Mathematics · Textbook solutions

Determinants

Every solved example, exercise, and miscellaneous question, in the order the textbook teaches them. Plus every board question asked from this chapter. · 92 questions

4.2 Determinant

18 q

Solved Examples

Worked · 5

4.1 Eg.1 $\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}$
4.1 Eg.2 $\begin{vmatrix} x & x+1 \\ x-1 & x \end{vmatrix}$
4.1 Eg.3 $\Delta = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}$
4.1 Eg.4 $\Delta = \begin{vmatrix} 0 & \sin\alpha & -\cos\alpha \\ -\sin\alpha & 0 & \sin\beta \\ \cos\alpha & -\sin\beta & 0 \end{vmatrix}$
4.1 Eg.5 $x$

Exercise 4.1

Practice · 13

Ex 4.1 Q1 $\begin{vmatrix} 2 & 4 \\ -5 & -1 \end{vmatrix}$
Evaluate the determinant.
Ex 4.1 Q2 (i) $\begin{vmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{vmatrix}$
Ex 4.1 Q2 (ii) $\begin{vmatrix} x^{2} - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}$
Ex 4.1 Q3 $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$
Ex 4.1 Q4 $A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{bmatrix}$
Evaluate the determinants.
Ex 4.1 Q5 (i) $\begin{vmatrix} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{vmatrix}$
Ex 4.1 Q5 (ii) $\begin{vmatrix} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{vmatrix}$
Ex 4.1 Q5 (iii) $\begin{vmatrix} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{vmatrix}$
Ex 4.1 Q5 (iv) $\begin{vmatrix} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{vmatrix}$
Ex 4.1 Q6 $A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{bmatrix}$
Find values of $x$ , if the following holds.
Ex 4.1 Q7 (i) $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$
Ex 4.1 Q7 (ii) $\begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix} = \begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}$
Ex 4.1 Q8 $\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}$
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4.3 Area of a Triangle

11 q

Solved Examples

Worked · 2

4.2 Eg.6 $(3, 8)$
4.2 Eg.7 $A(1, 3)$

Exercise 4.2

Practice · 9

Find area of the triangle with vertices at the point given in each of the following:
Ex 4.2 Q1 (i) $(1, 0)$
Ex 4.2 Q1 (ii) $(2, 7)$
Ex 4.2 Q1 (iii) $(-2, -3)$
Ex 4.2 Q2 $A(a, b + c)$
Find values of $k$ if area of triangle is 4 sq. units and vertices are:
Ex 4.2 Q3 (i) $(k, 0)$
Ex 4.2 Q3 (ii) $(-2, 0)$
Ex 4.2 Q4 (i) $(1, 2)$
Ex 4.2 Q4 (ii) $(3, 1)$
Ex 4.2 Q5 $(2, -6)$
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4.4 Minors and Cofactors

11 q

Solved Examples

Worked · 4

4.3 Eg.8 $\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$
4.3 Eg.9 $\begin{vmatrix} 1 & -2 \\ 4 & 3 \end{vmatrix}$
4.3 Eg.10 $a_{11}$
4.3 Eg.11 $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$

Exercise 4.3

Practice · 7

Write Minors and Cofactors of the elements of following determinants:
Ex 4.3 Q1 (i) $\begin{vmatrix} 2 & -4 \\ 0 & 3 \end{vmatrix}$
Ex 4.3 Q1 (ii) $\begin{vmatrix} a & c \\ b & d \end{vmatrix}$
Write Minors and Cofactors of the elements of following determinants:
Ex 4.3 Q2 (i) $\begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix}$
Ex 4.3 Q2 (ii) $\begin{vmatrix} 1 & 0 & 4 \\ 3 & 5 & -1 \\ 0 & 1 & 2 \end{vmatrix}$
Ex 4.3 Q3 $\Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix}$
Ex 4.3 Q4 $\Delta = \begin{vmatrix} 1 & x & yz \\ 1 & y & zx \\ 1 & z & xy \end{vmatrix}$
Ex 4.3 Q5 $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$
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4.5 Adjoint and Inverse of a Matrix

22 q

Solved Examples

Worked · 4

4.4 Eg.12 $\operatorname{adj} A$
4.4 Eg.13 $A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$
4.4 Eg.14 $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$
4.4 Eg.15 $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$

