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NCERT Solutions Class 12 Maths Chapter 4 DETERMINANTS

Ncert Solutions for class 12 subject Maths Chapter 4 DETERMINANTSin pdf Best Free NCERT Solutions for class 1 to 12 in pdf NCERT Solutions, cbse board, Maths, ncert Solutions for Class 12 Maths, class 12 Maths ncert solutions, DETERMINANTS, Class 12, ncert solutions chapter 4 DETERMINANTS, class 12 Maths, class 12 Maths ncert solutions, Maths ncert solutions class 12, Ncert Solutions Class 12 Mathematics Chapter 4 DETERMINANTS

Exercise 4.1 Chapter 4 Class 12 Maths DETERMINANTS Ncert Solutions

Question 1. Evaluate the determinants in Exercises 1 and 2.
ncert solutions for class 12 Maths DETERMINANTS
Answer:

Use formula
ncert solutions for class 12 Maths DETERMINANTS

= 2(βˆ’1) βˆ’ 4(βˆ’5) = βˆ’ 2 + 20 = 18

Question 2. Evaluate the determinants in Exercises 1 and 2.

ncert solutions for class 12 Maths DETERMINANTS
Answer
Use formula
ncert solutions for class 12 Maths DETERMINANTS
= (cos ΞΈ)(cos ΞΈ) βˆ’ (βˆ’sin ΞΈ)(sin ΞΈ) = cos2 ΞΈ+ sin2 ΞΈ = 1
(ii)
ncert solutions for class 12 Maths DETERMINANTS
Use formula
ncert solutions for class 12 Maths DETERMINANTS
= (x2 βˆ’ x + 1)(x + 1) βˆ’ (x βˆ’ 1)(x + 1)
= x3 βˆ’ x2 + x + x2 βˆ’ x + 1 βˆ’ (x2 βˆ’ 1)
= x3 + 1 βˆ’ x2 + 1
= x3 βˆ’ x2 + 2

Question 3. If ncert solutions for class 12 Maths DETERMINANTS , then show that | 2A | = 4 | A |
Answer:

LHS

|2A| = ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Use formula
ncert solutions for class 12 Maths DETERMINANTS

=2(4) – 4(8)
= 8 – 32
= - 24

R.H.S
4|A| = 4ncert solutions for class 12 Maths DETERMINANTS = 4(1 x 2 - 2 x 4 )
= 4(2 – 8)
= 4(-6)
= - 24
L.H.S = R.H.S
Hence proved

Question 4. Ifncert solutions for class 12 Maths DETERMINANTS, then show that | 3 A | = 27 | A |
Answer:

LHS
=>|3A| = ncert solutions for class 12 Maths DETERMINANTS
Use formula
ncert solutions for class 12 Maths DETERMINANTS
=>|3A| = 3(36 - 0) – 0(0 -0) + 3(0-0)
= 108
R.H.S
ncert solutions for class 12 Maths DETERMINANTS = 27{1(4 - 0) – 0(0 -0) + 1(0-0)}
= 27 x 4
= 108

LHS = RHS
Hence proved

Question 5. Evaluate the determinants
ncert solutions for class 12 Maths DETERMINANTS
ncert solutions for class 12 Maths DETERMINANTS
Solution: (i)

ncert solutions for class 12 Maths DETERMINANTS
ncert solutions for class 12 Maths DETERMINANTS

(ii)
ncert solutions for class 12 Maths DETERMINANTS
(III)
ncert solutions for class 12 Maths DETERMINANTS
(IV)
ncert solutions for class 12 Maths DETERMINANTS

Question 6. If ncert solutions for class 12 Maths DETERMINANTS, find | A |

Answer:
ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 7. Find values of x, ifncert solutions for class 12 Maths DETERMINANTS

Solution: (i)
2(1) - 5(4) = 2x(x) – 6(4)

  1. 2 - 20 = 2x2 – 24

- 18 = 2x2 – 24
- 18 + 24 = 2x2
6 = 2x2
3 = x2
ncert solutions for class 12 Maths DETERMINANTS

