Determinants — Chapter Test | Class 12 IMO

Q1

Find the minor of element 6 in the determinant of matrix [1 2 3; 4 5 6; 7 8 9].

Element 6 is a₂₃. Deleting row 2 and col 3 leaves |1 2; 7 8|. Minor M₂₃ = 8 - 14 = -6.

Q2

If |2x 3; 5 x| = |1 4; −2 3|, then x equals:

LHS = 2x² − 15. RHS = (1)(3) − (4)(−2) = 3 + 8 = 11. So 2x² − 15 = 11 → 2x² = 26 → x² = 13 → x = ±√13. Option 0 is ±√13. We adjust.

Q3

Ravi bought 3 pens and 2 pencils for ₹19. Priya bought 2 pens and 3 pencils for ₹16. The price of one pen is:

Let pen=x, pencil=y. 3x+2y=19, 2x+3y=16. Solving: det=9−4=5. Δₓ=57−32=25 → x=5.

Q4

The area of a parallelogram with adjacent sides given by vectors (2,1) and (1,3) using determinant is:

Area = |det[2 1; 1 3]| = |6 − 1| = 5.

Q5

If A and B are non-singular 3×3 matrices such that |A| = 2 and |B| = 3, then |2AB⁻¹| equals:

|2AB⁻¹| = 2³ |A| |B⁻¹| = 8 × 2 × (1/3) = 16/3.

Q6

If A is a 3×3 matrix with |A| = 2, find the value of |adj(adj A)|.

The formula is |adj(adj A)| = |A|^((n-1)²). Here n=3, so |adj(adj A)| = 2^((3-1)²) = 2⁴ = 16.

Q7

What is the determinant of an identity matrix of order n?

The identity matrix is a diagonal matrix with 1s on the diagonal. The product of these is always 1.

Q8

If |A| = 0 and (adj A)B ≠ O, the system of equations AX = B is:

This is the mathematical condition for an inconsistent system, meaning the lines or planes do not intersect at a common point.

Q9

The vertices of a triangular field are (0,0), (4,0) and (0,6). Its area is:

Area = (1/2) |0(0−6) + 4(6−0) + 0(0−0)| = (1/2) |24| = 12.

Q10

If two rows of a determinant are identical, the value of the determinant is:

If any two rows (or columns) of a determinant are identical, its value is always 0.

Q11

For a homogeneous system of equations AX = O, a non-trivial solution exists only if:

If |A| ≠ 0, only the trivial solution (X=O) exists. For non-trivial (non-zero) solutions, |A| must be 0.

Q12

The area of a triangular plot with vertices (2,5), (6,8) and (10,11) in sq units is:

Points are collinear as slope between any two is 3/4. Hence area = 0.

Q13

A triangle has vertices (a, b+c), (b, c+a) and (c, a+b). Its area is:

Area = (1/2) |a(c+a−a−b) + b(a+b−b−c) + c(b+c−c−a)| = (1/2) |a(c−b) + b(a−c) + c(b−a)| = (1/2)|ac−ab+ab−bc+bc−ac| = 0.

Q14

If A = [3 2; 1 4], then (A⁻¹)⁻¹ equals:

The inverse of the inverse is the original matrix itself.

Q15

The value of the determinant |2 3; 4 5| is:

For a 2×2 matrix |a b; c d|, determinant = ad − bc. So (2)(5) − (3)(4) = 10 − 12 = −2.