Exercise 4.4

Practice · 18

Find adjoint of each of the matrices in Exercises 1 and 2.
Ex 4.4 Q1 $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
Find adjoint of each of the matrices in Exercises 1 and 2.
Ex 4.4 Q2 $\begin{bmatrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{bmatrix}$
Verify $A(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I$ in Exercises 3 and 4.
Ex 4.4 Q3 $A(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I$
Verify $A(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I$ in Exercises 3 and 4.
Ex 4.4 Q4 $A(\operatorname{adj} A) = (\operatorname{adj} A) A = |A| I$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q5 $\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q6 $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q7 $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{bmatrix}$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q8 $\begin{bmatrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{bmatrix}$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q9 $\begin{bmatrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{bmatrix}$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q10 $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$
Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.
Ex 4.4 Q11 $\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\alpha & \sin\alpha \\ 0 & \sin\alpha & -\cos\alpha \end{bmatrix}$
Ex 4.4 Q12 $A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$
Ex 4.4 Q13 $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$
Ex 4.4 Q14 $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$
Ex 4.4 Q15 $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{bmatrix}$
Ex 4.4 Q16 $A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$
Ex 4.4 Q17 $A$
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Ex 4.4 Q18 $A$
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4.6 Applications of Determinants and Matrices

20 q

Solved Examples

Worked · 4

4.5 Eg.16 $2x + 5y = 1$
4.5 Eg.17 $3x - 2y + 3z = 8$
4.5 Eg.18 $The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.$
4.5 Eg.19 $\begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}\begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2 \end{bmatrix}$

Exercise 4.5

Practice · 16

Examine the consistency of the system of equations.
Ex 4.5 Q1 $x + 2y = 2$
Examine the consistency of the system of equations.
Ex 4.5 Q2 $2x - y = 5$
Examine the consistency of the system of equations.
Ex 4.5 Q3 $x + 3y = 5$
Examine the consistency of the system of equations.
Ex 4.5 Q4 $x + y + z = 1$
Examine the consistency of the system of equations.
Ex 4.5 Q5 $3x - y - 2z = 2$
Examine the consistency of the system of equations.
Ex 4.5 Q6 $5x - y + 4z = 5$
Solve system of linear equations, using matrix method.
Ex 4.5 Q7 $5x + 2y = 4$
Solve system of linear equations, using matrix method.
Ex 4.5 Q8 $2x - y = -2$
Solve system of linear equations, using matrix method.
Ex 4.5 Q9 $4x - 3y = 3$
Solve system of linear equations, using matrix method.
Ex 4.5 Q10 $5x + 2y = 3$
Solve system of linear equations, using matrix method.
Ex 4.5 Q11 $2x + y + z = 1$
Solve system of linear equations, using matrix method.
Ex 4.5 Q12 $x - y + z = 4$
Solve system of linear equations, using matrix method.
Ex 4.5 Q13 $2x + 3y + 3z = 5$
Solve system of linear equations, using matrix method.
Ex 4.5 Q14 $x - y + 2z = 7$
Ex 4.5 Q15 $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$
Ex 4.5 Q16 $The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.$

Miscellaneous Exercise on Chapter 4

10 q

Miscellaneous Exercise

Practice · 10

Misc Q1 $\begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$
Misc Q2 $\begin{vmatrix} \cos\alpha \cos\beta & \cos\alpha \sin\beta & -\sin\alpha \\ -\sin\beta & \cos\beta & 0 \\ \sin\alpha \cos\beta & \sin\alpha \sin\beta & \cos\alpha \end{vmatrix}$
Misc Q3 $A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}$
Let $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}$ . Verify that
Misc Q4 (i) $[adj\, A]^{-1} = adj\,(A^{-1})$
Misc Q4 (ii) $(A^{-1})^{-1} = A$
Misc Q5 $\begin{vmatrix} x & y & x+y \\ y & x+y & x \\ x+y & x & y \end{vmatrix}$
Misc Q6 $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$
Misc Q7 $\dfrac{2}{x} + \dfrac{3}{y} + \dfrac{10}{z} = 4$
Misc Q8 $x, y, z$
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Misc Q9 $A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}$
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What the board has asked from this chapter

Questions per paper · 21 papers · 2023 to 2026

Bars average the full papers of each year. 31 further questions come from set variants and are listed below.

