NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants

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The 4^th Chapter of NCERT Exemplar Solutions for Class 12 Mathematics is Determinants. This chapter covers topics such as introduction to determinants, 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 of NCERT Exemplar Solutions for Class 12 Maths Chapter 4 Determinants from the links given below to learn and understand the concepts of determinants in an easy manner.

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Exercise 4.3 Page No: 77

Short Answer (S.A.)

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

1.

Solution:

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

= x³ – 2x² + 2x + x² – 2x + 2

= x³ – x² + 2

2.

Solution:

3.

Solution:

4.

Solution:

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

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

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

5.

Solution:

6.

Solution:

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.

Solution:

8.

Solution:

9.

Solution:

10. If A + B + C = 0, then prove that

Solution:

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

(x₂, y₂), (x₃, y₃), then

Solution:

12. Find the value of q satisfying

Solution:

13. If , then find values of x.

Solution:

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

is independent of r.

Solution:

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.

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

Solution:

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

Solution:

Long Answer (L.A.)

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:

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

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

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

Solution:

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

Solution:

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

Solution:

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

Solution:

Objective Type Questions (M.C.Q.)

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

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

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,

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

27. The determinant equals

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

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

Solution:

Option (D)

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

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 (A) 0