DETERMINANTS questions

S.No.	DETERMINANTS
	EXERCISE 4.1
1-2	Evaluate the determinants in Exercises 1 and 2.
3	If A = ,then show that | 2A | = 4 | A |
4	If A = then show that | 3 A | = 27 | A |
5	Evaluate the determinants
6	If A =find | A |
7	Find values of x, if
8	If, then x is equal to (A) 6 (B) ± 6 (C) – 6 (D) 0
	EXERCISE 4.2
1-7	Using the property of determinants and without expanding in Exercises 1 to 7, provethat: Answer 1- 4  Answer 4- 7
8-14	By using properties of determinants, in Exercises 8 to 14, show that: Answer 8 - 10  Answer 11 - 12 Answer 13- 14
15	Let A be a square matrix of order 3 × 3, then | kA| is equal to (A) k| A| (B) k2 | A| (C) k3 | A| (D) 3k | A |
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
	EXERCISE 4.3
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)
2	Show that pointsA (a, b + c), B (b, c + a), C (c, a + b) are collinear.
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)
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.
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
	EXERCISE 4.4
1-2	Write Minors and Cofactors of the elements of following determinants:
3	Using Cofactors of elements of second row, evaluate Δ =
4	Using Cofactors of elements of third column, evaluate Δ
5	If Δ =and Aij is Cofactors of aij, then value of Δ is given by (A) a11 A31+ a12 A32 + a13 A33 (B) a11 A11+ a12 A21 + a13 A31 (C) a21 A11+ a22 A12 + a23 A13 (D) a11 A11+ a21 A21 + a31 A31
	EXERCISE 4.5
1-2	Find adjoint of each of the matrices in Exercises 1 and 2.
3-4	Verify A (adj A) = (adj A) A = |A| I in Exercises 3 and 4
5-11	Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
12	Let A = and B = Verify that (AB)–1 = B–1 A–1.
13	If A =, show that A2 – 5A + 7I = O. Hence find A–1.
14	For the matrix A =, find the numbers a and b such that A2 + aA + bI = O.
15	For the matrix A = Show that A3– 6A2 + 5A + 11 I = O. Hence, find A–1.
16	If A =Verify that A3 – 6A2 + 9A – 4I = O and hence find A–1
17	Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to (A) |A| (B) |A|2 (C) |A|3 (D) 3|A|
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
	EXERCISE 4.6
1-6	Examine the consistency of the system of equations in Exercises 1 to 6.
7-14	Solve system of linear equations, using matrix method, in Exercises 7 to 14. Question 7                    Question 8                         Question 9  Question 10                  Question 11                       Question 12  Question 13                  Question 14
15	If A =find A–1. Using A–1 solve the system of equations 2x – 3y + 5z = 11, 3x + 2y – 4z = – 5, x + y – 2z = – 3
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.
	Miscellaneous Exercises on Chapter 4
1	Prove that the determinant is independent of θ.
2	Without expanding the determinant, prove that
3	Evaluate
4	If a, b and c are real numbers, and Δ = Show that either a + b + c = 0 or a = b = c.
5	Solve the equation
6	Prove that
7	If A–1 =and Bfind AB -1
8	Verify that (i) [adj A]–1 = adj (A–1) (ii) (A–1)–1 = A
11-15	Using properties of determinants in Exercises 11 to 15, prove that:
16	Solve the system of equations
17	If a, b, c, are in A.P, then the determinant
18	If x, y, z are nonzero real numbers, then the inverse of matrix
19	Let A = , where 0 ≤ θ ≤ 2π. Then