2026

22 q

2 $A^2 = 4A + 3I$
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Adjoint and Inverse of a Matrix
4 $\Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$
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Determinant and its Properties
4 $If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :$
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Area of a Triangle
5 $If A is a non-singular matrix, then which of the following is not true ?$
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Adjoint and Inverse of a Matrix
5 $(3A)^{-1} =$
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Adjoint and Inverse of a Matrix
5 $x$
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Determinant and its Properties
6 $B(\text{adj } B) = \begin{bmatrix} \dfrac{1}{3} & 0 & 0 \\ 0 & \dfrac{1}{3} & 0 \\ 0 & 0 & \dfrac{1}{3} \end{bmatrix}$
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Adjoint and Inverse of a Matrix
6 $\begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86$
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Determinant and its Properties
7 $\Delta$
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Area of a Triangle
10 $A(\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
17 $A(\text{adj } A) = \begin{bmatrix} 2026 & 0 & 0 \\ 0 & 2026 & 0 \\ 0 & 0 & 2026 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
32 $A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.$
Solving System of Linear Equations
32 $Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each type of book for a total of ₹ 21. B buys 4 art books, 3 story books and 2 puzzle solving books for ₹ 60. C buys 6 art books, 2 story books and 3 puzzle solving books and pays ₹ 10 more than B. Use matrix method to find the cost of each type of book.$
Solving System of Linear Equations
32 (a) $P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$
Solving System of Linear Equations
32 $Find cost (per kg) of each fertilizer A, B and C that the farmer needs to buy, such that 1 kg each of fertilizer A and C added to 2 kg of B costs him ₹ 400. Also, cost of each kg of fertilizer B and C added together is equal to cost of 1 kg of fertilizer A. However, cost of 3 kg of fertilizer B added to ₹ 200 is the same as cost of 1 kg of fertilizer A and C together. Use matrix method to find the solution.$
Solving System of Linear Equations
32 (b) $\Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix}$
Determinant and its Properties
34 (a) $On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.$
Solving System of Linear Equations
33 (a) $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$
Solving System of Linear Equations
33 (b) $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$
Determinant and its Properties
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
37 (i) $AX = B$
Solving System of Linear Equations
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
37 (ii) $A^{-1}$
Adjoint and Inverse of a Matrix
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
37 (iii) (a) $A^{-1}$
Adjoint and Inverse of a Matrix

2025

28 q

1 $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
3 $A = kB$
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Determinant and its Properties
4 $\begin{vmatrix} 2x & 5 \\ 12 & x \end{vmatrix} = \begin{vmatrix} 6 & -5 \\ 4 & 3 \end{vmatrix}$
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Determinant and its Properties
4 $\det(P) = \alpha$
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Determinant and its Properties
5 $|A| = 3$
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Determinant and its Properties
6 $If A is a square matrix of order 2 such that det (A) = 4, then det (4 adj A) is equal to :$
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Adjoint and Inverse of a Matrix
6 $A = [a_{ij}]$
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Determinant and its Properties
6 $\det(9\,A^{-1})$
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Adjoint and Inverse of a Matrix
6 $|A| = 5$
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Adjoint and Inverse of a Matrix
7 $3 \times 3$
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Determinant and its Properties
7 $If M and N are square matrices of order 3 such that det (M) = m and MN = mI, then det (N) is equal to :$
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Determinant and its Properties
7 $\begin{vmatrix} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{vmatrix} = -57$
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Determinant and its Properties
8 $\begin{bmatrix} 4+x & x-1 \\ -2 & 3 \end{bmatrix}$
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Determinant and its Properties
33 $An amount of ₹ 10,000 is put into three investments at the rate of 10%, 12% and 15% per annum. The combined annual income of all three investments is ₹ 1,310, however the combined annual income of the first and the second investments is ₹ 190 short of the income from the third. Use matrix method and find the investment amount in each at the beginning of the year.$
Solving System of Linear Equations
11 $A system of linear equations is represented as AX = B, where A is coefficient matrix, X is variable matrix and B is the constant matrix. Then the system of equations is$
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Solving System of Linear Equations
15 $If A and B are invertible matrices, then which of the following is not correct ?$
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Adjoint and Inverse of a Matrix
33 $Three students run on a racing track such that their speeds add up to 6 km/h. However, double the speed of the third runner added to the speed of the first results in 7 km/h. If thrice the speed of the first runner is added to the original speeds of the other two, the result is 12 km/h. Using matrix method, find the original speed of each runner.$
Solving System of Linear Equations
21 $5x - ky = 2$
Solving System of Linear Equations
21 $\det (A) = 3$
Determinant and its Properties
27 (a) $2x + 5y - 1 = 0$
Solving System of Linear Equations
32 $3 \times 3$
Adjoint and Inverse of a Matrix
33 $A furniture workshop produces three types of furniture - chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is 45. It was also found that production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice the production of tables. Determine the units produced of each type of furniture, using matrix method.$
Solving System of Linear Equations
35 (a) $A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$
Solving System of Linear Equations
35 $A school wants to allocate students into three clubs : Sports, Music and Drama, under following conditions : The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.$
Solving System of Linear Equations
35 (b) $A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$
Adjoint and Inverse of a Matrix
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions :
36 (i) $AX = B$
Solving System of Linear Equations
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions :
36 (ii) $|A|$
Adjoint and Inverse of a Matrix
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following questions :
36 (iii) (a) $A^{-1}$
Adjoint and Inverse of a Matrix

