Are these practice questions for Determinants based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Determinants free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Determinants mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Determinants for full coverage.

How many practice questions are there for Determinants?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Determinants Class 12 Mathematics Practice Questions with Answers

easy

In the determinant

\Delta = \left|\begin{array}{ccc} 2 & -3 & 5 \ 6 & 0 & 4 \ 1 & 5 & -7 \end{array}\right|

, calculate the minor of the element

a_{23}

.

Try solving it in your notebook first, then check the solution.

easy

Solve for

x

given that

\left|\begin{array}{cc} 2 & 3 \ 4 & 5 \end{array}\right| = \left|\begin{array}{cc} x & 3 \ 2x & 5 \end{array}\right|

.

Try solving it in your notebook first, then check the solution.

easy

Explain what is meant by a 'consistent' and an 'inconsistent' system of linear equations.

Try solving it in your notebook first, then check the solution.

easy

Write the formula for the cofactor

A_{ij}

of an element

a_{ij}

in terms of its minor

M_{ij}

.

Try solving it in your notebook first, then check the solution.

easy

Create a non-zero

2 \times 2

singular matrix whose trace (sum of diagonal elements) is 5. Justify your answer.

Try solving it in your notebook first, then check the solution.

easy

Define a singular matrix.

Try solving it in your notebook first, then check the solution.

easy

List the six ways a determinant of order 3 can be expanded to find its value.

Try solving it in your notebook first, then check the solution.

easy

If A is a square matrix of order 3 and

|A| = -4

, then calculate the value of

|2A|

.

Try solving it in your notebook first, then check the solution.

easy

What is the value of the determinant of a matrix of order 1,

A = [a]

?

Try solving it in your notebook first, then check the solution.

easy

State the condition for a square matrix A to be invertible.

Try solving it in your notebook first, then check the solution.

medium

Explain the difference between

|A|

for a matrix A and the modulus of a number.

Try solving it in your notebook first, then check the solution.

medium

If

A = \left[\begin{array}{cc} 3 & -2 \ 4 & -2 \end{array}\right]

, find the value of

k

such that

A^2 = kA - 2I

, where I is the identity matrix of order 2.

Try solving it in your notebook first, then check the solution.

medium

Justify why the inverse of a diagonal matrix

D = \begin{pmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{pmatrix}

is

D^{-1} = \begin{pmatrix} 1/d_1 & 0 & 0 \ 0 & 1/d_2 & 0 \ 0 & 0 & 1/d_3 \end{pmatrix}

, provided no diagonal elements are zero.

Try solving it in your notebook first, then check the solution.

medium

For what value of

k

is the matrix

A = \left[\begin{array}{cc} k & 2 \ 3 & 4 \end{array}\right]

singular?

Try solving it in your notebook first, then check the solution.

medium

Critique the following argument and identify the flaw: To solve the equation

\begin{vmatrix} x & 2 \ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \ 18 & 6 \end{vmatrix}

, we can equate the corresponding elements. Thus,

x=6

. Justify your critique with the correct method.

Try solving it in your notebook first, then check the solution.

medium

Solve the following system of linear equations using the matrix method:

3x + 2y = 8

2x - y = 3

Try solving it in your notebook first, then check the solution.

medium

If

A = \left[\begin{array}{cc} 2 & 3 \ 1 & 2 \end{array}\right]

, demonstrate that

A^2 - 4A + I = O

, where I is the identity matrix and O is the zero matrix of order 2.

Try solving it in your notebook first, then check the solution.

medium

Calculate the area of the triangle with vertices at the points

A(-2, -3)

,

B(3, 2)

, and

C(-1, -8)

using determinants.

Try solving it in your notebook first, then check the solution.

medium

State the property that relates the product of a square matrix A and its adjoint with its determinant.

Try solving it in your notebook first, then check the solution.

medium

Let

A

be a square matrix of order 3. Justify the relationship between

|\text{adj}(A)|

and

|A|

. Use this to prove that if

A

is a skew-symmetric matrix of order 3, then

|\text{adj}(A)|=0

.

Try solving it in your notebook first, then check the solution.

medium

Calculate the value of the determinant

\left|\begin{array}{cc} \cos 15^{\circ} & \sin 15^{\circ} \ \sin 75^{\circ} & \cos 75^{\circ} \end{array}\right|

.

Try solving it in your notebook first, then check the solution.

medium

Calculate the inverse of the matrix

A = \left[\begin{array}{ccc} 2 & -1 & 3 \ 1 & 3 & -1 \ 3 & 2 & 1 \end{array}\right]

.

Try solving it in your notebook first, then check the solution.

medium

Derive the condition for three points

(x_1, y_1)

,

(x_2, y_2)

, and

(x_3, y_3)

to be collinear using the concept of the area of a triangle expressed as a determinant. Justify your derivation.

Try solving it in your notebook first, then check the solution.

medium

Justify, without expanding, why the determinant of the matrix

A = \begin{vmatrix} \sin^2 x & 1 & \cos^2 x \ \sin^2 y & 1 & \cos^2 y \ \sin^2 z & 1 & \cos^2 z \end{vmatrix}

is zero.

Try solving it in your notebook first, then check the solution.

medium

Given the vertices of a triangle

(x_1, y_1)

,

(x_2, y_2)

, and

(x_3, y_3)

, write the expression for its area in determinant form. Also, state the condition for the three points to be collinear.

