JEE Mathematics Determinants MCQs Set 14

Answer: (b) \( \frac{-\Delta(b,c,d)}{\Delta(a,b,c)} \)

Explanation:
1. Rewrite the equations as a \( x + b y + c z = -d \). By Cramer's rule \( x = \Delta(-d, b, c)/\Delta(a, b, c) = -\Delta(d, b, c)/\Delta(a, b, c) \).
2. Moving the column \( d \) from first to last position is a cyclic shift of three columns, an even permutation, so \( \Delta(d, b, c) = \Delta(b, c, d) \).
3. Hence \( x = -\Delta(b, c, d)/\Delta(a, b, c) \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (b) \( 8\cos^3\theta-4\cos\theta+6 \)

Explanation:
1. Expand along the first row: \( C(C^2 - 1) - 1(C - 6) = C^3 - 2C + 6 \).
2. Substitute \( C = 2 \cos \theta \): \( 8 \cos^3 \theta - 4 \cos \theta + 6 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) 0

Explanation:
1. The matrix equals \( J + x y^T \) where \( J \) is the all-ones matrix (rank 1) and \( x y^T \) has rank 1.
2. So the matrix has rank at most 2, and a \( 3 x 3 \) matrix of rank at most 2 has determinant 0.
3. \( \Delta = 0 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) 21

Explanation:
1. \( a_0 \) is the constant term, which equals the determinant at \( x = 0 \).
2. At \( x = 0 \): rows (0, -1, 3), (1, 2, -3), (-3, 4, 0).
3. Determinant = \( 0(0 + 12) - (-1)(0 - 9) + 3(4 + 6) = -9 + 30 = 21 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (a) 1 only

Explanation:
1. \( |B| = 1(0 - 10) - 2(0 + 4) + 3(-15 - 0) = -10 + 8 - 45 = -47 \).
2. An orthogonal matrix has \( |A| = 1 \) or \( -1 \), so |AB| = \( |A||B| \) = plus or minus 47. Statement 1 is true.
3. Matrix multiplication is not commutative in general, and there is no reason for AB = BA here. Statement 2 is not true.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (c) \( [-\sqrt2,\sqrt2] \)

Explanation:
1. A non-trivial solution exists when the determinant of coefficients is zero.
2. Determinant = \( \lambda(-\cos^2 \alpha - \sin^2 \alpha) - \sin \alpha(-\cos \alpha + \sin \alpha) + \cos \alpha(\sin \alpha + \cos \alpha) = -\lambda + \sin 2 \alpha + \cos 2 \alpha \).
3. So \( \lambda = \sin 2 \alpha + \cos 2 \alpha \), which takes every value in \( [-sqrt2, sqrt2] \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) k is not 0

Explanation:
1. Unique solution needs the coefficient determinant to be non-zero.
2. Determinant = \( 1(k + 2) - 1(2k + 3) + 1(4 - 3) = -k \).
3. So \( k \) is not equal to 0.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (a) \( [z] \)

Explanation:
1. From the ranges: \( [x] = -1, [y] = 0, [z] = 1 \).
2. The determinant is \( |[[0, 0, 1], [-1, 1, 1], [-1, 0, 2]]| = 0 - 0 + 1((-1)(0) - (1)(-1)) = 1 \).
3. This equals \( [z] \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (b) \( \theta \) only

Explanation:
1. Expand along the first row: \( x(-x^2 - 1) - \sin \theta(-x \sin \theta - \cos \theta) + \cos \theta(-\sin \theta + x \cos \theta) \).
2. = \( -x^3 - x + x \sin^2 \theta + \sin \theta \cos \theta - \sin \theta \cos \theta + x \cos^2 \theta = -x^3 - x + x = -x^3 \).
3. The value \( -x^3 \) does not contain \( \theta \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) \( -\frac12 \)

Explanation:
1. \( \det(A^{-1}) = 1(2) - (-2)(-2) = 2 - 4 = -2 \).
2. \( \det(A) = 1/\det(A^{-1}) = -1/2 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (a) \( (14)^4 \)

Explanation:
1. \( |A| = 1(1 + 2) - 2(-1 - 4) + (-1)(1 - 2) = 3 + 10 + 1 = 14 \).
2. For an \( n x n \) matrix, \( |adj A| = |A|^{n-1} \), so \( |adj(adj A)| = |A|^((n-1)^2) \).
3. With \( n = 3 \): \( 14^4 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (b) -2

Explanation:
1. Expand: \( f(x) = \cos x(x^2 - 2x^2) - x(2 \sin x - 2x \tan x) + (2x \sin x - x^2 \tan x) \).
2. Simplifying: \( f(x) = -x^2 \cos x + x^2 \tan x \).
3. Then \( f'(x) = -2x \cos x + x^2 \sin x + 2x \tan x + x^2 \sec^2 x \), so \( f'(x)/x = -2 \cos x + x \sin x + 2 \tan x + x \sec^2 x \), which tends to \( -2 \) as \( x \) tends to 0.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (b) 1

Explanation:
1. The determinant \( (k + 1)(k + 3) - 8k = k^2 - 4k + 3 = (k - 1)(k - 3) \) must be zero.
2. \( k = 1 \): equations \( 2x + 8y = 4 \) and \( x + 4y = 2 \) are the same line, so infinitely many solutions.
3. \( k = 3 \): \( 4x + 8y = 12 \) and \( 3x + 6y = 8 \) are parallel but different lines, so no solution. Only one value, \( k = 1 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) are in H.P.

