ denote the subspace corresponding to $L$ . Answer the given sub questions.

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Correct answer: 1

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Question 12

+2 marksNumerical answer

Consider the system of linear equations $AX = b$ , where $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{pmatrix}$ , $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $b = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ .

Let $L$ denote the set of all solutions of the above system. Clearly, $L$ forms an affine space. Let $W$ denote the subspace corresponding to $L$ . Answer the given sub questions.

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Correct answer: 1

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Question 13

+1 markNumerical answer

Consider the system of linear equations $AX = b$ , where $A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 1 \end{pmatrix}$ , $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ and $b = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ .

Let $L$ denote the set of all solutions of the above system. Clearly, $L$ forms an affine space. Let $W$ denote the subspace corresponding to $L$ . Answer the given sub questions.

Show answer

Correct answer: 3

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