# Exam Question for Class 12 Mathematics Chapter 4 Determinants

Please refer to below Exam Question for Class 12 Mathematics Chapter 4 Determinants. These questions and answers have been prepared by expert Class 12 Mathematics teachers based on the latest NCERT Book for Class 12 Mathematics and examination guidelines issued by CBSE, NCERT, and KVS. We have provided Class 12 Mathematics exam questions for all chapters in your textbooks. You will be able to easily learn problems and solutions which are expected to come in the upcoming class tests and exams for standard 10th.

## Chapter 4 Determinants Class 12 Mathematics Exam Question

All questions and answers provided below for Exam Question Class 12 Mathematics Chapter 4 Determinants are very important and should be revised daily.

**Exam Question Class 12 Mathematics Chapter 4 Determinants**

**Very Short Answer Type Questions**

**Question**. Find the maximum value of∴**Answer.**

⇒ Δ = 1[(1 + sinθ) (1 + cos θ) – 1]

– 1 (1 + cos θ – 1) +1 (1 – 1 – sin θ)

= 1 + cos θ + sin θ + sin θ cos θ – 1 – cos θ – sin θ

= sin θ cos θ

∴ Maximum value of Δ is 1/2

**Question.**

**Answer.**

⇒ (x + 3) (2x) – (–2) (–3x) = 8

⇒ 2x^{2} + 6x – 6x = 8 ⇒ 2x^{2} = 8

⇒ x^{2} = 4 ⇒ x = 2 [x ≠ –2 Q x ∈ N]

**Question**.

**Answer.**

⇒ (x + 1)(x + 2) – (x – 3)(x – 1) = 4 × 3 – (1) × (–1)

⇒ x^{2} + x + 2x + 2 – (x^{2} – 3x – x + 3) = 12 + 1

⇒ x^{2} + 3x + 2 – x^{2} +x^{2}– 3 = 13

⇒ 7x = 13 + 1 ⇒ x = 2

**Question**.

**Answer.**

⇒ 2x(x + 1) – 2(x + 1) (x + 3) = 3 – 15

⇒ 2x^{2}+ 2x – 2(x^{2} + 4x + 3) = – 12

⇒ 2x^{2}+ 2x – 2x^{2} – 8x – 6 = – 12

⇒ 6x = – 12 + 6 ⇒ x = -6/-6

**Question**. Evaluate :

**Answer.**

= cos 15° cos75° – sin 75° sin 15°

= cos(15° + 75°) = cos 90° = 0

**Question**. Evaluate :

**Answer.**

**Question**. Evaluate :

**Answer.**

= (a + ib) (a – ib) – (c + id) (– c + id)

= (a^{2} + b^{2}) – (– c^{2} – d^{2}) {**∵** i2 = –1}

= a^{2} + b^{2} + c^{2} + d^{2}

**Question**. Evaluate :

**Answer.**

= sin30°cos60° + sin60°cos30°.

= sin(30° + 60°) = sin90° = 1

**Question**. Evaluate :

**Answer.**

**Question**.

**Answer.**

⇒ x^{2} – x = 6 – 4 ⇒ x^{2 }– x – 2 = 0

⇒ (x – 2) (x + 1) = 0 ⇒ x = 2, x = –1

**Question**. Write the value of

**Answer.**

**Question**. If A is a 3 × 3 matrix, **|**A**|** ≠ 0 and **|**3A**|** = k **|**A**|**, then write the value of k.**Answer.** We have, |3A| = k |A|

⇒ 3^{3} |A| = k |A|

[Using |mA| = m^{n} |A|, where n is order of A]

⇒ k = 27.

**Question**. Let A be a square matrix of order 3 × 3. Write**the value of |2A|, where |A| = 4.****Answer.** A is a 3 × 3 matrix and |A| = 4

⇒ |2A| = 2^{3} · |A| = 8 × 4 = 32.

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

**Answer.** R_{1} and R_{3} are proportional.

Therefore, value of the determinant is zero.

**Question**. Show that the following determinant vanishes

**Answer.**

**Question**. Without expanding, show that

**Answer.**

**Question**.

**then write the cofactor of the element a _{21} of its 2^{nd} row.**

**Answer.**

**Question**. If A_{ij }is the cofactor of the element a_{ij} of the**determinant**

**then write the value of a _{32 }· A_{32.}**

**Answer.**

**Question**.

**Answer.**

**Question**.

**Answer.**

**Question.**

**Answer.**

**Question.** Find the minor of the element of second row**and third column (a _{23}) in the determinant**

**Answer.**

**Question.** If A is a square matrix of order 3 such that**|adj A| = 64, find |A|. ****Answer.** A is a 3 × 3 matrix s.t. |adj A | = 64.

We know that

|adj A|= |A|^{n – 1}

⇒ 64 = |A|^{3 – 1} ⇒|A|^{2} = 64 ⇒ |A| =± 8.