2024

31 q

1 $\begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix}$
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Determinant and its Properties
2 $\mathrm{A} = \begin{bmatrix} 2 & -1 & 1 \\ \lambda & 2 & 0 \\ 1 & -2 & 3 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
2 $\left|\mathrm{A}\right| = -2$
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Determinant and its Properties
2 $A^{-1} = \dfrac{1}{7}\begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
3 $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$
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Adjoint and Inverse of a Matrix
3 $\left|\mathrm{adj}{\cdot}\mathrm{A}\right| = 8$
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Adjoint and Inverse of a Matrix
3 $\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc$
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Determinant and its Properties
4 $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
4 $A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
4 $A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}$
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Minors and Cofactors
6 $a_{ij}$
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Minors and Cofactors
7 $\mathrm{f}(x) = \begin{vmatrix} x^2 & \sin x \\ \mathrm{p} & -1 \end{vmatrix}$
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Determinant and its Properties
12 $\begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix}$
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Determinant and its Properties
13 $\begin{vmatrix} 1 & 3 & 1 \\ k & 0 & 1 \\ 0 & 0 & 1 \end{vmatrix} = \pm 6$
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Determinant and its Properties
18 $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
19 $A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$
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Determinant and its Properties
32 (a) $\mathrm{A} = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$
Solving System of Linear Equations
32 (a) $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$
Solving System of Linear Equations
32 (b) $A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix}$
Adjoint and Inverse of a Matrix
32 (b) $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{bmatrix}$
Solving System of Linear Equations
33 (a) $A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}$
Adjoint and Inverse of a Matrix
33 (b) $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 1 & 1 & 2 \end{bmatrix}$
Solving System of Linear Equations
34 (a) $\dfrac{2}{x} + \dfrac{3}{y} + \dfrac{10}{z} = 4$
Solving System of Linear Equations
34 $A = \begin{bmatrix} 2 & 1 & -3 \\ 3 & 2 & 1 \\ 1 & 2 & -1 \end{bmatrix}$
Solving System of Linear Equations
34 $\begin{bmatrix} 1 & 2 & -3 \\ 3 & 2 & -2 \\ 2 & -1 & 1 \end{bmatrix}\begin{bmatrix} 0 & 1 & 2 \\ -7 & 7 & -7 \\ -7 & 5 & -4 \end{bmatrix}$
Solving System of Linear Equations
34 (b) $A = \begin{bmatrix} 1 & \cot x \\ -\cot x & 1 \end{bmatrix}$
Adjoint and Inverse of a Matrix
34 $A^{-1}$
Solving System of Linear Equations
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022 – 23, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports. In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000. Based on the above information, answer the following questions :
38 (i) $Express the given information algebraically using matrices.$
Solving System of Linear Equations
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022 – 23, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports. In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000. Based on the above information, answer the following questions :
38 (ii) $Check whether the system of matrix equations so obtained is consistent or not.$
Solving System of Linear Equations
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022 – 23, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports. In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000. Based on the above information, answer the following questions :
38 (iii) (a) $Find the number of scholarships of each kind given by the school, using matrices.$
Solving System of Linear Equations
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious achievers based on certain criteria. In the session 2022 – 23, the school offered monthly scholarship of ₹ 3,000 each to some girl students and ₹ 4,000 each to meritorious achievers in academics as well as sports. In all, 50 students were given the scholarships and monthly expenditure incurred by the school on scholarships was ₹ 1,80,000. Based on the above information, answer the following questions :
38 (iii) (b) $Had the amount of scholarship given to each girl child and meritorious student been interchanged, what would be the monthly expenditure incurred by the school ?$
Solving System of Linear Equations

