Fill in the blanks with correct number.
3245 = 3 × 51 − 2 × 4 = 15 − 8 = 7
Solution — Question 1
Step 1 : Use the determinant formula
For abcd, value = ad − bc.
3245 = 3 × 5 − 2 × 4
= 15 − 8
= 7
Therefore, the correct number is 7.
Find the values of following determinants.
(1) −1724
Solution — Q.2 (1)
Step 1 : Apply the formula
For abcd, D = ad − bc.
(−1 × 4) − (7 × 2) = −4 − 14 = −18
Therefore, the value of the determinant is −18.
(2) 53−70
Solution — Q.2 (2)
Step 1 : Apply the formula
For abcd, D = ad − bc.
(5 × 0) − (3 × (−7)) = 0 + 21 = 21
Therefore, the value of the determinant is 21.
(3) 73533212
Solution — Q.2 (3)
Step 1 : Apply the formula
For abcd, D = ad − bc.
73 × 12 − 53 × 32 = 76 − 156 = −86 = −43
Therefore, the value of the determinant is −43.
Solve the following simultaneous equations using Cramer’s rule.
(1) 3x − 4y = 10;4x + 3y = 5
Solution — Q.3 (1)
Step 1 : Write the equations in standard form
3x − 4y = 10
4x + 3y = 5
Step 2 : Find D
Formula: D = a₁b₁a₂b₂ = a₁ × b₂ − b₁ × a₂
3−443 = (3) × (3) − (−4) × (4)
= 9 − (−16)
= 25
Step 3 : Find Dx
Formula: Dx = c₁b₁c₂b₂ = c₁ × b₂ − b₁ × c₂
10−453 = (10) × (3) − (−4) × (5)
= 30 − (−20)
= 50
Step 4 : Find Dy
Formula: Dy = a₁c₁a₂c₂ = a₁ × c₂ − c₁ × a₂
31045 = (3) × (5) − (10) × (4)
= 15 − 40
= −25
Step 5 : Find x and y
Formula: x = DxD and y = DyD.
x = 5025 = 2
y = −2525 = −1
Hence, the solution is x = 2 and y = −1.
(2) 4x + 3y − 4 = 0;6x = 8 − 5y
Solution — Q.3 (2)
Step 1 : Write the equations in standard form
4x + 3y = 4
6x + 5y = 8
Step 2 : Find D
Formula: D = a₁b₁a₂b₂ = a₁ × b₂ − b₁ × a₂
4365 = (4) × (5) − (3) × (6)
= 20 − 18
= 2
Step 3 : Find Dx
Formula: Dx = c₁b₁c₂b₂ = c₁ × b₂ − b₁ × c₂
4385 = (4) × (5) − (3) × (8)
= 20 − 24
= −4
Step 4 : Find Dy
Formula: Dy = a₁c₁a₂c₂ = a₁ × c₂ − c₁ × a₂
4468 = (4) × (8) − (4) × (6)
= 32 − 24
= 8
Step 5 : Find x and y
Formula: x = DxD and y = DyD.
x = −42 = −2
y = 82 = 4
Hence, the solution is x = −2 and y = 4.
(3) x + 2y = −1;2x − 3y = 12
Solution — Q.3 (3)
Step 1 : Write the equations in standard form
x + 2y = −1
2x − 3y = 12
Step 2 : Find D
Formula: D = a₁b₁a₂b₂ = a₁ × b₂ − b₁ × a₂
122−3 = (1) × (−3) − (2) × (2)
= −3 − 4
= −7
Step 3 : Find Dx
Formula: Dx = c₁b₁c₂b₂ = c₁ × b₂ − b₁ × c₂
−1212−3 = (−1) × (−3) − (2) × (12)
= 3 − 24
= −21
Step 4 : Find Dy
Formula: Dy = a₁c₁a₂c₂ = a₁ × c₂ − c₁ × a₂
1−1212 = (1) × (12) − (−1) × (2)
= 12 − (−2)
= 14
Step 5 : Find x and y
Formula: x = DxD and y = DyD.
x = −21−7 = 3
y = 14−7 = −2
Hence, the solution is x = 3 and y = −2.
(4) 6x − 4y = −12;8x − 3y = −2
Solution — Q.3 (4)
Step 1 : Write the equations in standard form
6x − 4y = −12
8x − 3y = −2
Step 2 : Find D
Formula: D = a₁b₁a₂b₂ = a₁ × b₂ − b₁ × a₂
6−48−3 = (6) × (−3) − (−4) × (8)
= −18 − (−32)
= 14
Step 3 : Find Dx
Formula: Dx = c₁b₁c₂b₂ = c₁ × b₂ − b₁ × c₂
−12−4−2−3 = (−12) × (−3) − (−4) × (−2)
= 36 − 8
= 28
Step 4 : Find Dy
Formula: Dy = a₁c₁a₂c₂ = a₁ × c₂ − c₁ × a₂
6−128−2 = (6) × (−2) − (−12) × (8)
= −12 − (−96)
= 84
Step 5 : Find x and y
Formula: x = DxD and y = DyD.
x = 2814 = 2
y = 8414 = 6
Hence, the solution is x = 2 and y = 6.
(5) 4m + 6n = 54;3m + 2n = 28
Solution — Q.3 (5)
Step 1 : Write the equations in standard form
4m + 6n = 54
3m + 2n = 28
Step 2 : Find D
Formula: D = a₁b₁a₂b₂ = a₁ × b₂ − b₁ × a₂
4632 = (4) × (2) − (6) × (3)
= 8 − 18
= −10
Step 3 : Find Dx
Formula: Dx = c₁b₁c₂b₂ = c₁ × b₂ − b₁ × c₂
546282 = (54) × (2) − (6) × (28)
= 108 − 168
= −60
Step 4 : Find Dy
Formula: Dy = a₁c₁a₂c₂ = a₁ × c₂ − c₁ × a₂
454328 = (4) × (28) − (54) × (3)
= 112 − 162
= −50
Step 5 : Find m and n
Formula: m = DxD and n = DyD.
m = −60−10 = 6
n = −50−10 = 5
Hence, the solution is m = 6 and n = 5.
(6) 2x + 3y = 2;x − y2 = 12
Solution — Q.3 (6)
Step 1 : Write the equations in standard form
2x + 3y = 2
x − y2 = 12
Step 2 : Find D
Formula: D = a₁b₁a₂b₂ = a₁ × b₂ − b₁ × a₂
231−12 = (2) × (−12) − (3) × (1)
= −1 − 3
= −4
Step 3 : Find Dx
Formula: Dx = c₁b₁c₂b₂ = c₁ × b₂ − b₁ × c₂
2312−12 = (2) × (−12) − (3) × (12)
= −1 − 32
= −52
Step 4 : Find Dy
Formula: Dy = a₁c₁a₂c₂ = a₁ × c₂ − c₁ × a₂
22112 = (2) × (12) − (2) × (1)
= 1 − 2
= −1
Step 5 : Find x and y
Formula: x = DxD and y = DyD.
x = −52−4 = 58
y = −1−4 = 14
Hence, the solution is x = 58 and y = 14.