Students can Download Maths Chapter 4 Determinants Ex 4.1 Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

## Karnataka 2nd PUC Maths Question Bank Chapter 4 Determinants Ex 4.1

### 2nd PUC Maths Determinants NCERT Text Book Questions and Answers Ex 4.1

Evaluate the determinations

Question 1.

\(\left|\begin{array}{rr}{2} & {4} \\{-5} & {-1}\end{array}\right|\)

Answer:

2 x (-1) -4x (-5)

= -2 + 20 = 18

Question 2.

(i)

\(\left|\begin{array}{rr}{\cos \theta} & {-\sin \theta} \\{\sin \theta} & {\cos \theta}\end{array}\right|\)

Answer:

Cos θ x Cos θ – sin θ x sin θ

= cos^{2} θ + sin^{2} θ = 1

(ii)

\(\left|\begin{array}{cc}{\mathbf{x}^{2}-\mathbf{x}+\mathbf{1}} & {\mathbf{x}-\mathbf{1}} \\{\mathbf{x}+\mathbf{1}} & {\mathbf{x}+\mathbf{1}}\end{array}\right|\)

Answer:

(x^{2} – x + 1) (x + 1) – (x – 1) (x + 1)

= x^{3} + 1 – (x^{2} – 1)

= x^{3} – x^{2} + 2

Question 3.

\(\text { If } \mathbf{A}=\left[\begin{array}{ll}{1} & {2} \\{4} & {2}\end{array}\right]\),then show that |2A| = 4 |A|

Answer:

Question 4.

\(\text { If } \mathbf{A}=\left[\begin{array}{lll}{\mathbf{1}} & {\mathbf{0}} & {\mathbf{1}} \\{\mathbf{0}} & {\mathbf{1}} & {\mathbf{2}} \\{\mathbf{0}} & {\mathbf{0}} & {\mathbf{4}}\end{array}\right]\) then show that |3A| = 27 |A|

Answer:

Question 5.

Evaluate the determinant

(i)

\(\left|\begin{array}{rrr}{3} & {-4} & {5} \\{1} & {1} & {-2} \\{2} & {3} &{1}\end{array}\right|\)

Answer:

Expanding along R_{2} (Second Row where two elements are zeroes)

(ii)

\(\left|\begin{array}{rrr}{3} & {-4} & {5} \\{1} & {1} & {-2} \\{2} & {3} & {1}\end{array}\right|\)

Answer:

Expanding along R_{1
}

(iii)

\(\left|\begin{array}{ccc}{0} & {1} & {2} \\{-1} & {0} & {-3} \\{-2} & {3} & {0}\end{array}\right|\)

Answer:

Expanding along R_{1
}

(iv)

\(\left|\begin{array}{ccc}{2} & {-1} & {-2} \\{0} & {2} & {-1} \\{3} & {-5} &{0}\end{array}\right|\)

Answer:

Expanding along R_{3
}

Question 6.

\(\text { If } A=\left[\begin{array}{ccc}{1} & {1} & {-2} \\{2} & {1} & {-3} \\{5} & {4} & {-9}\end{array}\right] \text { find }|A|\)

Answer:

Question 7.

find X if

(i)

\(\left|\begin{array}{cc}{2} & {4} \\{5} & {1}\end{array}\right|=\left|\begin{array}{cc}{2 x} & {4} \\{6} & {x}\end{array}\right|\)

Answer:

2 x 1 -4 x 5 = 2 × x × -4 × 6

2 – 20 = 2x^{2} – 24 – 18

= 2x^{2 }– 24

2x^{2} = 6,

x^{2} = 3

\(x=\pm \sqrt{3}\)

(ii)

\(\left|\begin{array}{cc}{2} & {3} \\{4} & {5}\end{array}\right|=\left|\begin{array}{cc}{x} & {3} \\{2 x} & {5}\end{array}\right|\)

2 x 5 -3 x 4=5 x x – 6x

10 – 12 = 5x – 6x

-2 =-x

x = 2

Question 8.

\(\text { If }\left|\begin{array}{cc}{x} & {2} \\{18} {x}\end{array}\right|=\left|\begin{array}{cc}{6} & {2} \\{18} & {6}\end{array}\right|\)

(A) 6

(B) ± 6

(C) -6

(D) 0

Answer:

2 x 18 = 6 x 6-18 x 2

x^{2 }– 36 = 36 – 36

x^{2} – 36 = 0

x^{2} = 36

x= ±6

hence B is the correct answer.

Note: Two determinants ms equal does not mean corresponding elements are equal so do not equate the elements like we are doing in matrix.