Archives

If [latex]\left|\begin{array}{ccc} x^{2}+x & 3 x-1 & -x+3 \\ 2 x+1 & 2+x^{2} & x^{3}-3 \\ x-3 & x^{2}+4 & 3 x \end{array}\right|=a_{0}+a_{1}x+……+a_{7}x^{7}[/latex] then the value of a₀ is

1) 35 2) 24 3) 23 4) 22 5) 21 Solution: (5) 21... View Article

The value of the determinant [latex]\left|\begin{array}{ccc} 15 ! & 16 ! & 17 ! \\ 16 ! & 17 ! & 18 ! \\ 17 ! & 18 ! & 19 ! \end{array}\right|[/latex] =

1) 15 ! + 16 1 2) 2(15!) (16!) (17!) 3) 15! + 161 + 17! 4) 16! + 17! Solution: (2) 2(15!) (16!) (17!)... View Article

If A is a 2 × 2 matrix with non – zero entries let A² = I, where I is 2 × 2 identity matrix. Define tr (A) = Sum of diagonal element of A and |A| = Determinant of matrix A. Statement I tr (A) = 0. Statement II |A| = i

1) Statement I is correct, Statement Ii is correct; Statement Ii is the correct explanation for Statement I 2) Statement I is correct, statement... View Article

The number of 3 × 3 non-singular matrices, with four entries as I and all other entries as 0 is

1) less than 4 2) 5 3) 6 4) at least 7 Solution: (4) at least 7... View Article

[latex]\begin{bmatrix} 1 &1 &1 \\ a&b &c \\ a^{2}-bc&b^{2}-ca &c^{2}-ab \end{bmatrix}[/latex] =

1) 0 2) 1 3) abc 4) (a– b) (b – c) (c – a) Solution: (1) 0... View Article

If Z = [latex]begin{vmatrix} 1 &1+2i &-5i \ 1-2i&-3 &5+3i \ 5i&5-3i &7 end{vmatrix}[/latex] then (I = – √-1)

1) z is purely real 2) z is purely imaginary 3) 4) is purely imaginary Solution: (1) z is purely real... View Article

If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then |3AB| is equal to

1) – 9 2) –81 3) – 27 4) 81 Solution: (2) –81... View Article

Statement I. Determine if a skew-symmetric matrix of order 3 is zero. Statement II. For any matrix A , det (A^T) = det (A) and det (-A) = – det (A). Where det (B) denotes the determinant of matrix B. Then,

1) Statement I is correct, Statement II is incorrect 2) both statements are correct 3) both statements are incorrect 4) Statement I is... View Article

If ω ≠ 1 is a cube root of unity and S is the set of all non – singular matrices of the form if [latex]A=\begin{bmatrix} 1 &a &b \\ \omega&1 &c \\ \omega^{2}&\omega &1 \end{bmatrix}[/latex] where each of a, b and c is either 𝜔 or 𝜔². Then the number of distinct matrices in the set S is

1) 2 2) 6 3) 4 4) 8 Solution: (1) 2... View Article

Let P and Q be 3 × 3 matrices, P ≠ Q If P³ = Q³ and P²Q = Q²P, then the determinant of (P² + Q²) is equal to

1) –2 2) 1 3) 0 4) – 1 Solution: (3) 0... View Article

If [latex]\begin{bmatrix} 1 &2 &1 \\ 1&3 &1 \end{bmatrix}[/latex] and Q = PP^T, then the value of Q is

1) 2 2) - 2 3) 1 4) 0 Solution: (1) 2... View Article

If P, Q and R are angles of ΔPQR, then the value of [latex]\begin{vmatrix} -1 &cosR &cosQ \\ cosR&-1 &cosP \\ cosQ&cosP &-1 \end{vmatrix}[/latex] =

1) – 1 2) 0 3) 1/2 4) 1 Solution: (2) 0... View Article

The value of [latex]\begin{vmatrix} a^{2}+1 &ab &ac \\ ba&b^{2}+1 &bc \\ ca&cb &c^{2}+1 \end{vmatrix}[/latex] is

The value of is 1) (a + b + c)2 2) a2 + b2 + c2 3) a2 + b2 + c2 + 1 4) 1 Solution: (3) a2 + b2 + c2 + 1... View Article

Let P = [a_ij] bc a 3 × 3 matrix and Q = [b_ij], where b_ij = 2^i+j for 1 ≤ i, j ≤ 3, if the determinant of P is 2, then the determinant of the matrix Q is

1) 210 2) 211 3) 212 4) 213 Solution: (4) 213... View Article

The value of [latex]\begin{vmatrix} sin\alpha &cos\alpha &sin(\alpha+\gamma) \\ sin\beta&cos\beta &sin(\beta+\gamma) \\ sin\delta&cos\delta &sin(\gamma+\delta) \end{vmatrix}[/latex] is

1) sinα sinβ sinδ 2) cosα cosβ cosδ 3) 1 4) 0 Solution: (4) 0... View Article

The value of 9 ∈ [0, π / 2] and satisfying the equation [latex]\left|\begin{array}{ccc} 1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 4 \theta \end{array}\right|=0[/latex] is

1) 11π/24 2) 17π/24 3) 5π/24 4) π/24 Solution: (1) 11π/24... View Article

If three-digit numbers A28, 3B9 and 62C, where A, B and C are integers between 0 and 9, are divisible by a fixed integer k, then the determinant [latex]begin{vmatrix} A &3 &6 \ 8&9 &C \ 2&B &2 end{vmatrix}[/latex] is

1) divisible by k 2) divisible by k2 3) divisible by 2k 4) None of these Solution: (1) divisible by k... View Article