Please refer to Determinants Class 12 Mathematics Exam Questions provided below. These questions and answers for Class 12 Mathematics have been designed based on the past trend of questions and important topics in your Class 12 Mathematics books. You should go through all Class 12 Mathematics Important Questions provided by our teachers which will help you to get more marks in upcoming exams.

**Class 12 Mathematics** **Exam Questions Determinants**

Class 12 Mathematics students should read and understand the important questions and answers provided below for Determinants which will help them to understand all important and difficult topics.

**Very Short Answer Type Questions**

**Question. If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is**

A. 12 B. −2 C. −12, −2 D. 12, −2**Answer.** 12, -2

**Question. Find the inverse of the matrix**

**Answer.**

**Question. Write Minors and Cofactors of the elements of following determinant:**

** Answer.** By the definition of minors and cofactors, we have:

A_{11} = cofactor of a_{11}= (−1)^{1+1}M_{11}= 1

A_{12} = cofactor of a_{12}= (−1)^{1+2} M_{12}= 0

A_{13} = cofactor of a_{13}= (−1)^{1+3} M_{13}= 0

A_{21} = cofactor of a_{21}= (−1)^{2+1} M_{21}= 0

A_{22} = cofactor of a_{22}= (−1)^{2+2} M_{22}= 1

A_{23 }= cofactor of a_{23}= (−1)^{2+3} M_{23}= 0

A_{31} = cofactor of a_{31}= (−1)^{3+1} M_{31}= 0

A_{32 }= cofactor of a_{32}= (−1)^{3+2} M_{32}= 0

A_{33 }= cofactor of a_{33}= (−1)^{3+3 }M_{33} = 1

**Question.**

**Question. Evaluate the determinant**

**Answer.**

= (x^{2} − x + 1)(x + 1) − (x − 1)(x + 1)

= x^{3} − x^{2} + x + x^{2} − x + 1 − (x^{2} − 1)

= x^{3} + 1 − x^{2} + 1

= x^{3} − x^{2} + 2

**Question. If**

**Answer.** The given matrix is

It can be observed that in the first column, two entries are zero. Thus, we expand along the first column (C1) for easier calculation

**Question. Solve the system of linear equations, using matrix method .5x+2y=47x+3y=5Answer. **The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.

**Question. Find the inverse of the matrix**

**Answer.**

**Question. Find the area of the angle with vertices at the point given:(1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)Answer. **The area of the triangle with vertices (1, 0), (6, 0), (4, 3) is given by there elation,

(i)

(ii)

**Question. Find adjoint of each of the matrices:**

**Answer.**

**Question. **Write the value of the determinant

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question. **Write the value of the following determinant :

**Answer.**

**Question.**

**Answer.**

**Short Answer Type Questions** **-II**

**Question.** By using properties of determinants, prove the**following :**

**Answer.**

**Question.****The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method.****Answer.** The monthly incomes of Aryan and Babban are ₹ 3x and ₹ 4x respectively and the monthly expenditures of Aryan and Babban are ₹ 5y and ₹ 7y respectively

∴ ₹ 3x – 5y = 15000

4x – 7y = 15000

Writing in matrix form

The monthly income of Aryan = ₹ 3x = ₹ 90,000

The monthly income of Babban = ₹ 4x = ₹ 120,000

**Question.** A trust invested some money in two types of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ₹ 2,800 as interest. However, if trust had interchanged money in bonds, they would have got ₹ 100 less as interest. Using matrix method, find the amount invested by the trust.**Answer.** Let ₹ x be invested in first bond (10%)

Let ₹ y be invested in second bond (12%)

Invested in first bond (10%) = x = ₹ 10,000

Invested in second bond (12%) = y = ₹ 15,000

The amount invested by the trust = x + y = ₹ 25000

**Question.** Using properties of determinants, prove the following :

**Answer.**

**Expanding along C _{1}, we get**

**Question.** Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to its 3, 2 and 1 students with a total award money of ₹ 1,000. School Q wants to spend ₹ 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is ₹ 600, using matrices, find the award money for each value.**Answer.**

₹ x = ₹100, y = ₹ 200, z = ₹ 300 is the award money for each value

** Question. A total amount of ₹ 7,000 is deposited in three different savings bank accounts with annual interest rates of 5%, 8% and 8 (1/2)% respectively.The total annual interest from these three accounts is ₹ 550. Equal amounts have been deposited in the 5% and 8% savings accounts. Find the amount deposited in each of the three accounts, with the help of matrices.Answer.** Let the amount deposited in 5% p.a = ₹ x

Let the amount deposited in 8% p.a = ₹ y

Let the amount deposited in 8 (1/2)% p.a = ₹ z

**Question.**

**Answer.**

**Long Answer Type Questions**

**Question.** Using properties of determinants, prove the**following :**

**Answer.**

** Question. Using matrices, solve the following system of equations :4x + 3y + 2z = 60, x + 2y + 3z = 45,6x + 2y + 3z = 70.Answer. **The given equations can be written as

**Question.** Using matrix method, solve the following system**of equations :**

**Answer.** Given equations can be written as

**Question.** If a, b, c are positive and unequal, show that the following determinant is negative :

**Answer.**

**Question.**

**Hence solve the system of equations :x – 2y = 10; 2x + y + 3z = 8 and – 2y + z = 7.Answer.**

**Question.**

**Answer.**

**Question.**

**following system of equations :2x – y + z = – 3, 3x – z = 0, 2x + 6y = 2.Answer.**

**Question.**

x + 2y – 3z = –4; 2x + 3y + 2z = 2;

3x – 3y – 4z = 11.

