Chapter 4 Determinants | class 12th | Important Question for Maths

Class 12 Mathematics Important Questions Chapter 4 – Determinants

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1 Mark Questions

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

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

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3. Evaluate 
Ans. 

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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

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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

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

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7. Evaluate 
Ans. 

[R₁ and R₃ are identical]

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8. Show that 
Ans. 

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

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10. Find adj A for 
Ans. adJ A = 

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11. Without expanding, prove that 
Ans. 

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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

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





Taking (1 – x) common from R₁ and R₂

Expending along C¹

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

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

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16. Without expanding, prove that

Ans. 


Hence Prove

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17. Verify that det A = det 
Ans. 



Hence prove.

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18. If then show that 
Ans. 
 
 

Hence Prove

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

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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 = 

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21. 
Ans. 

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22. Show that using properties of det.
Ans. 

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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

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2. 
Ans. 




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




Expending along R₁

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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

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







Hence prove.

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

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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



 

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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

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8. 
Ans. 

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9. Verify that

Ans. 


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

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

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11. Using matrices solve the following system of equation



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

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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

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

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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
= 

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

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16. Evaluate 
Ans. 

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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

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18. Show that using properties of det. 

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




 


Expending along R₁

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





x, y, z all are different

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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

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

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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

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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

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24. 
Ans. 





Expending along R₁

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

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

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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

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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