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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)... 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 \(\begin{array}{l}\begin{aligned} &\left|\begin{array}{ccc} 1 & a & b \\ w... 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 \(\begin{array}{l}\begin{array}{l} \mathrm{P} \text { is } 3 \times... 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 \(\begin{array}{l}\text { Given, } \begin{aligned} P &=\left[\begin{array}{lll} 1... 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 \(\begin{array}{l}\begin{aligned} &\text { Multiply } C \text {... 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 \(\begin{array}{l}\begin{vmatrix} a^{2}+1 &ab &ac \\ ba&b^{2}+1 &bc \\ ca&cb &c^{2}+1 \end{vmatrix}\end{array} \) is 1) (a + b... 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 \(\begin{array}{l}\begin{array}{l} \mathrm{P}=\left[\mathrm{a}_{\mathrm{i}}\right] ; \mathrm{Q}=\left[\mathrm{b}_{\mathrm{i}}\right] ; \text { where }... 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 \(\begin{array}{l}\Delta=\left|\begin{array}{lll}\sin \alpha & \cos \alpha... 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 \(\begin{array}{l}\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4... View Article

Find the area of the triangle with vertices (2,3), (0,1) and (1,2)

1) 1/2 sq unit 2) 0 sq unit 3) 2 sq units 4) 2 (1/2) sq units Solution: (2) 0... View Article

The determinant [latex]begin{vmatrix} x &y &x+y \ y&x+y &x \ x+y&x &y end{vmatrix}[/latex] is divisible by

1) x – y 2) x2 – y2 + xy 3) x2 + xy + y2 4) x2 – xy... View Article

If K > 1 and the determinant of the matrix A², where matrix A is k², then |ɑ| =

\(\begin{array}{l}A=\begin{bmatrix} k &k\alpha &\alpha \\ 0&\alpha &k\alpha \\ 0&0 &k \end{bmatrix}\end{array} \) 1) 1/K2 2) k 3) K2 4) 1/k... 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... View Article

If x, y and z are all different and [latex]\left|\begin{array}{ccc} x & x^{2} & 1+x^{3} \\ y & y^{2} & 1+y^{3} \\ z & z^{2} & 1+z^{3} \end{array}\right|[/latex] = 0 then

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

If a, b and c are non-zero and different from 1, then the value of [latex]\left|\begin{array}{ccc} \log _{a} 1 & \log _{a} b & \log _{a} c \\ \log _{a}\left(\frac{1}{b}\right) & \log _{b} 1 & \log _{a}\left(\frac{1}{c}\right) \\ \log _{a}\left(\frac{1}{c}\right) & \log _{a} c & \log _{c} 1 \end{array}\right|[/latex] is

1) 0 2) 1 + loga(a + b + c) 3) loga(a + b + c) 4) 1 Solution: (1)... View Article

If [latex]\begin{vmatrix} 2a &x_{1} &y_{1} \\ 2b&x_{2} &y_{2} \\ 2c&x_{3} &y_{3} \end{vmatrix}[/latex] = abc / 2 ≠ 0, then the area of the triangle whose vertices are [(x₁ / a), (y₁ / a)], [(x₂ / b), (y₂ / b)] and [(x₃ / c), (y₃ / c)] is

1) 1/4(abc) 2) 1/8 (abc) 3) 1/4 4) 1/8 5) 1/12 Solution: (4) ⅛ \(\begin{array}{l}\begin{aligned} &\left|\begin{array}{ccc} 2 a & x_{1}... View Article

If a₁, a₂, a₃ …… are in AP, then the value of [latex]\begin{vmatrix} a_{1} &a_{2} &1 \\ a_{2}&a_{3} &1 \\ a_{3}&a_{4} &1 \end{vmatrix}[/latex] is

1) a4 – a1 2) [a1 + a4] / 2 3) 1 4) [a2 + a3] / 2 5) 0... View Article

If I denotes the 3 × 3 identity matrix and P is a matrix obtained by rearranging the columns of I. Then,

1) there are six distinct choices for P and det(P) = 1 2) there are six distinct choices for P... View Article

Consider the following statements: I. If any two rows or columns of a determinant are identical, then the value of the determinant is zero. II. If the corresponding rows and columns of a determinant are interchanged, then the value of the determinant does not change. III. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant in sign. Which of these is correct?

1) I and II 2) II and III 3) I, II and III 4) I and III Solution: (3) I,... View Article

If a, b and c are in AP, then the value of [latex]\begin{vmatrix} x+2 &x+3 &x+a \\ x+4&x+5 &x+b \\ x+6&x+7 &x+c \end{vmatrix}[/latex] is

1) 0 2) x – (a + B + c) 3) a + b + c 4) 9x2 + a... View Article

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