Answer. Given equations can be written as

**Question. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category.Answer.** Sol. x + y + z = 12

2x + 3(y + z) = 33 ⇒ 2x + 3y + 3z = 33

x + z = 2y ⇒ x – 2y + z = 0

Writing in matrix form

**Question.**

**system of equations :x + 2y + z = 4; – x + y + z = 0; x – 3y + z = 2**

**Answer.**|A|= 1(1 + 3) + 1(2 + 3) + 1(2 – 1)

= 4 + 5 + 1

= 10 ≠ 0 ∴ A–1 exists

A

_{11}= 1 + 3 = 4, A

_{12 }= – (–1 – 1) = 2

A

_{13}= 3 – 1 = 2, A

_{21}= – (2 + 3) = –5

A

_{22 }= 1 – 1 = 0, A

_{23}= – (–3 – 2) = 5

A

_{31 }= 2 – 1 = 1, A

_{32}= – (1 + 1) = –2

A

_{33}= 1 + 2 = 3

**Question. A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of ₹ 6,000. Three times the award money for Hard work added to that given for honesty amounts to ₹ 11,000. The award money given for Honesty and Hard work together is double the one given for Regularity.****Represent the above situation algebraically and find the award money for each value, using matrix method.****Answer.** Let the award money for Honesty = ₹ x,

The award money for Regularity = ₹ y

and the award money for Hard work = ₹ z

According to the question,

x + y + z = ₹ 6,000 …(i)

x + 3z = ₹ 11,000 …(ii)

x + z = 2 (y)

x – 2y + z = 0 …(iii)

Writing in matrix form

∴ x = ₹ 500, y = ₹ 2,000, z = ₹ 3,500

Award money for Honesty, x = ₹ 500

Award money for Regularity, y = ₹ 2,000

Award money for Hard work, z = ₹ 3,500

**Question.** **Using elementary transformations, find the inverse of the matrix**

**and use it to solve the following system of linear equations :8x + 4y + 3z = 19 2x + y + z = 5x + 2y + 2z = 7.Answer.**

**Question.**

**divisible by (x + y + z), and hence find the quotient.Answer.**

**Question. Using properties of determinants, show that ΔABC is isosceles if :**

**Answer.**

Expanding along R1, we have

(cos B – cos A) (cos C – cos A) .

[cos C + cos A – cos B – cos A] = 0

(cos B – cos A) (cos C – cos A) (cos C – cos B) = 0

Either cos B – cos A = 0 or cos C – cos A = 0

or cos C – cos B = 0

⇒ cos B = cos A or cos C = cos A

or cos C = cos B

⇒ B = A or C = A or C = B

L.i. ABC is isosceles.

**Question. Using properties of determinants, prove that :**

**Answer.**

Expanding along R_{1}, we have

= (ab + bc + ca) 1[(ab + bc + ca)^{2} – 0]

= (ab + bc + ca)3 = R.H.S

**Question.**

**the system of equations x + 3z = 9,– x + 2y – 2z = 4, 2x – 3y + 4z = – 3.Answer.**

**Question.**

**the system of equations2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.Answer.**

**Question.. A shopkeeper has 3 varieties of pens ‘A’, ‘B’ and ‘C’. Meenu purchased 1 pen of each variety for a** **total of ₹ 21. Jeevan purchased 4 pens of ‘A’** **variety, 3 pens of ‘B’ variety and 2 pens of ‘C’** **variety for ₹ 60. While Shikha purchased 6 pens** **of ‘A’ variety, 2 pens of ‘B’ variety and 3 pens of ‘C’ variety for ₹ 70. Using matrix method, find** **cost of each variety of pen.****Answer.** Let the cost of pens ‘A’, ‘B’ and ‘C’ be ₹ x, ₹ y and ₹z respectively.

Then the system of equations is

x + y + z = 21

4x + 3y + 2z = 60

6x + 2y + 3z = 70

Writing in matrix form

Cost of pen ‘A’ = x = ₹ 5

Cost of pen ‘B’ = y = ₹ 8

Cost of pen ‘C’ = z = ₹ 8

**CASE STUDY:**

Manjit wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50m, then its area will remain same, but if length is decreased by 10m and breadth is decreased by 20m, then its area will decrease by 5300 m2 Based on the information given above, answer the following questions:

**Question. The equations in terms of X and Y are**a. x-y=50, 2x-y=550

b. x-y=50, 2x+y=550

c. x + y = 50, 2x + y=550

d. x +y = 50, 2x + y=550

## Answer

B

**Question**. Which of the following matrix equation is represented by the given information

## Answer

A

** Question. The value of x (length of rectangular field) is**a. 150m

b. 400m

c. 200m

d. 320m

## Answer

C

** Question. The value of y (breadth of rectangular field) is**a. 150m.

b. 200m.

c. 430m.

d. 350m

## Answer

A

** Question. How much is the area of rectangular field?**a. 60000Sq.m.

b. 30000Sq.m.

c. 30000m

d. 3000m

## Answer

B