NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants

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The 4th Chapter of NCERT Exemplar Solutions for Class 12 Mathematics is Determinants. This chapter covers topics such as introduction to determinants, then a determinant matrix of order one, two and three, properties of the determinant, minors and cofactor, adjoint and the inverse of a matrix, applications of matrices and solution of the system of linear equations using the inverse of a matrix. Further, students can make use of the solutions PDF ofNCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants from the link given below to learn and understand all the concepts of determinants in an easy manner.

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Access answers to NCERT Exemplar Solutions For Class 12 Maths Chapter 4 Determinants

Exercise 4.3 Page No: 77

Short Answer (S.A.)

Using the properties of determinants in Exercises 1 to 6, evaluate:

1.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 1

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 2

= (x2 – 2x + 2) . (x + 1) – (x – 1) . 0

= x3 – 2x2 + 2x + x2 – 2x + 2

= x3 – x2 + 2

2.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 3

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 4

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 5

3.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 6

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 7

4.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 8

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 9

[Expanding along first column]

= (x + y + z) . 1[3y(3z + x) + (3z)(x – y)]

= (x + y + z)(3yz + 3yx + 3xz)

= 3(x + y + z)(xy + yz + zx)

5.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 10

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 11

6.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 12

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 13

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 14

Lastly, expanding along R1, we have

= (a + b + c) [1 x 0 + (a + b + c)2]

= (a + b + c)3

Using the properties of determinants in Exercises 7 to 9, prove that:

7.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 15

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 16

8.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 17

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 18

9.
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 19

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 20

10. If A + B + C = 0, then prove that
NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 21

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 22

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 23

11. If the co-ordinates of the vertices of an equilateral triangle with sides of length ‘a’ are (x1, y1),

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 24

(x2, y2), (x3, y3), then

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 25

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 26

12. Find the value of q satisfying

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 27

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 28

13. If , then find values of x.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 29

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 30

14. If a1, a2, a3, …, ar are in G.P., then prove that the determinant

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 31

is independent of r.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 32

Hence, the determinant is independent of r.

15. Show that the points (a + 5, a – 4), (a – 2, a + 3) and (a, a) do not lie on a straight line for any value of a.

Solution:

Given points are (a + 5, a – 4), (a – 2, a + 3) and (a, a).

Now, we have to prove that these points do not lie on a straight line.

So, if we prove that these points form a triangle then it can’t line on a straight line.

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 33

Hence, the given points form a triangle and can’t lie on a straight line.

16. Show that the DABC is an isosceles triangle if the determinant

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 34

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 35

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 36

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 37

17. Find A–1 if and show that A-1 = (A2 – 3I)/ 2.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 38

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 39

Long Answer (L.A.)

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 40

18. If A = , find A-1.

Using A–1, solve the system of linear equations

x – 2y = 10 , 2x y z = 8 , –2y + z = 7.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 41

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 42

19. Using matrix method, solve the system of equations

3x + 2y – 2z = 3, x + 2y + 3z = 6, 2x y + z = 2.

Solution:

Given system of equations are:

3x + 2y – 2z = 3

x + 2y + 3z = 6 and

2x y + z = 2

Or,

AX = B

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 43

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 44

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 45

20. Given find BA and use this to solve the system of

equations y + 2z = 7, x y = 3, 2x + 3y + 4z = 17.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 46

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 47

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 48

21. If a + b + c ¹ 0 and then prove that a = b = c.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 49

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 50

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 51

22. Prove that is divisible by a + b + c and find the quotient.

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 52

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 53

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 54

23. If x + y + z = 0, prove that

Solution:

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 55

Objective Type Questions (M.C.Q.)

Choose the correct answer from given four options in each of the Exercises from 24 to 37.

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 56

24. If then, value of x is

(A) 3 (B) ± 3 (C) ± 6 (D) 6

Solution:

Option (C) ± 6

From the given,

On equating the determinants, we have

2x2 – 40 = 18 + 14

2x2 = 72

x2 = 36

Thus, x = ± 6

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 57

25. The value of determinant

(A) a3 + b3 + c3 (B) 3 bc (C) a3 + b3 + c3 – 3abc (D) none of these

Solution:

Option (C) a3 + b3 + c3 – 3abc

Given,

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 58

26. The area of a triangle with vertices (–3, 0), (3, 0) and (0, k) is 9 sq. units. The value of k will be

(A) 9 (B) 3 (C) – 9 (D) 6

Solution:

Option (B) 3

We know that, the area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 59

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 60

27. The determinant equals

(A) abc (bc) (c a) (a b) (B) (bc) (c a) (a b)

(C) (a + b + c) (b c) (c a) (a b) (D) None of these

Solution:

Option (D)

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 61

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 62

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 63

28. The number of distinct real roots of = 0 in the interval -π/4 ≤ x ≤ π/4 is

(A) 0 (B) 2 (C) 1 (D) 3

Solution:

Option (C) 1

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 64

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 65

29. If A, B and C are angles of a triangle, then the determinant is equal to

(A) 0 (B) -1 (C) 1 (D) None of these

Solution:

Option (C) 0

NCERT Exemplar Solutions Class 12 Mathematics Chapter 4 - 66

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