Mathematics Test 277

Result

Mathematics Test 277

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Weekly Quiz Competition

Question 1
4 / -1

If A and B are two matrices, such that, AB = B and BA = A, then A² + B² is equal to

A
2AB

B
2BA

C
A + B

D
AB

Solution

We have:

A² + B² = (BA)² + (AB)²

= (BA)(BA) + (AB)(AB)

= B(AB)A + A(BA)B

= B(BA) + A(AB) = BA + AB = A + B
Question 2
4 / -1

If R is a relation from a non – empty set A to a non – empty set B, then

A
R ⊆ A × B.

B
R = A ∪ B

C
R = A ∩ B

D
R =A × B

Solution

Let A and B be two sets. Then a relation R from set A to set B is a subset of A × B. Thus, R is a relation from A to B ⇔ R ⊆ A × B.
Question 3
4 / -1

Let R be the relation over the set of straight lines of a plane such that l₁ R l₂ ⇔ l₁ ⊥ l₂. Then, R is

A
Symmetric

B
Reflexive

C
Transitive

D
An equivalence relation

Solution

To be reflexive, a line must be perpendicular to itself, but which is not true. So, R is not reflexive

For symmetric, if l₁ R l₂ ⇒ l₁ ⊥ l₂.

⇒ l₂ ⊥ l₁ ⇒ l₁ R l₂ hence symmetric

For transitive, if l₁ R l₂ and l₂ R l₃

⇒ l₁ R l₂ and l₂ R l₃ does not imply that l₁ ⊥ l₃ hence not transitive.
Question 4
4 / -1

If cos^-¹ x = tan^-¹ x, then sin(cos^-¹ x) is equal to

A
x

B
x²

C
1/x

D
1/x²

Solution
Question 5
4 / -1

The order of the single matrix obtained from is

A
2 × 3

B
2 × 2

C
3 × 3

D
3 × 2

Solution
Question 6
4 / -1

If ω is a complex cube root of unity, then the matrix

A
Singular matrix

B
Non-singular matrix

C
Skew symmetric matrix

D
None of these

Solution
Question 7
4 / -1

Let A = {X = (x, y, z)^T : PX = 0 and x² + y² + z² = 1}, where then the set A

A
Is an empty set

B
Contains more than two elements

C
Contains exactly two elements

D
Is a singleton

Solution
Question 8
4 / -1

In a skew symmetric matrix, the diagonal elements are all

A
Different from each other

B
Zero

C
One

D
None of these

Solution

A skew-symmetric matrix is a square matrix A that satisfies the condition:

A^T = −A

This means that the transpose of the matrix is equal to the negative of the matrix. As a result, the elements on the diagonal of a skew-symmetric matrix must be zero. This is because for any diagonal element a_ii, the condition a_ii = −a_iimust hold, which is only true if a_ii = 0.

Therefore, the diagonal elements of a skew-symmetric matrix are all zero.

Thus, the correct answer is B: Zero.
Question 9
4 / -1

The void relation (a subset of A x A) on a non empty set A is:

A
Anti symmetric

B
Transitive

C
Reflexive

D
Transitive and symmetric

Solution

The relation { } ⊂ A x A on a is surely not reflexive. However, neither symmetry nor transitivity is contradicted. So { } is a transitive and symmetry relation on A.
Question 10
4 / -1

A relation R in a set A is called reflexive,

A
If (b, b) ∈ R, for every a ∈ A

B
If (b, a) ∈ R, for every a, b ∈ A

C
If (a, b) ∈ R, for every a, b ∈ A

D
If (a, a) ∈ R, for every a ∈ A

Solution

A relation R on a non empty set A is said to be reflexive if fx Rx for all x ∈ R, Therefore, R is reflexive.