Class 12 Determinants Important Questions: Your Path to Excellence
Class 12 determinants important questions form the backbone of effective board exam preparation. These carefully selected problems help students master properties and applications.
Determinants require conceptual clarity and consistent practice. Students often struggle with expansion techniques and properties. However, regular practice builds confidence. Moreover, it enhances problem-solving speed.
Smart Preparation with Curated Resources
Targeted practice yields remarkable results. Students should focus on frequently tested concepts. Understanding step-wise marking schemes is equally vital. This approach ensures comprehensive syllabus coverage.
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Consistent revision helps retain formulas. Solving diverse problems builds adaptability. Consequently, students approach exams with greater assurance and readiness.
Determinants HOTS & Important Questions Class 12 Mathematics
1. If \( A \) is a square matrix of order 3 and \( |A| = 5 \), find the value of \( |adj(adj A)| \).
2. Without expanding the determinant at any stage, prove that:
\[ \begin{vmatrix} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0 \end{vmatrix} = 0 \]3. If \( a, b, c \) are in Arithmetic Progression (A.P.), find the value of the determinant:
\[ \begin{vmatrix} 2y+4 & 5y+7 & 8y+a \\ 3y+5 & 6y+8 & 9y+b \\ 4y+6 & 7y+9 & 10y+c \end{vmatrix} \]4. If \( A = \begin{bmatrix} 2 & -3 \\ 3 & 4 \end{bmatrix} \), show that \( A^2 – 6A + 17I = O \). Hence, find \( A^{-1} \).
5. Let \( A \) be a non-singular square matrix of order \( n \). Prove that \( |adj A| = |A|^{n-1} \).
6. If \( x, y, z \) are non-zero real numbers, find the inverse of the matrix:
\[ A = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix} \]7. Using properties of determinants, prove that:
\[ \begin{vmatrix} a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b \end{vmatrix} = 2(a+b+c)^3 \]8. If \( A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix} \), where \( 0 \le \theta \le 2\pi \), then find the range of \( |A| \).
9. Solve the following system of equations using matrix method:
\[ \frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4; \quad \frac{4}{x} – \frac{6}{y} + \frac{5}{z} = 1; \quad \frac{6}{x} + \frac{9}{y} – \frac{20}{z} = 2 \]10. If \( a, b, c \) are real numbers and \( \Delta = \begin{vmatrix} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{vmatrix} = 0 \), show that either \( a+b+c=0 \) or \( a=b=c \).
11. Find the equation of the line joining \( A(1, 3) \) and \( B(0, 0) \) using determinants and find \( k \) if \( D(k, 0) \) is a point such that area of \( \triangle ABD \) is 3 sq. units.
12. If \( A \) is an invertible matrix of order 2, then prove that \( \det(A^{-1}) = \frac{1}{\det(A)} \).
13. For what value of \( k \) is the following system of equations inconsistent?
\[ 4x + 6y = 11; \quad 2x + ky = 7 \]14. If \( A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} \), then verify that \( A(adj A) = |A|I \).
15. Using properties of determinants, solve for \( x \):
\[ \begin{vmatrix} x+a & b & c \\ a & x+b & c \\ a & b & x+c \end{vmatrix} = 0 \]Answers and Hints
1. Answer: \(5^4 = 625\). Hint: Use the property \(|adj A| = |A|^{n-1}\) for a square matrix of order \(n\). Then, \(|adj(adj A)| = |adj A|^{n-1} = |A|^{(n-1)^2}\).
2. Hint: Observe that the matrix is skew-symmetric of odd order, so its determinant is zero.
3. Answer: 0. Hint: Since \(a, b, c\) are in A.P., \(2b = a + c\). Show that the rows/columns are linearly dependent.
4. Answer: \(A^{-1} = \frac{1}{17} \begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}\). Hint: Verify \(A^2 – 6A + 17I = O\) and express \(A^{-1}\) as \(\frac{1}{17}(6I – A)\).
5. Hint: Use the definition of the adjugate matrix and the property \(|adj A| = |A|^{n-1}\).
6. Answer: \(\text{diag}(x^{-1}, y^{-1}, z^{-1})\). Hint: The inverse of a diagonal matrix is the diagonal matrix of reciprocals.
7. Hint: Use row operations to simplify the determinant. Factor out \((a+b+c)\) from each row/column.
8. Answer: \([2, 4]\). Hint: Expand the determinant and simplify to \(4 – 2\sin^2 \theta\).
9. Answer: \(x=2, y=3, z=5\). Hint: Let \(u = \frac{1}{x}\), \(v = \frac{1}{y}\), \(w = \frac{1}{z}\) and solve the linear system.
10. Hint: Expand the determinant and factorize to show \((a+b+c)^2 (2a-b-c) = 0\).
11. Answer: \(y = 3x\) and \(k = \pm 2\). Hint: Use the area formula for a triangle with vertices \((1,3)\), \((0,0)\), and \((k,0)\).
12. Hint: Use the property \(A^{-1} = \frac{adj A}{|A|}\) and take determinants on both sides.
13. Answer: \(k = 3\). Hint: The system is inconsistent if the determinant of the coefficient matrix is zero and the equations are not proportional.
14. Hint: Compute \(adj A\) and multiply by \(A\). Verify that the result is \(|A|I\).
15. Answer: \(x = 0\) or \(x = -(a+b+c)\). Hint: Expand the determinant and factorize.
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