CBSE Class 12 Maths Chapter-4 Important Questions – Free PDF Download
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CBSE Class 12 Mathematics Important Questions Chapter 4 – Determinants
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. (x2 – x + 1) (x + 1) – (x + 1) (x – 1)
= x3 – x2 + x + x2 – x + 1 – (x2 – 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 R3 are identical]
8. Show that 
Ans. 
9. Find value of x, if 
Ans. (2 – 20) = (2x2 – 24)
-18 = 2×2 – 24
-2x2 = -24 + 18
-2x2 = 6
2x2 = 6
x2 = 3
10. Find adj A for 
Ans. adJ A = 
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
A2 – 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
