CBSE Class 12 Maths 2024-25: Chapter 4 Determinants Important Competency-Based Questions with Answers; Download Free PDF (2025)

As the CBSE Class 12board examsget closer, it’s important for students to understand the newexam pattern. Starting in the2024-25 school year, CBSE will include50% more competency-based questions. These questions will be bothmultiple choiceandwritten, focusing on how to use what students have learned inreal-life situations.

This article exploresChapter 4Determinantshighlights keycompetency-based questionsand provides answers to help students succeed.

Understanding Competency-Based Questions in Chapter 4Determinants

Competency-based questionsare designed to see how well students can apply their knowledge in everyday life. They can come in different forms, such ascase studies,true-false questions,gap-filling tasks, andlong or short answer questions.

These questions are different from regularmemorization. They encourage students to think critically and solve problems, helping them understand the concepts inChapter 4Determinants better.

CBSE Class 12Maths Chapter 4Determinants Important Competency-Based Questions

Q.1 U and V are two non-singular matrices of order n and k is any scalar.

Given below are two statements based on the above information - one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).

Assertion (A) : det( k U) = k ⁿ × det(U)

Reason (R) : If W is a matrix obtained by multiplying any one row or column of V by the scalar k, then det(W) = k × det(V).

1. Both (A) and (R) are true and (R) is the correct explanation for (A).
2. Both (A) and (R) are true but (R) is not the correct explanation for (A).
3. (A) is true but (R) is false.
4. (A) is false but (R) is true.

Answer. 1

Q.2

Answer.Expands the determinant as:

n ( n - 1)! + n ( n !)

Simplifies the above expression as:

0.5 n ! + n ( n !) = n !( n + 1) = ( n + 1)!

Q.3Given below is a system of linear equations in three variables.

tx + 3 y - 2z = 1
x + 2 y + z = 2
- tx + y + 2 z = -1

For what value of t , will the system fail to have a unique solution? Show your work.

Answer.Writes the coefficient matrix and finds its determinant as:

Solves the equation -4 t - 8 = 0 for t and writes that for t = (-2), the system fails to have a unique solution.

Q.4P( k, -1), Q(3, -5) and R(6, -1) are vertices of ΔPQR. The lengths of side QR and altitude PS are 5 units and 4 units respectively.

Use determinants to find the value(s) of k. Show your work.

Answer.Uses QR and PS to find the area of the triangle as 10 sq units. The working may lookas follows:

Equates the area found to that using determinants as follows:

Expands the above determinant to write the equation as:

Solves the above equation to find the values of k as 1 or 11

Q.5

identify what type of matrix is (Adj A) x (Adj B). Show your work.

Answer.Finds (Adj A) as:

Finds (Adj B) as:

Finds the product of (Adj A) and (Adj B) as:

Identifies the matrix (Adj A) × (Adj B) as a scalar matrix.

(Award full marks if diagonal matrix is written instead of scalar matrix.)

Q.6

The co-factor of element 6 is (-11) and the minor of element 2 is 0.

Find the values of p and q. Show your work.

Answer.Frames an equation in p using the co-factor of element 6 as

Solves the above equation to find the value of p as 3.

Frames an equation in q using the minor of element 3 as

Solves the above equation to find the value of q as (-1).

Q.7Without expanding the determinant, show that Δ is a perfect square for any integer values of a, b and c .

Answer.Performs the row operation R ₁ -> R₁ - R ₂ - R ₃ on Δ to get:

Takes (-2) common from the first row and performs the row operation R 1 ₃ -> R ₃ - R ₂ on Δ to get:

Performs the row operation R 1 2 -> R 2 - R1 on Δ to get:

Performs the row operation R 1 ₃ -> R ₃ + R ₂ on Δ to get:

Expands the determinant to get:

1 Δ = (-2)[- c ² ( b ² a ² ) + b ² (- c ² a ² )]
= 4 a ² b ² c ²
= (2 abc ) ²

Hence concludes that Δ is a perfect square for any any integer value of a,b and c.