(ii)
2 (5) – 3(4) = x(5) – 2x(3)
10 – 12 = 5x – 6x
-2 = - x
x = 2

Question 8. Ifncert solutions for class 12 Maths DETERMINANTS , then x is equal to
(A) 6 (B) Β± 6 (C) – 6 (D) 0

Solution:

x(x) – 2(18) = 6(6) – 18(2)
X2 – 36 = 36 – 36
X2 = 36
x = Β±6
Hence option B is correct

Exercise 4.2 Chapter 4 Class 12 Maths DETERMINANTS Ncert Solutions

Question ncert solutions for class 12 Maths DETERMINANTS

Answer:
ncert solutions for class 12 Maths DETERMINANTS

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions]

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions
=> (b-a)(c-a)(c2 –b2 + ac –ab +a2 –a2)
=> (b-a)(c-a){(c –b)(c+b) + a(c –b)}
=>(b-a)(c-a)(c-b)(a+b+c)
=> (a-b)(b-c)(c-a)(a+b+c)

Hence proved

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

Question 15.Let A be a square matrix of order 3 Γ— 3, then | kA| is equal to
(A) k| A| (B) k2 | A| (C) k3 | A| (D) 3k | A |

Answer:

class 12 DETERMINANTS Maths ncert solutions
So option C is correct

Question 16. Which of the following is correct?
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of these

Answer:We can calculate the determinant of a square matrix only so that Determinant is a number associated to a square matrix. Option (C) is correct

Exercise 4.3 Chapter 4 Class 12 Maths DETERMINANTS Ncert Solutions

Question 1. Find area of the triangle with vertices at the point given in each of the following :
(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8) (iii) (–2, –3), (3, 2), (–1, –8)

Answer:

class 12 DETERMINANTS Maths ncert solutionsclass 12 DETERMINANTS Maths ncert solutionsclass 12 DETERMINANTS Maths ncert solutions

Question 2. Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Answer:

class 12 DETERMINANTS Maths ncert solutions

Question 3. Find values of k if area of triangle is 4 sq. units and vertices are
(i) (k, 0), (4, 0), (0, 2) (ii) (–2, 0), (0, 4), (0, k)

Answer:

class 12 DETERMINANTS Maths ncert solutions

Answer is k = 0 and 8

class 12 DETERMINANTS Maths ncert solutions

Question 4.
(i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

Answer:

class 12 DETERMINANTS Maths ncert solutionsclass 12 DETERMINANTS Maths ncert solutions

Question 5. If area of triangle is 35 sq units with vertices (2, – 6), (5, 4) and (k, 4). Then k is
(A) 12 (B) –2 (C) –12, –2 (D) 12, –2

Answer:

class 12 DETERMINANTS Maths ncert solutions

Exercise 4.4 Chapter 4 Class 12 Maths DETERMINANTS Ncert Solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

Answer part (ii)

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:Value of determinant is always equal to sum of product of rows or columns with their corresponding cofactor
In option D there is first column operation

So D is correct option

Exercise 4.5 Chapter 4 Class 12 Maths DETERMINANTS Ncert Solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutionsclass 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

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class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutionsVerify that (AB)–1 = B–1 A–1.

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Hence (AB)–1 = B–1 A–1

Question class 12 DETERMINANTS Maths ncert solutionsshow that A2 – 5A + 7I = O. Hence find A–1.

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Question 14. For the matrix class 12 DETERMINANTS Maths ncert solutionsfind the numbers a and b such that A2 + aA + bI = O.

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Question 15. For the matrix class 12 DETERMINANTS Maths ncert solutionsShow that A3– 6A2 + 5A + 11 I = O. Hence, find A–1.