**Question.****If A is an invertible square matrix of order 3 and |A| = 5, then find the value of |adj A|.****Answer.** Given, |A | = 5 and A is invertible matrix of order 3.

We know that |adj A|= |A|^{n – 1}

⇒ |adj A| = (5)^{3 – 1} ⇒ |adj A| = 25.

**Question.** For what value of x, is the given matrix

**Answer.**

**Question.****For what value of x, the matrix**

**is a singular matrix?****Answer.**

**Question.** Write A^{–1} for

**Answer.**

**Question.** For what value of x, the matrix

**Answer.**

**Question.**

**Answer.**

**Question.** For what value of x is the matrix

**Answer.**

**Question.** For what value of x is

**Answer.**

⇒ 2(x + 1) (x – 2) – 2x^{2} = 0

⇒ 2(x^{2} – x – 2) – 2x^{2} = 0 ⇒ 2 (– x – 2) = 0

⇒ x = – 2

**Question.** Write the adjoint of the following matrix :

**Answer.**

**Question.**

**Answer.**

**Question.** If |A| = 2, where A is a 2 × 2 matrix, find |adj A|.** Answer.** |adjA| = |A|

^{n–1 }= (2)

^{2– 1}= 2.

**Question.** If A is a non-singular matrix of order 3 and**|adj A| = |A| ^{k}, then write the value of k.**

**Answer.**We know that |adjA| = |A|

^{n}

^{– 1}

⇒ |adjA| = |A|

^{3 – 1 }= |A|

^{2}= |A|

^{k}

⇒ k= 2

**Short Answer Type Questions**

**Question.**

**using properties of determinants find the value of f(2x) – f(x).****Answer.**

⇒ f(x) = a[a(x + a) + (x^{2} + ax)]

⇒ f(x) = a(a^{2} + ax + ax + x^{2}) = a(a^{2} + 2ax + x^{2})

Now, f(2x) = a{a^{2} + 2a (2x) + (2x)^{2}} = a(a^{2} + 4ax + 4x^{2})

∴ f(2x) – f(x) = a( a^{2} + 4ax + 4x^{2} – a^{2} – 2ax – x^{2})

= ax(3x + 2a)

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

**Answer.**

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

**Answer.**

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

**Answer.**

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

**Answer.**

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

**Answer.**

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

**Answer.**

**Question.**

**Answer.**

Since each element of third column of determinant is the sum of two elements, therefore determinants can be expressed as the sum of two determinants given by

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

**Answer.**

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

**Answer.**

**Question.** Using properties of determinants, solve for x :

**Answer.**

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

**Answer.**

**Question.** Find the equation of the line joining A(1, 3) and B(0, 0) using determinants and find the value of k if D(k, 0) is a point such that area of ΔABD is**3 square units.****Answer.** Using determinants, the line joining A(1, 3)

**Question.**

**Answer.**

**Question.** Find the adjoint of the matrix

**and hence show that A·(adj A) = |A|I _{3}.**

**Answer.**

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

**Answer.**

**Question.** Prove the following using properties of**determinants :**

**Answer.**

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

**Answer.**

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

**Answer.**

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

**Answer.**

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

**112.**

**Answer.**

**Question.** Show that the matrix

**satisfies the equation A ^{2} – 5A + 7I = O. Hence, find A^{–1}.**

**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. This problem reflects which value? **Answer.** Let the monthly income of Aryan be ₹ 3x and that of Babban be ₹ 4x

Also, let monthly expenditure of Aryan be ₹ 5y and

that of Babban be ₹ 7y

According to question,

3x – 5y = 15000

4x – 7y = 15000

These equations can be written as

AX = B

⇒ x = 30000 and y = 15000

So, monthly income of Aryan = 3 × 30000 = `90000

Monthly income of Babban = 4 × 30000 = `120000

From this question we are encouraged to save a part of money every month.

**Question.** Using properties of determinants, show that

**Answer.**

Since each element in the first column of determinant is the sum of two elements, therefore, determinant can be expressed as the sum of two determinants given by

Taking x common from R_{1}, R_{2}, R_{3} in first determinant and x common from C_{2}, C_{3}, y common from C_{1} in second determinant, we get

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

**Answer.**

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

**Answer.**

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

**Answer.**

**Long Answer Type Questions**

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

**Answer.** We have,

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

⇒ cos B = cos A ⇒ B = A

or cos C = cos A ⇒ C = A or cosC = cosB ⇒ C = B

∴ ΔABC is an isosceles triangle.

**Question.****If a, b and c are all non-zero and**

**Answer.**

**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. Apart from the above three values, suggest one more value for awards.** Answer.** According to question, we have,

3x + 2y + z = 1000 …(i)

4x + y + 3z = 1500 …(ii)

x + y + z = 600 …(iii)

The given system of equations can be written as AX = B

∴ A is invertible and system of equations has a unique solution given by X = A^{–}1 B

Now, A_{11} = –2, A_{12} = –1, A_{13} = 3,

A_{21} = –1, A_{22} = 2, A_{23 }= –1,

A_{31 }= 5, A_{32 }= –5, A_{33 }= –5

⇒ x = 100, y = 200, z = 300

Hence the money awarded for discipline, politeness and punctuality are ₹ 100, ₹ 200 and ₹ 300 respectively.