Q.8 A ball is thrown from a balcony. Its height above the ground after t seconds is given by h ( t ) = pt ² + qt + r , where p , q , and r ∈ R, and h ( t ) is in meters.

Shown below is the trajectory of the ball with its height from the ground at t = 0 s, t = 1 s and t = 5 s.

Solve for p , q and r and find h(t) using the matrix method. Show your work and give valid reasons.

Answer.Finds r = 30 by substituting t = 0 in h ( t ).

Frames the following equations in p and q by substituting t = 1 and t = 5respectively:

p + q = 7
5 p + q = 3

Writes the above system of equations in the matrix form using AX = B as

Finds |A| as -4 (≠ 0). Hence, writes that A^-1 exists and the system has a unique solution.

Finds adj A as:

Finds A 1 -1 using |A| and adj A as

Writes that X = A ^-1 B and finds X as:

Concludes that p = -1 and q = 8.

Writes h ( t ) = -t 0.5 2 +8 t +30.

Q.9A and B are invertible matrices of same order such that:

Find B. Show your work.

Answer.Finds B^-1 as (AB) ^-1 A as follows:

Finds adj (B 1.5 -1 ) as follows:


Finds the matrix B as follows:

Q.10P and Q are square matrices of order 4. The determinants of P and Q are 5 and 4 respectively.

Find the determinant of the matrix 3P2 Q. Show your work.

Answer.Finds (3P² Q| as 3⁴ x 5 x 5 x 4 = 34 (100)=8100.

Q.11Shown below is a quadrilateral with A(1, 2), B(5, 1), C(4, 6) and D(2, 6) as its vertices.

Find the area of ABCD using determinants. Show your steps.

Answer.Divides the quadrilateral into two triangles ABC and ACD and writes an expression for the area of the quadrilateral using determinants. For example:

Q.12A theatre has two categories of tickets, one for adults and one for children.

Bindu's family paid Rs 400 to buy 6 tickets for adults and 4 tickets for children.
Nalini's family paid Rs 325 to buy 5 tickets for adults and 3 tickets for children.

Find the cost of each category of the ticket by matrix method. Show your steps.

Answer.Assumes the cost of each adult ticket as x and each child's ticket as y. Writes the equations that represent the given scenario as follows:

6 x + 4 y = 400
5x + 3y = 325

Writes the above system of equations in the matrix form as AX = B where:

Finds A❘ as 18 - 20-20. Hence writes that A¹ exists and the system has a unique solution.

Finds A¹ as:

Writes that X = A-¹ B and finds X as:

Concludes that the cost of each adult and child's ticket is Rs 50 and Rs 25 respectively.

Q.13 Find the x-coordinate of the location of team Charlie. Use the determinant method and show your steps.

Answer.Uses the determinant method to write the expression for the area of a triangle as:

Expands the LHS of the above equation and simplifies it to get:

-58 - 2 x = ±74

Finds the value of x as (-66) or 8 and concludes that x = 8.

Q.14The head of the camp plans to have a medical centre on the line joining the coordinates of teams Alpha and Beta such that its x -coordinate is 5.

Find the y -coordinate of the medical centre using the determinant method. Show your steps and give a valid reason.

Answer.Writes that the Alpha team, medical centre and the Beta team are on the same line and hence the area of the triangle formed by three collinear points will be zero.

Writes the relation as:

Simplifies the above equation to find the value of the y -coordinate as 68/8 or 8.5 units.

Q.15Find the coordinates of the secret room. Show your steps.

Answer.Finds the adjoint of the given matrix as:

Writes the coordinates of the location of the secret room as (-11, -8).

----

👉 Read Also -CBSE Class 12 Half-Yearly/Mid Term 2024-25 : Most Important Questions with Answers; PDF Download (All Subjects)

👉 Read Also -How CBSE’s New Exam Pattern Will Impact Class 11 and 12 Students

👉CBSE Class 12 Study Materials