2023

26 q

2 $A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
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Adjoint and Inverse of a Matrix
2 $(2, -6)$
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Area of a Triangle
2 $|A| = x$
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Determinant and its Properties
2 $|A| = 2$
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Determinant and its Properties
3 $3 \times 3$
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Adjoint and Inverse of a Matrix
3 $\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{bmatrix}$
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Determinant and its Properties
3 $A^2 - A + I = O$
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Adjoint and Inverse of a Matrix
4 $|A| = |kA|$
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Determinant and its Properties
5 $(x_1, y_1)$
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Area of a Triangle
5 $\begin{vmatrix} 2 & 7 & 1 \\ 1 & 1 & 1 \\ 10 & 8 & 1 \end{vmatrix}$
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Determinant and its Properties
5 $\begin{vmatrix} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{vmatrix} = 0$
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Determinant and its Properties
5 $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$
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Determinant and its Properties
5 $\begin{vmatrix} 6 & 0 & -1 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{vmatrix}$
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Determinant and its Properties
11 $\begin{bmatrix} x & 2 \\ 3 & x - 1 \end{bmatrix}$
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Determinant and its Properties
14 $\Delta ABC$
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Area of a Triangle
16 $\left|\frac{A^{-1}}{2}\right| = \frac{1}{k|A|}$
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Determinant and its Properties
26 $\Delta PQR$
Area of a Triangle
26 $\begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$
Determinant and its Properties
32 (a) $A = \begin{bmatrix} -3 & -2 & -4 \\ 2 & 1 & 2 \\ 2 & 1 & 3 \end{bmatrix}$
Solving System of Linear Equations
32 $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$
Solving System of Linear Equations
33 (a) $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$
Adjoint and Inverse of a Matrix
33 (b) $x + 2y + 3z = 6$
Solving System of Linear Equations
34 (b) $A = \begin{bmatrix} 3 & 2 \\ 5 & -7 \end{bmatrix}$
Solving System of Linear Equations
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250. Based on the above information, answer the following questions :
37 (i) $Convert the given above situation into a matrix equation of the form AX = B.$
Solving System of Linear Equations
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250. Based on the above information, answer the following questions :
37 (ii) $|A|$
Determinant and its Properties
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250. Based on the above information, answer the following questions :
37 (iii) (a) $A^{-1}$
Adjoint and Inverse of a Matrix

The 10 board questions to practise first

The questions the board asked again come first, most asked first. Each one says how often and when.

Question 1
Asked 4 times: 2023*, 2024, 2025, 2026 (* = numbers or wording changed that year)
33 (a) $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$
Question 2
Asked 2 times: 2025, 2026
32 $A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.$
Question 3
Asked 2 times: 2023*, 2025 (* = numbers or wording changed that year)
4 $\det(P) = \alpha$
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Question 4
Asked 2 times: 2023*, 2024 (* = numbers or wording changed that year)
33 (a) $A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}$
Question 5
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
33 (b) $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$
Question 6
This type asked 4 times: 2023, 2024, 2025, 2026. This one: 2026
5 $x$
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Question 7 (case study, 4 parts)
This type asked 3 times: 2023, 2024, 2026. This one: 2026
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length. On the basis of the above information, answer the following questions :
37 (i) $AX = B$
37 (ii) $A^{-1}$
37 (iii) (a) $A^{-1}$
37 (iii) (b) $A^2 + 7\,I$
Question 8
This type asked 3 times: 2024, 2025, 2026. This one: 2026
32 (a) $P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$
Question 9
This type asked 3 times: 2024, 2025, 2026. This one: 2026
5 $If A is a non-singular matrix, then which of the following is not true ?$
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Question 10
This type asked 3 times: 2023, 2024, 2026. This one: 2026
32 (b) $\Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix}$

4.2 Determinant

Solved Examples

Exercise 4.1

4.3 Area of a Triangle

Solved Examples

Exercise 4.2

4.4 Minors and Cofactors

Solved Examples

Exercise 4.3

4.5 Adjoint and Inverse of a Matrix

Solved Examples

Exercise 4.4

4.6 Applications of Determinants and Matrices

Solved Examples

Exercise 4.5

Miscellaneous Exercise on Chapter 4

Miscellaneous Exercise

What the board has asked from this chapter

2026

2025

2024

2023

The 10 board questions to practise first

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7 (case study, 4 parts)

Question 8

Question 9

Question 10