Determinants Test

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Determinants Test - 9

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Question 1
1 / -0.25

The solution of the following system of equation is
2x + 3y = 5
5x –2y = 3

A
x = 2, y = 3

B
x = -1, y = -1

C
x = 1, y = 1

D
x = 3, y = 2

Solution

2x + 3y = 5
Multiply equation with ‘5 ’, we get
10x + 15y = 25 ……………………(1)
5x –2y = 3
Multiply equation with ‘2 ’, we get
10x - 4y = 6 ………………………(2)
Subtracting (1) from (2), we get
19y = 19
y = 1
Put the value of y in eq(1)
10x + 15(1) = 25
10x = 10
x = 1
Question 2
1 / -0.25

One third of sum of two angles is 60 °and one fourth of their difference is 28 °. The angles are

A
30 °and 44 °

B
34 °and 90 °

C
146 °and 34 °

D
130 °and 60 °

Solution

Solution:-
Let the two angles be 'x 'and 'y '.
So, according to the question,
One third of the sum of two angles :- 1/3(x+y) = 60
x/3 + y/3 = 60    ......(1)
Quarter of their difference :- 1/4(x-y) = 28
x/4 - y/4 = 28...........(2)
Multiplying the equation (1) by 3 and equation (2) by 4, we get
x + y =180 .......(3)   x - y = 112 ......(4)
Subtracting (4) from (3),
x + y = 180
x - y = 112
-   +     -
_________
    2y = 68
_________
2y = 68
y = 34
Putting the value of y = 34 in the equation (3)
x + y = 180
x + 34 = 180
x = 180 - 34
x = 146
The two angles are 146 °and 34 °.
Question 3
1 / -0.25

For a square matrix A in a matrix equation AX = B, if │A │≠0, then

A
There exists a unique solution

B
There exists no solution

C
There exists infinite number of solutions

D
The system may or may not be consistent

Solution

Given, AX = B, where A is a square matrix.
If A is invertible (i.e., A 0), then there exists a unique solution for X.
Explanation: When A is invertible, it means that there exists a unique matrix A-1 such that A-1A = I, where I is the identity matrix.
Now, if we multiply both sides of the given equation by A-1, we get: A-1AX = A-1B ⇒IX = A-1B (using A-1A = I) ⇒X = A-1B
Hence, we get a unique solution for X, which is X = A-1B. This is because the inverse of a matrix is unique, and so there can be only one solution for X.
Therefore, the correct option is (A) - There exists a unique solution.
Question 4
1 / -0.25

The following system of equations has
x + 3y + 3z = 2
x + 4y + 3z = 1
x + 3y + 4z = 2

A
Infinite Solution

B
Trivial Solution

C
Unique Solution

D
No Solution

Solution

Let A = {(1,3,3) (1,4,3) (1,3,4)}
|A| = 1(16-9) -3(4-3) +3(3-4)
|A| = 1(7) -3(1) +3(-1)
= 7 - 3 - 3
= 1
Therefore, A is not equal to zero, it has unique solution.
Question 5
1 / -0.25

Inverse of a matrix A is given by

A
| A |

B

C
A^-1 = AdjA

D
A^-1 = Matrix of cofactors of A

Solution

Inverse of matrix (A^-1 ) = (adj A)/|A|
Question 6
1 / -0.25

If , then A^-1 =

A
-A

B
I

C
A

D
-I

Solution

The inverse of matrix \( A \), which is the identity matrix \( I \), is itself. So, the answer is ( A^{-1} = A ), which corresponds to option C.
Question 7
1 / -0.25

The system of equations kx + 2y –z = 1,
(k –1)y –2z = 2
(k + 2)z = 3 has a unique solution, if k is

A
–1

B
–2

C
1

D
0

Solution

This system of equations has a unique solution, if
Question 8
1 / -0.25

System of equations AX = B is inconsistent if

A
│B │= 0

B
(adj A) B = 0

C
(adj A) B ≠0

D
│A │≠0

Solution

If (adj A) B ≠0 (zero matrix), then the solution does not exist. The system of equations is inconsistent. Else, if (adj A) B = 0 then the system will either have infinitely many solutions (consistent system) or no solution (inconsistent system).
Question 9
1 / -0.25

Inverse of, is

A

B

C

D

Solution

A = {(6,7) (8,9)}
|A| = (6 * 9) - (8 * 7)
= 54 - 56
|A| = -2
A^-1 = -½{(9,-7) (-8,6)}
A^-1 = {(-9/2, 7/2) (4,-3)}
Question 10
1 / -0.25

A system of linear equations AX = B is said to be inconsistent, if the system of equations has

A
Trivial Solution

B
Infinite Solutions

C
No Solution

D
Unique Solutions

Solution

A linear system is said to be consistent if it has at least one solution; and is said to be inconsistent if it has no solution. have no solution, a unique solution, and infinitely many solutions, respectively.