Apart from the above three values schools can award children for sincerity.

**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

`x,`

y and ` z be deposited at the rates of interest 5%, 8% and 8(1/2)% respectively.According to question,

x + y + z = 7000

x – y = 0

∴ A^{–1} exists. So, system of equations has a unique solution and it is given by X = A^{–1}B

Now, A_{11 }= –17, A_{12} = –17, A_{13} = 26,

A_{21} = –1, A_{22 }= 7, A_{23 }= –6,

A_{31} = 1, A_{32} = 1, A_{33 }= –2

**Question.****Using matrices, solve the following system of** **linear equations :****x – y + 2z = 7, 3x + 4y – 5z = – 5****2x – y + 3z = 12 **** Answer.** We have,

x – y + 2z = 7

3x + 4y – 5z = – 5

2x – y + 3z = 12

The given system of equations can be written as

AX = B

= 1 (12 – 5) – 3(– 3 + 2) + 2(5 – 8)

= 7 + 3 – 6 = 4 0.

∴ A^{–1} exists. So, system of equations has a unique

solution given by X = A–1B

Now, A_{11} = 7, A_{12 }= –19, A_{13} = – 11, A_{21} = 1, A_{22} = –1,

A_{23 }= –1, A_{31 }= –3, A_{32} = 11, A_{33 }= 7

**Question.****Using matrices, solve the following system of** **equations :****2x + 3y + 3z = 5, x – 2y + z = – 4,****3x – y – 2z = 3**** Answer.** We have,

2x + 3y + 3z = 5

x – 2y + z =- 4

3x – y – 3z = 3

The given system of equations can be written as AX = B

= 2(4 + 1) – 3(– 2 – 3) + 3(– 1 + 6)

= 10 + 15 + 15 = 40 ≠ 0

∴ A^{–1} exists. So, system of equations has a unique solution and it is given by X = A^{–1}B

Now, A_{11} = 5, A_{12} = 5, A_{13} = 5, A_{21} = 3, A_{22} = –13,

A_{23} = 11, A_{31} = 9, A_{32} = 1, A_{33} = –7

**Question.** Using matrices, solve the following system of**equations :****3x + 4y + 7z = 4; 2x – y + 3z = – 3;****x + 2y – 3z = 8**** Answer.** We, have

3x + 4y + 7z = 4

2x – y + 3z = – 3

3x + 2y – 3z = 8

The system of equations can be written as AX = B

= 3(3 – 6) – 4(– 6 – 3) + 7(4 + 1)

= – 9 + 36 + 35 = 62 ≠ 0

∴ A^{–1} exists. So, system of equations has a unique

solution given by X = A^{–1}B

Now, A_{11} = –3, A_{12} = 9, A_{13} = 5, A_{21} = 26, A_{22} = –16,

A_{23} = –2, A_{31} = 19, A_{32 }= 5, A_{33 }= –11

**Question.** Two schools, P and Q, want to award their selected students for the values of sincerity, truthfulness and hard work at the rate of ₹ x, ₹ y and ₹ z for each respective value per student. School P awards its 2, 3 and 4 students on the above respective values with a total prize money of ₹ 4,600. School Q wants to award its 3, 2 and 3 students on the respective values with a total award money of ₹ 4,100. If the total amount of award money for one prize on each value is ₹ 1,500, using matrices find the award money for each value. Suggest one other value which the school can consider for awarding the students. ** Answer.** According to question, we have

x + y + z = 1500

2x + 3y + 4z = 4600

3x + 2y + 3z = 4100

The system of equations can be written as AX = B

= 1(9 – 8) – 1(6 – 12) + 1(4 – 9)

= 1 + 6 – 5 = 2 0

∴ A^{–1} exists and so, system of equations has a

unique solution given by X = A^{–1}B

Now, A_{11} = 1, A_{12} = 6, A_{13 }= –5,

A_{21} = –1, A_{22} = 0, A_{23} = 1,

A_{31} = 1, A_{32} = –2, A_{33} = 1

⇒ x = 500; y = 400; z = 600.

Apart from sincerity, truthfulness and hard work,

the schools can include an award for regularity.

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

**Answer.**

= (a + b + c) [(c – b) (b – c) – (a – b) (a – c)]

= (a + b + c) [(bc – c^{2 }– b^{2}+ bc ) – (a^{2} – ac – ab + bc)]

= (a + b + c) [2bc – c^{2} – b^{2} – a^{2} + ac + ab – bc]

= (a + b + c) [–a^{2} – b^{2} – c^{2} + ab + bc +ca]

= –(a + b + c) [a^{2} + b^{2} + c^{2} – ab – bc – ca]