Try solving it in your notebook first, then check the solution.

medium

Design a solution using the matrix method to find the cost of three food items. A family of 4 bought 3 kg of rice, 2 kg of sugar, and 5 kg of flour and paid a total of ₹850. Another family of 3 bought 2 kg of rice, 3 kg of sugar, and 4 kg of flour for ₹780. A third family of 5 bought 4 kg of rice, 3 kg of sugar, and 6 kg of flour for ₹1150. Formulate the problem as a system of linear equations and create the complete solution to find the price per kg for each item.

Try solving it in your notebook first, then check the solution.

medium

Describe in detail the steps to find the determinant of a

3 \times 3

matrix

A = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix}

by expanding along the first row (

R_1

). Define the terms 'minor' and 'cofactor' as part of your explanation.

Try solving it in your notebook first, then check the solution.

medium

Justify that the value of the determinant

\Delta = \begin{vmatrix} 1 & a & a^2 \ 1 & b & b^2 \ 1 & c & c^2 \end{vmatrix}

is

(a-b)(b-c)(c-a)

.

Try solving it in your notebook first, then check the solution.

medium

Summarize the conditions used to check the consistency of a system of linear equations

AX = B

. Explain the cases for when the determinant of

A

is zero and when it is not zero.

Try solving it in your notebook first, then check the solution.

medium

If

A

is a non-singular square matrix of order

n

, what is the formula for

|\text{adj } A|

in terms of

|A|

?

Try solving it in your notebook first, then check the solution.

medium

Describe the process of finding the adjoint of a

3 \times 3

matrix

A = [a_{ij}]

.

Try solving it in your notebook first, then check the solution.

medium

Evaluate the statement: "If A is a square matrix of order 3 and

|A| = 4

, then

|2 \text{adj}(A)| = 128

". Justify your reasoning.

Try solving it in your notebook first, then check the solution.

medium

Formulate a system of linear equations for the following problem and justify the existence of a unique solution: "The sum of three numbers is 20. If we multiply the first number by 2, add the second number to the result, and subtract the third number, we get 23. If we multiply the first number by 3 and add the second and third numbers to it, we get 46." Do not solve the system.

Try solving it in your notebook first, then check the solution.

hard

Justify that for any skew-symmetric matrix

A

of odd order

n

, its determinant must be zero. Critique the statement for a skew-symmetric matrix of even order.

Try solving it in your notebook first, then check the solution.

hard

Solve the following system of linear equations using the matrix method:

x + y - z = 3

2x + 3y + z = 10

3x - y - 7z = 1

Try solving it in your notebook first, then check the solution.

hard

Calculate the adjoint of the matrix

A = \left[\begin{array}{ccc} 1 & -2 & 3 \ 0 & 2 & 1 \ -4 & 5 & 2 \end{array}\right]

.

Try solving it in your notebook first, then check the solution.

hard

If

A

is a non-singular matrix such that

A^2 - 5A + 7I = O

, design a method to express

A^{-1}

in terms of

A

and

I

. Justify each step of your design.

Try solving it in your notebook first, then check the solution.

hard

Explain the complete procedure for finding the inverse of a

3 \times 3

non-singular matrix A. List all the necessary definitions and formulas involved.

Try solving it in your notebook first, then check the solution.

hard

Let

A = \left[\begin{array}{ccc} 1 & -2 & 0 \ 2 & 1 & 3 \ 0 & -2 & 1 \end{array}\right]

. First, calculate

A^{-1}

. Then, use the result to solve the following system of linear equations:

x - 2y = 10

2x + y + 3z = 8

-2y + z = 7

Try solving it in your notebook first, then check the solution.

hard

Summarize the steps to solve a system of linear equations in two variables,

a_1x + b_1y = d_1

and

a_2x + b_2y = d_2

, using the matrix method.

Try solving it in your notebook first, then check the solution.

hard

Consider two matrices

P = \begin{pmatrix} 1 & -1 & 0 \ 2 & 3 & 4 \ 0 & 1 & 2 \end{pmatrix}

and

Q = \begin{pmatrix} 2 & 2 & -4 \ -4 & 2 & -4 \ 2 & -1 & 5 \end{pmatrix}

. First, evaluate the product

PQ

. Then, using this result, design a method to solve the following system of linear equations without finding the inverse of the coefficient matrix directly.

x - y = 3

2x + 3y + 4z = 17

y + 2z = 7

Justify your method.

Try solving it in your notebook first, then check the solution.

hard

Let

A

be a square matrix of order

n

. Prove the property

A(\text{adj } A) = (\text{adj } A)A = |A|I

. Using this result, justify the formula for the inverse of a non-singular matrix,

A^{-1} = \frac{1}{|A|}(\text{adj } A)

.

Try solving it in your notebook first, then check the solution.

hard

The sum of three numbers is 2. If we subtract the second number from twice the first number, we get 3. The sum of the first and third numbers is equal to the second number. Represent this problem as a system of linear equations and solve for the numbers using the matrix method.

Try solving it in your notebook first, then check the solution.

hard

Propose a condition on the matrix

B

for the system of linear equations

AX=B

to be inconsistent, given that the coefficient matrix

A

is singular.

Try solving it in your notebook first, then check the solution.

DeterminantsClass 12 Mathematics Practice Questions