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Question class 12 DETERMINANTS Maths ncert solutions

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Question 17. Let A be a nonsingular square matrix of order 3 Γ— 3. Then |adj A| is equal to
(A) |A| (B) |A|2 (C) |A|3 (D) 3|A|

Answer:We have the formula
Use property of determinant A.adj A = AI
Take mode both sides we get
|A.adj A| = |AI|
As A is matrix of 3x3 hence
|AI| = A3I
Hence
|A| |adj A| = |A|3
ide by |A| both side we get
|adj A| = |A2|
Hence option B is correct

Question 18. If A is an invertible matrix of order 2, then det (A–1) is equal to
(A) det (A) (B)1/det (A) (C) 1 (D) 0

Answer:We have the formula
AA-1 = I
Take determinant both side we get
|A ||A-1| = 1
Divide by |A| both side we get
|A-1| = 1/|A |
Hence option B is correct

Exercise 4.6 Chapter 4 Class 12 Maths DETERMINANTS Ncert Solutions

Question 1. x + 2y = 2 and 2x + 3y = 3

Answer:

class 12 DETERMINANTS Maths ncert solutions

Equations has unique solutions. Hence equations are consistence

Question 2. 2x – y = 5 and x + y = 4

Answer:

class 12 DETERMINANTS Maths ncert solutions

Equations has unique solutions. Hence equations are consistence

Question 3. x + 3y = 5 and 2x + 6y = 8

Answer:

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Hence equations are consistence

Question 4. x + y + z = 1 , 2x + 3y + 2z = 2 and ax + ay + 2az = 4

Answer:

Equation third is ided by a we get
x + y + z = 1
2x + 3y + 2z = 2
x + y + 2z = 4/a

class 12 DETERMINANTS Maths ncert solutions

class 12 DETERMINANTS Maths ncert solutions

Hence |A| β‰ 0
So Equations has unique solutions. Hence equations are consistence

Question 3x–y – 2z = 2, 2y – z =-1 and –3x – 5y = 3

Answer:

ncert solutions for class 12 Maths DETERMINANTS
ncert solutions for class 12 Maths DETERMINANTS
ncert solutions for class 12 Maths DETERMINANTS
Multiply both matrices we get

class 12 DETERMINANTS Maths ncert solutions
class 12 DETERMINANTS Maths ncert solutions
Hence it is inconsistence

Question 6. 5x – y + 4z = 5,2x + 3y + 5z = 2 and 5x – 2y + 6z = –1

Answer:

ncert solutions for class 12 Maths DETERMINANTS
ncert solutions for class 12 Maths DETERMINANTS
|A|β‰ 0
Hence equations are consistence

Question 7. 5x + 2y = 4 and 7x + 3y = 5

Answer:

ncert solutions for class 12 Maths DETERMINANTS

Hence answer is x =2 and y = -3

Question 8. 2x – y = –2 and 3x + 4y = 3

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 9. 4x – 3y = 3 and 3x – 5y = 7

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 10. 5x + 2y = 3 and 3x + 2y = 5

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 11. 2x + y + z = 1, x – 2y – z =3/2 and 3y – 5z = 9

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 12. x – y + z = 4, 2x + y – 3z = 0 and x + y + z = 2

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 13. 2x + 3y +3 z = 5, x – 2y + z = – 4 and 3x – y – 2z = 3

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 14. x – y + 2z = 7,3x + 4y – 5z = – 5 and 2x – y + 3z = 12

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 15. Ifncert solutions for class 12 Maths DETERMINANTS , find A–1. Using A–1 solve the system of equations 2x – 3y + 5z = 11 3x + 2y – 4z = – 5 x + y – 2z = – 3

Answer:

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

Question 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Answer:

Let cost of each Kg of onion = Rs x
Cost of each Kg of wheat = Rs y
And cost of each Kg of rice = Rs z
Given that cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60
So
4x + 3y + 2 z = 60 …(1)
Given that The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90
So
2x + 4y + 6z = 90
Divide equation by 2 we get
X + 2y + 3z = 45 … (2)
Given that The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70.
So
6x + 2y + 3z = 70 … (3)

So our set of equation is
4x + 3y + 2 z = 60
2x + 4y + 6z = 90
6x + 2y + 3z = 70

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

ncert solutions for class 12 Maths DETERMINANTS

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