# NCERT Solutions for Class 12 Chemistry

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# Determinants

Exercise 1

1.

2.

3.     If  then show that |2A| = 4|A|.

4.    If  then show that |3A| = 27|A|.

5.    Evaluate the determinants:

6.   If  find |A|.

7.   Find the value of x, if:

8.    If  then x is equal to:

(A) +

(B) ±6

(C) -6

(D) 0

Exercise 2

Using the properties of determinants and without expanding in Exercise 9 to 15, prove that:

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

Choose the correct answer in Exercises 23 and 24.

23.  Let A be a square matrix of order 3 x 3, then is equal to:

(A) k|A|

(B) k2|A|

(C) k3|A|

(D) 3k|A|

24.  Which is 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.

Exercise 3

25.  Find the area of the triangle with vertices at the points 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)

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

27.  Find values of 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)

28.  (i) Find the equation of the line joining (1, 2) and (3, 6) using determinants.

(ii) Find the equation of the line joining (3, 1) and (9, 3) using determinants.

29.  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

Exercise 4

30.  Write minors and cofactors of the elements of the following determinants:

31.  Write minors and cofactors of the elements of the following determinants:

32.  Using cofactors of elements of second row, evaluate:

33.  Using cofactors of elements of third column, evaluate:

34.  If  and Aij is cofactor of aij, then value of Δ is given by:

(A) a11A31 + a12A32 + a13A33

(B) a11A11 + a12A21 + a13A31

(C) a21A11 + a22A12 + a23A13

(D) a11A11 + a21A21 + a31A31

Exercise 5

Find adjoint of each of the matrices in Exercise 35 and 36.

35.

36.

Verify A (adj. A) = (adj. A) A = |A|I in Exercise 37 and 38.

37.

38.

Find the inverse of the matrix (if it exists) given in Exercise 39 to 45.

39.

40.

41.

42.

43.

44.

45.

46.  Let   and  verify that (AB)-1 = B-1A-1.

47.  If  show that A2 – 5A + 7I = 0. Hence find A-1.

48.  For the matrix  find numbers a and b such that A2 + aA + bI = 0.

49.  For the matrix  show that A3 – 6A2 + 5A + 11I = 0. Hence find A-1.

50.  If  verify that A3 – 6A2 + 9A + 4I = 0 and hence find A-1.

51.  Let A be a non-singular matrix of order 3 x 3. Then |adj.A| is equal to:

(A) |A|

(B) |A|2

(C) |A|3

(D) 3|A|

52.  If A is an invertible matrix of order 2, then det (A-1) is equal to:

(A) det A

(B)

(C) 1

(D) 0

Exercise 6

Examine the consistency of the system of equations in Exercises 53 to 55.

53.  x + 2y = 2;    2x + 3y = 3

54.  2x – y = 5;     x + y = 4

55.  x + 3y = 5;     2x + 6y = 8

Examine the consistency of the system of equations in Exercises 56 to 58.

56.  x + y + z = 1;    2x + 3y + 2z = 2;     ax + ay + 2az = 4

57.  3x – y – 2z = 2;     2y – z = -1;    3x – 5y = 3

58.  5x – y + 4z = 5;     2x + 3y + 5z = 2;   5x – 2y + 6z = -1

Solve the system of linear equations, using matrix method, in Exercise 59 to 62.

59.  5x + 2y = 4;     7x + 3y = 5

60.  2x – y = -2;     3x + 4y = 3

61.  4x – 3y = 3;     3x – 5y = 7

62.  5x + 2y = 3;    3x + 2y = 5

Solve the system of linear equations, using matrix method, in Exercise 63 to 66.

63.

64.  x – y + z = 4;   2x + y – 3z = 0;   x + y + z = 2

65.  2x + 3y + 3z = 5;  x – 2y + z = -4;   3x – y – 2z = 3

66.  x – y + 2z = 7;  3x + 4y – 5z = -5;  2x – y + 3z = 12

67.  If  find A. Using A solve the system of equations 2x – 3y + 5z = 11; 3x + 2y – 4z = -5; x + y – 2z = -3.

68.  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 2 kg rice is Rs 90. The cost of 6 kg onion, 2 k wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Exercise 7

69.  Prove that the determinant  is independent of θ.

70.  Without expanding the determinants, prove that:

71.  Evaluate:

72.  If a, b, c are real numbers and  Show that either a+b+c = 0 or a=b=c

73.  Solve the equation:

74.  Prove that:

75.  If

76.  Let  verify that:

77.  Evaluate:

78.  Evaluate:

Using properties of determinants in Exercises 80 to 84, prove that:

79.

80.

81.

82.

83.

84.  Solve the system of the following equations: (Using matrices):

Choose the correct answer in Exercise 86 to 88.

85.  If a, b, c are in A.P., then the determinant

(A) 0

(B) 1

(C) x

(D) 2x

86.  If x, y, z are non-zero real numbers, then the inverse of matrix  is:

87.  Let  where 0 ≤ θ ≤ 2π. Then:

(A) Det (A) = 0

(B) Det (A) ∈ (2, ∞)

(C) Det (A) ∈ (2, 4)

(D) Det (A) ∈ [2, 4]

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