# Important Questions for CBSE Class 12 Maths Chapter 4 - Determinants

## CBSE Class 12 Maths Chapter-4 Important Questions – Free PDF Download

Free PDF download of Important Questions for CBSE Class 12 Maths Chapter 4 – Determinants prepared by expert Maths teachers from latest edition of CBSE(NCERT) books, On CoolGyan.Org to score more marks in CBSE board examination.

## 1 Mark Questions

1.Find values of x for which .
Ans. (3 – x)2 = 3 – 8
3 – x2 = 3 – 8
-x2 = -8

2. A be a square matrix of order 3 3, there is equal to
Ans.
N=3

3. Evaluate
Ans.

4. Let find all the possible value of x and y if x and y are natural numbers.
Ans. 4 – xy = 4 -8
xy = 8
of x = 1 x = 4 x = 8
y = 8 y =1 y = 1

5. Solve
Ans. (x– x + 1) (x + 1) – (x + 1) (x – 1)
= x3 – x+ x + x2 – x + 1 – (x– 1)
= x3 + 1 – x2 + 1
= x3 – x2 + x2

6. Find minors and cofactors of all the elements of the det.
Ans.

7. Evaluate
Ans.

[R1 and Rare identical]

8. Show that
Ans.

9. Find value of x, if
Ans. (2 – 20) = (2x– 24)
-18 = 2×2 – 24
-2x2 = -24 + 18
-2x2 = 6
2x2 = 6
x2 = 3

11. Without expanding, prove that
Ans.

12. If matrix is singular, find x.
Ans. For singular |A| = 0
1(-6 -2) + 2(-3 -x) + 3 (2 -2x) = 0
-8 – 6 – 2x + 6 – 6x = 0
-8x = + 8
x = -1

13. Show that, using properties if det.
Ans.

Taking (1 – x) common from R1 and R2

Expending along C1

14. If than x is equal to
Ans. x2 – 36 = 36 – 36
x2 = 36

15. is singular or not
Ans.
= 8 – 8
= 0
Hence A is singular

16. Without expanding, prove that

Ans.

Hence Prove

17. Verify that det A = det
Ans.

Hence prove.

18. If then show that
Ans.

Hence Prove

19. A be a non – singular square matrix of order 3 3. Then is equal to
Ans.
N=3

20. If A is an invertible matrix of order 2, then det is equal (A-1) to
Ans. A is invertible AA-1 =
det (AA-1) = det (I)
det A.(det A-1) = det ()
det A-1 =

21.
Ans.

22. Show that using properties of det.
Ans.

## 4 Marks Questions

1. Show that, using properties of determinants.

OR

Ans. Multiplying R1 R2 and R3 by a, b, c respectively

Taking a, b, c, common from c1, c2, and c3

Expending along R1

OR {solve it}
{hint : }
Taking common 3 (a+b) from C1

2.
Ans.

Taking (x + y + z) common from c2 and C3

Expending along R1

3. Find the equation of line joining (3, 1) and (9, 3) using determinants.
Ans. Let (x, y) be any point on the line containing (3, 1) and (9, 3)

x-3y=0

4. If
then verify that (AB)-1 = B-1 A-1
Ans.

Hence prove.

5. Using cofactors of elements of third column, evaluate
Ans.

6. If
find A-1, using A-1 solve the system of equations
2x – 3y + 5z = 11
3x + 2y – 4z = -5
x + y -2z = -3
Ans.

The given system of equation can be written is Ax = B, X = A-1B

7. Show that, using properties of determinants.

Ans.

Taking common (1 + a2 + b2) from R1

Taking (1 + a2 + b2) common from R2

Expending entry R1

8.
Ans.

9. Verify that

Ans.

=2 (-12) + (-3) (22) +5 (18)
= 0 Hence prove.

10.If, find matrix B such that AB = I
Ans.
Therefore A-1 exists
AB = I
A-1 AB = A-1I
B = A-1

11. Using matrices solve the following system of equation

Ans. Let
24 + 3v + 10v = 4
44 – 64 + 5w = 1
64 + 9v – 20w = 2

12.Given

find AB and use this result in solving the following system of equation.

OR
Use product

To solve the system of equations.
x – y + 2z = 1
2y – 3z = 1
3x – 2y + 4z = 2
Ans.

Let

OR

x = 0 y = 5 z = 3

13. If a, b, c is in A.P, and then finds the value of
Ans.

14.
Find the no. a and b such that A2 + aA + bI = 0 Hence find A-1
Ans.

a = -4, b =1
A– 4A + I = 0
A2 – 4A = -I
AAA-1 – 4AA-1 = -IA-1
A – 4I = -A-1
A-1 = 4I – A

15. Find the area of whose vertices are (3, 8) (-4, 2) and (5, 1)
Ans.

16. Evaluate
Ans.

17. Solve by matrix method
x – y + z = 4
2x + y – 3 z = 0
x + y + z = 2
Ans.

System of equation can be written is

18. Show that using properties of det.

Ans. Taking a, b, c common from R1, R2 and R3

Expending along R1

19. If x, y, z are different and then show that 1 + xyz = 0 ans.
Ans:

x, y, z all are different

20. Find the equation of the line joining A (1, 30 and B (0, 0) using det. Find K if D (K, 0) is a point such then area of ABC is 3 square unit
Ans. Let P (x, y) be any point on AB. Then area of ABP is zero

Area ABD =3 square unit

21. Show that the matrix satisfies the equation A2 – 4A + I = 0. Using this equation, find A-1
Ans.

22. Solve by matrix method.
3x – 2y + 3z = 8
2x + y – z = 1
4x – 3y + 2z = 4
Ans. The system of equation be written in the form AX = B, whose

23. The sum of three no. is 6. If we multiply third no. by 3 and add second no. to it, we get II. By adding first and third no. we get double of the second no. represent it algebraically and find the no. using matrix method.
Ans. I = x II = y II = z
x + y + z = 6
y + 3z = 11
x + z = 2y
This system can be written as AX = B whose

24.
Ans.

Expending along R1

25. Find values of K if area of triangle is 35 square. Unit and vertices are (2, -6), (5, 4), (K, 4)
Ans.

26. Using cofactors of elements of second row, evaluate
Ans.

27. If Show that A2 – 5A + 7I = 0. Hence find A-1
Ans.

Prove.
A2 – 5A + 7I = 0 (given)
A2 – 5A = -7I
A2A-1-5AA-1 = -7IA-1
AAA-1 – 5AA-1 = -7IA-1
A – 5I = -7A-1
7A-1 = 5I – A

28. The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
Ans. cost of 1kg onion = x
cost of 1kg wheat = y
cost of 1kg rise = z
4x + 3y + 2z = 60
2x + 4y + 6z = 90
6x + 2y + 3z = 70