Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 7 Matrices and Determinants Ex 7.4 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 7 Matrices and Determinants Ex 7.4

Question 1.
Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3)
Answer:
The given points are (0, 0), (1, 2) and (4, 3)
Area of the triangle with vertices
(x1, y1), (x2, y2) and (x3, y3) is
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 1
∴ The area of the triangle with vertices
(0, 0), (1, 2) and (4, 3) is
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 2
Area cannot be negative. Taking positive value, we have
Required area Δ = \(\frac{5}{2}\) sq.units.

Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4

Question 2.
If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.
Answer:
Given Area of the triangle with vertices (k, 2), (2, 4) and (3, 2) is 4 square units.
The area of the triangle with vertices
(x1, y1) , (x2, y2) and (x3, y3) is
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 3
Given Δ = 4, (x1, y1) = (k , 2), (x2, y2) = (2 , 4) and (x3, y3) = (3 , 2)
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 4
± 4 = k(4 – 2) – 2 (2 – 3) + 1(4 – 12)
± 4 = k × 2 – 2 × – 1 – 8
± 4 = 2k + 2 – 8
± 4 = 2k – 6
2k – 6 = 4 or 2k – 6 = -4
2k = 4 + 6 or 2k = – 4 + 6
2k = 10 or 2k = 2
k = 5 or k = 1
Required values of k are 1, 5.

Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4

Question 3.
Identify the singular and non – singular matrices.
(i) \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right] \)
(ii) \(\left[ \begin{matrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{matrix} \right] \)
(iii) \(\left[ \begin{matrix} 0 & a\quad -\quad b & k \\ b-\quad a & 0 & 5 \\ -k & -5 & 0 \end{matrix} \right] \)
Answer:
(i) \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right] \)
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 5
|A| = 1 (45 – 48) – 2(36 – 42) + 3(32 – 35)
|Al = – 3 – 2 × – 6 + 3 × – 3
|A| = – 3 + 12 – 9
|A| = – 12 + 12 = 0
∴ A is a singular matrix.

Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4

(ii) \(\left[ \begin{matrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{matrix} \right] \)
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 6
|B| = 2(0 – 20) + 3 (- 42 – 4) + 5(30 – 0)
|B| = -40 + 3 × – 46 + 150
|B| = -40 – 138 + 150
|B| = -178 + 150 ≠ 0
∴ B is non singular.

(iii) \(\left[ \begin{matrix} 0 & a\quad -\quad b & k \\ b-\quad a & 0 & 5 \\ -k & -5 & 0 \end{matrix} \right] \)
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 7
|C| = 0 – (a – b) (0 + 5k) + k(-5 (b – a) – 0)
|C| = -5k (a – b) – 5k (b – a)
|C| = -5k (a – b) + 5k(a – b)
|C| = o
∴ C is a singular matrix.

Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4

Question 4.
Determine the values of a and b so that the following matrices are singular:
(i) A = \(\left[ \begin{matrix} 7 & 3 \\ -2 & a \end{matrix} \right] \)
(ii) B = \(\left[ \begin{matrix} b\quad -\quad 1 & 2 & 3 \\ 3 & 1 & 2 \\ 1 & -2 & 4 \end{matrix} \right] \)
Answer:
(i) A = \(\left[ \begin{matrix} 7 & 3 \\ -2 & a \end{matrix} \right] \)
|A| = \(\left[ \begin{matrix} 7 & 3 \\ -2 & a \end{matrix} \right] \)
|A| = 7a + 6
Given that A is singular
∴ |A| = 0
7a + 6 = 0 ⇒ a = \(\frac{-6}{7}\)

(ii) B = \(\left[ \begin{matrix} b\quad -\quad 1 & 2 & 3 \\ 3 & 1 & 2 \\ 1 & -2 & 4 \end{matrix} \right] \)
|B| = \(\left[ \begin{matrix} b\quad -\quad 1 & 2 & 3 \\ 3 & 1 & 2 \\ 1 & -2 & 4 \end{matrix} \right] \)
= (b – 1 )(4 + 4) – 2(12 – 2) + 3(- 6 – 1)
= 8 (b – 1) – 20 – 21
= 8b – 8 – 41
|B| = 8b -49
Given that B is singular
∴ |B| = 0
8b – 49 = 0 ⇒ b = \(\frac{49}{8}\)

Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4

Question 5.
If cos 2θ = 0, determine
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 8
Answer:
Given cos 2θ = 0
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 9

Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4

Question 6.
Find the value of the product
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 10
Answer:
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 11
Samacheer Kalvi 11th Maths Guide Chapter 7 Matrices and Determinants Ex 7.4 12

Leave a Reply