CBSE Class 12 Mathematics Determinants Assignment Set 02

This assignment has 15 short practice questions on Chapter 4 Determinants for Class 12 Mathematics (CBSE), on adjoints, cofactors, inverses, area of a triangle and values of determinants. The answer to each question is given below it for quick self-checking.

Practice Questions

Question 1. Let A be a non-singular matrix of the order 3 × 3, and |A| = −4, then find the value of |adj A|.
Answer:
16

Question 2. If \( \begin{vmatrix} 2x & 5 \\ 8 & x \end{vmatrix} = \begin{vmatrix} 6 & 5 \\ 8 & 3 \end{vmatrix} \), then find the value of x.
Answer:
x = ±3

Question 3. If \( \Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \) and A_ij is the cofactor of element a_ij, then find the sum a₁₁A₁₂ + a₂₁A₂₂ + a₃₁A₃₂.
Answer:
0

Question 4. If \( A = \begin{bmatrix} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{bmatrix} \), then find the value of λ for which A⁻¹ does not exist.
Answer:
\( \lambda = -\frac{8}{5} \)

Question 5. If the value of a third order determinant is 12, then find the value of the determinant formed by replacing each element by its corresponding cofactor.
Answer:
144

Question 6. The area of triangle with vertices (−3, 0), (3, 0) and (0, k) is 18 square units, then find the value of k.
Answer:
k = ±6

Question 7. If \( A = \begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix} \), then find A⁻¹ in terms of A.
Answer:
\( A^{-1} = \frac{1}{19}A \)

Question 8. If A_ij is the cofactor of the element a_ij of the determinant \( \begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix} \), then write the value of a₃₂A₃₂.
Answer:
110

Question 9. If A is a skew-symmetric matrix of order 3, then find the value of |A|.
Answer:
0

Question 10. Write the sum of the products of elements of any row of a determinant with the corresponding cofactors of another row.
Answer:
0

Question 11. If there are two values of a which make the determinant \( \Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86 \), then find their sum.
Answer:
−4

Question 12. If A and B are square matrices of order 3 such that |A| = −1, |B| = 3, then find the value of |3AB|.
Answer:
−81

Question 13. Find the inverse of the matrix \( \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} \).
Answer:
\( \begin{bmatrix} \frac{4}{11} & \frac{1}{11} \\ -\frac{3}{11} & \frac{2}{11} \end{bmatrix} \)

Question 14. If the points (a, 0), (0, b) and (1, 1) are collinear, then using determinant, prove that a + b = ab.
Answer:
Since the points are collinear, \( \frac{1}{2}\begin{vmatrix} a & 0 & 1 \\ 0 & b & 1 \\ 1 & 1 & 1 \end{vmatrix} = 0 \)
⇒ a(b − 1) − 0 + 1(0 − b) = 0 ⇒ ab − a − b = 0 ⇒ a + b = ab

Question 15. If \( A = \begin{bmatrix} 0 & 3 \\ 2 & 0 \end{bmatrix} \) and A⁻¹ = λ(adj A), then find the value of λ.
Answer:
\( \lambda = -\frac{1}{6} \)