Explanation:
1. A non-zero solution requires the determinant \( |[[1, 2a, a], [1, 3b, b], [1, 4c, c]]| = 0 \).
2. Expanding: (3bc - 4bc) \( - 2a(c - b) + a(4c - 3b) \) = -bc + 2ac - ab = 0.
3. So 2ac = ab + bc; dividing by abc: \( 2/b = 1/c + 1/a \). So \( 1/a, 1/b, 1/c \) are in \( A.P \)., meaning a, \( b, c \) are in \( H.P \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (b) 0

Explanation:
1. Write \( \cos(A - P) = \cos A \cos P + \sin A \sin P \).
2. The matrix is the product of \( [[\cos A, \sin A], [\cos B, \sin B], [\cos C, \sin C]] (3 x 2) \) and \( [[\cos P, \cos Q, \cos R], [\sin P, \sin Q, \sin R]] (2 x 3) \).
3. A product through a 2-dimensional space has rank at most 2, so the \( 3 x 3 \) determinant is 0.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) \( -f(\alpha)f(\beta)f(\gamma) \)

Explanation:
1. The roots are \( 1, \omega, \omega^2 \), and \( f(1) f(\omega) f(\omega^2) = (a + b + c)(a + b \omega + c \omega^2)(a + b \omega^2 + c \omega) = a^3 + b^3 + c^3 \) - 3abc.
2. The given determinant expands to \( 3abc - a^{3} - b^{3} - c^{3} = -(a^{3} + b^{3} + c^{3} - 3abc) \).
3. So it equals \( -f(\alpha) f(\beta) f(\gamma) \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) 24

Explanation:
1. Splitting each column independently gives \( 2 x 3 x 4 \) choices.
2. So \( n = 24 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (c) \( |A|^{n-2}A \)

Explanation:
1. \( adj(adj A) = |A|^{n-2} A \) for an invertible \( n x n \) matrix \( A \) (and holds in general).
2. Check with \( n = 2 \): \( adj(adj A) = A \), matching \( |A|^0 A \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) \( \Delta'=\Delta(1+pqr) \)

Explanation:
1. The new columns are \( C1 + p C2, C2 + q C3, C3 + r C1 \).
2. This is the original matrix times \( [[1, 0, r], [p, 1, 0], [0, q, 1]] \).
3. The determinant of that factor is 1(1 - 0) - 0 + r(pq - 0) = 1 + pqr. So Delta' = Delta(1 + pqr).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (d) \( (2014)^2 \)

Explanation:
1. \( \det(M_r) = r^2 - (r - 1)^2 = 2r - 1 \).
2. The sum of \( 2r - 1 \) for \( r = 1 \) to 2014 is the sum of the first 2014 odd numbers = \( 2014^2 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (c) negative

Explanation:
1. The determinant equals \( -(a^{3} + b^{3} + c^{3} - 3abc) = -(a + b + c)(a^{2} + b^{2} + c^{2} - ab - bc - ca) \).
2. For distinct positive numbers, \( a + b + c \gt 0 \) and \( a^2 + b^2 + c^2 \) - ab - bc - ca > 0, so the determinant is negative.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (a) \( D'=D^2 \)

Explanation:
1. The cofactor matrix is the transpose of \( adj A \), and \( |adj A| = |A|^{n-1} \).
2. For \( n = 3 \): \( D' = D^2 \).

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (b) \( \begin{bmatrix}2&6\\0&8\end{bmatrix} \)

Explanation:
1. \( A^2 = [[1, 8], [0, 9]] \).
2. \( f(A) = A^2 - A + 2I = [[1 - 1 + 2, 8 - 2 + 0], [0, 9 - 3 + 2]] \).
3. = [[2, 6], [0, 8]].

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (c) 2

Explanation:
1. Expanding, the determinant equals \( -(x - 2)^2 \) (using \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \)).
2. Setting it to zero gives \( x = 2, a \) repeated root.
3. So one root is 2.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.

Answer: (c) 6

Explanation:
1. Subtract row 3 from row 1 and row 2: rows 1 and 2 become (1, 0, -1) and (0, 1, -1).
2. Expanding along the first row: \( 1[1(1 + 4 \sin 2x) + \cos^2 x] - 0 + (-1)[0 - \sin^2 x] = 1 + 4 \sin 2x + \cos^2 x + \sin^2 x = 2 + 4 \sin 2x \).
3. The maximum of \( \sin 2x \) is 1, so the maximum of \( f(x) \) is 6.

Teacher's Note:
1. Use row or column operations to simplify a determinant before expanding it.
2. A determinant is zero when two rows or columns are identical or proportional, or when the matrix has rank less than its order.