Determinants

37 solved questions from Class 12 Mathematics, each with the full working. Tap any question to see the step-by-step solution.

• 1 What is the domain, range, or fundamental theorem introduced in 'Determinants' (Class 12 Mathematics)? medium MCQ Functions & Theorems
• 2 What is the derivative or integral formula associated with 'Determinants'? medium MCQ Calculus & Formulae
• 3 How do you find the solution set or optimum value in 'Determinants'? medium MCQ Problem Solving & Optimization
• 4 Let A(x) = x 2 3 4. Find d/dxdet(A(x)). hard 3 marks Derivative of determinant with variable
• 5 Compute the determinant of the matrix 3 5 2 7 easy MCQ 1 mark Determinant of a 2×2 matrix
• 6 If det(AB)=12 and det(A)=3, find det(B). hard 3 marks Determinant of product of matrices
• 7 For what value of (k) is the determinant of k 4 6 8 zero? medium 2 marks Determinant zero condition
• 8 For what value of k is the determinant of 5 2 kend zero? easy 1 mark Determinant zero condition
• 9 Compute the determinant of x 1 1 x 1 1 xend and evaluate it for x=3. hard 4 marks Determinant of a matrix with constant off-diagonal elements
• 10 Find the determinant of 2 1 3 4 -1 0 2. medium 2 marks Cofactor expansion for 3×3 determinants
• 11 Let A be the 3×3 matrix with entries a_ij=i+j (i,j=1,2,3). Compute det(A). hard 3 marks Row operations to detect linear dependence
• 12 Calculate the determinant of the 2×2 matrix 7 3 4 9. Give your answer as an integer. easy 2 marks Determinant of a 2×2 matrix
• 13 For what value of k does the determinant of the matrix k 4 1 2 k 5 3 6 k become zero? Choose the correct option. medium MCQ 3 marks Condition for zero determinant (singular matrix)
• 14 Consider the matrix M(t)=t 1 2 3 t 4 5 6 t.\n(a) Find the determinant det M(t) as a polynomial in t.\n(b) Determine all real values of t for which M(t) is singular (i.e., det M(t)=0). hard 5 marks Determinant of a 3×3 matrix with a parameter
• 15 If det(AB)=12 and det(A)=3, find det(B). Give your answer as a single integer. medium 2 marks Multiplicative property of determinants
• 16 Let A(t)=t 1 0 2 t 3 4 5 t.\n(a) Find det A(t) as a polynomial in t.\n(b) Determine all real values of t for which A(t) is singular.\n(c) Compute the sum of all such real values of t. hard 5 marks Parameterised determinant and roots of cubic
• 17 Find the determinant of the 2×2 matrix 12 5 7 9. easy 1 mark Determinant of $2\times2$ matrix
• 18 If det P = -3 for a 3×3 matrix P, find det(5 P^-1). medium 2 marks Determinant of scalar multiple and inverse
• 19 Let B be a 3×3 matrix with det B = -2. For which value of k does the matrix C = k B have determinant 54?\n\nA) -3\nB) 3\nC) -2\nD) 2 medium MCQ 2 marks Scaling property of determinants
• 20 Compute the determinant of the 2×2 matrix 8 5 7 6. easy 1 mark Determinant of a 2×2 matrix
• 21 If A is a 4×4 matrix with det A = 5, find det(adj A). (Here adj A denotes the adjoint of A.) medium 2 marks Determinant of the adjoint matrix
• 22 Consider the 4×4 matrix E(λ)=λ 1 1 1 1 λ 1 1 1 1 λ 1 1 1 1 λ.\n\n(a) Find det E(λ) as a polynomial in λ.\n(b) Determine all real values of λ for which E(λ) is singular. hard 4 marks Determinant of a matrix with a parameter (using eigenvalue method or row operations)
• 23 Compute the determinant of the 2×2 matrix \begin{bmatrix}13 & 7\\ 5 & 9\end{bmatrix}. easy 1 mark Determinant of a $2\times2$ matrix
• 24 Let P be a 3×3 matrix with det P = -5. Find the determinant of the matrix Q = (-2 P)^-1. medium 2 marks Determinant of scalar multiple and of an inverse matrix
• 25 Define F(λ)=λ 2 3 2 λ 4 3 4 λ.\n\n(a) Find det F(λ) as a polynomial in λ.\n(b) Determine all real values of λ for which F(λ) is singular.\n(c) Let G=adj F(1). Compute det G. hard 5 marks Determinant of a parameter matrix and determinant of an adjoint
• 26 Let A=1 0 2 3 -1 4 5 2 -3 and let B be the matrix obtained by interchanging the first and third rows of A. If det A = 7, what is det B? medium MCQ 2 marks Effect of row interchange on determinant
• 27 Let A be a 4×4 matrix with det A = -6. Define B = (3 A^-1)^T. What is det B? hard MCQ 3 marks Determinant of a scalar multiple of an inverse matrix
• 28 For the matrix A(t)=t 1 2 3 t 4 5 6 t, find all real values of t for which A(t) is singular (i.e., det A(t)=0). medium 2 marks Determinant as a polynomial in a parameter
• 29 Find the determinant of the matrix \begin{bmatrix}4 & 2 & 1\\ 0 & -3 & 5\\ 7 & 0 & 2\end{bmatrix}. easy 1 mark Determinant of 3×3 matrix
• 30 Let P=2 -1 0 1 3 4 5 2 -2. Compute det(adj P). hard 3 marks Determinant of an adjoint matrix
• 31 If det C = 12 for a 3×3 matrix C, what is det (5 C^-1)? medium MCQ 2 marks Determinant of scalar multiple of inverse matrix
• 32 A 3×3 matrix K satisfies det K = -8. Find det((2K^T)^-1 (3K)). medium 3 marks Determinant of product with scalar multiples and transpose
• 33 Find the determinant of the matrix \begin{bmatrix}6 & 2 & 1\\0 & -3 & 4\\5 & 7 & -2\end{bmatrix}. easy 1 mark Determinant of 3×3 matrix by cofactor expansion
• 34 Let H(λ)=λ 2 2 2 2 λ 2 2 2 2 λ 2 2 2 2 λ. hard 5 marks Determinant of matrix $aI+bJ$ and adjoint of singular matrix
• 35 Compute the determinant of the matrix \begin{bmatrix}5 & -2\\ 3 & 4\end{bmatrix}. easy 1 mark Determinant of a 2x2 matrix
• 36 Let B be a 4×4 matrix with det B = 3. Define C = (2B^-1)^T·(5B). Find det C. hard 5 marks Determinant properties for transpose, inverse, scalar multiplication, and product
• 37 Let A be a 3×3 matrix with det A = 7. Find det(adj A). medium 2 marks Determinant of adjoint matrix