Determinant of Third Order Class 12 OP Malhotra Exe-5B ISC Maths Solutions Ch-5. In this article you would learn how to find the value of third order Determinant Questions / Problems with solutions / answer using formula. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.
Determinant of Third Order Class 12 OP Malhotra Exe-5B ISC Maths Solutions Ch-5
| Board | ISC |
| Publications | S Chand |
| Subject | Maths |
| Class | 12th |
| Chapter-5 | Determinants |
| Writer | OP Malhotra |
| Exe-5(B) | Determinant of Third Order and its Value |
How to Find Value of Third Order Determinant
Class 12 OP Malhotra Exe-5B ISC Maths Solutions Ch-5
Que-1: Write minors and cofactors of elements of determinant ……………
Sol: Let |A|
∴ M11 = d ; M12 = b;
M21 = c & M22 = a
Thus factors are :
A11 = (-1)1+1 M11 = M11 = d ;
A12 = (-1)1+2 M12 = – M12 = – b ;
A21 = (-1)2+1 M21 = – M21 = – c
A22 = (-1)2+2 M22 = M22 = a
Que-2: Write down the minors of – 2 and 4 in …………..
Sol: Minor of – 2 = M22 = 8 – 3 = 5
and Minor of 4 = M33 = 2 x (- 2) – 1 x 1
= – 4 – 1 = – 5
Que-3: Write down the co-factors of 3 and – 2 of the determinant ……………..
Sol:
Que-4: (i) Write the co-factors of elements of the second row of the determinant …………………
(ii) Using the cofactors of elements of second row evaluate ∆ = …………………
Sol:
(ii)
Que-5: Write the minors and co-factors of each element of the first column of the following determinants and evaluate the determinant in each case.
(i) ……………….. (iii)
Sol: The element of C1 are 5 & 0
i.e. C11 = 5 ; C21 = 0
∴ M11 = – 1 ; M21 = 20
Thus, A11 = (- 1)1+1 M11 = M11 = – 1 ;
A21 = (- 1)2+1 M21 = – M21 = – 20
∴ ∆ = C11A11 + C21A21
= 5 x (- 1) + 0 x (- 20) = – 5
(ii) The element of C11 are ; a11 = 1 ; a21= 1 ; a21 = 1;
M11 = 1/2 = ab² – c²a = a(b² – c²)
M21 = 1/2 = a²b – bc² = b(a² – c²)
M31 = 1/2 = a²c – b²c = c(a² – b²)
∴ Cofactors of first column are ;
A11 = (- 1)1+1 M11 = M11 = a(b² – c²)
A21 = (- 1)2+1 M21 = – M21 = – b(a² – c²)
A31 = (- 1)3+1 M31 = M31 = c(a² – b²)
∴ Value of determinant
= a11A11 + a21A21 + a31A31
= a(b² – c²) – b(a² – c²) + c(a² – b²)
= ab{b – a) + bc(c – b) + ca(a – c)
(iii) The element of C1 are ; a11 = 0 ; a21= 1 ; a31 = 3;
The minors of C1 are ;
M11 = 5 – 0 = 5;
M21 = 2 – 42 = – 40
and M31 = 0 – 30 = – 30
The cofactors of C, are ;
A11 = (-1)1+1 M11 = + M11 = 5 ;
A31 = (-1)3+1 M31 = M31 = – 30 ;
A21 = (-1)2+1 M21 = – M21 = – 40
Thus, Value = a11A11 + a21A21 + a31A31
= 0 x 5 + 1 x 40 + 3 x (- 30) = – 50
Que-6: Evaluate ……………..
Sol: Let ∆ = ……………..; expanding along C3
= 0 + 1(10 + 3) + 0 (15 – 2) = 13
Que-7: Find the value of the determinants
(i)……….(ii)………….(iii)………….
Sol: (i) Let ∆ = ………………..;
expanding along R1
Que-8: Expand the determinants by minors of the given row or column
(i)…………………….(iii)
Sol:
Que-9: Solve for x :
(i) ……………………= 3
(ii) …………………..= 28
Sol: (i) Given ……………. = 3
expanding along R1, we have
⇒ x(4 – 3) – 0(8 – 0) + 0(2 – 0) = 3
⇒ – x = 3
(ii) Given ……………… = 28
expanding along C1
x² (8 – 1) – 0(4x – 1) + 3(x – 2) = 28
⇒ 7x² + 3x – 6 – 28 = 0
⇒ 7x² + 3x – 34 = 0
⇒ (x – 2) (7x + 17) = 0
⇒ x = 2, −17/7
Que-10: Show that………………..
Sol: L.H.S. = ……………….
expanding along R1
= 1(1 + c²) – a(- a + bc) + b(ac + b)
= 1 + c² + a² + b² = R.H.S
Que-11: Expand the simplify the following :
(i)…………………………(iii)
Sol:
Que-12: If one root of ………………… is x = – 9, find the other roots.
Sol: Given
expanding along R1
⇒ 7(7x – 6) – 6(14 – 2x) + x (6 – x²) = 0
⇒ 49x – 42 – 84 + 12x + 6x – x³ = 0
⇒ 67x – x³ – 126 = 0
⇒ x³ – 67x + 126 = 0
⇒ (x – 2)(x² + 2x – 63) = 0
⇒ (x – 2)(x – 7)(x + 9) = 0
⇒ x = 2, 7, – 9
Since one of the root of given eqn be x = – 9
Thus, the other two roots are 2 & 7
–: End of Determinant of Third Order Class 12 OP Malhotra Exe-5B ISC Maths Solutions :–
Return to :- OP Malhotra ISC Class-12 S Chand Publication Maths Solutions
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