If a, b and c are cube roots of unity, is
\(\begin{array}{l}\left|\begin{array}{ccc} e^{a} & e^{2 a} & e^{3 a} \\ e^{b} & e^{2 b} & e^{3 b} \\ e^{c} & e^{2... View Article
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If [latex]\Delta_{r}=\left|\begin{array}{ccc} 1 & n & n \\ 2 r & n^{2}+n+1 & n^{2}+n \\ 2 r-1 & n^{2} & n^{2}+n+1 \end{array}\right|[/latex] and [latex]\sum _{r=0}^{m}\Delta_{r}=56[/latex], then n =
1) 4 2) 6 3) 7 4) 8 Solution: (3) 7
The value of the determinant is
\(\begin{array}{l}\begin{vmatrix} sin\alpha &cos\alpha &sin(\alpha+\gamma) \\ sin\beta&cos\beta &sin(\beta+\gamma) \\ sin\delta&cos\delta &sin(\gamma+\delta) \end{vmatrix}\end{array} \) 1) sinα sinβ sinδ 2) cosα cosβ cosδ... View Article
If [latex]\left|\begin{array}{ccc} x+y+2 z & x & y \\ z & y+z+2 x & y \\ z & x & z+x+2 y \end{array}\right|=[/latex] = k (x + y + z)3, then the value of k is
1) 1 2) 2 3) 4 4) 8 Solution: (2) 2
If the area of the triangle on the complex plane formed by the points z , z + iz is 200, then the value of 3|z| be equal to
1) 20 2) 40 3) 60 4) 80 Solution: (3) 60 \(\begin{array}{l}\begin{array}{l} Z=x+i y \text { then } z+i z=x+i... View Article
[latex]If \: \Delta_{r} = \left|\begin{array}{ccc} 2 r-1 & { }^{m} C_{r} & 1 \\ m^{2}-1 & 2^{m} & m+1 \\ \sin ^{2}\left(m \right) & \sin ^{2}(m) & \sin ^{2}(m+1) \end{array}\right| then\: the \: value \: \sum_{r=0}^{m}\Delta _{r}=[/latex]
1) 1 2) 0 3) 2 4) None of these Solution: (2) 0
If ɑ, β ≠ 0, f (n) = ɑn + βn and [latex]\left|\begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2)+1+f(3) & 1+f(3) & 1+f(4) \end{array}\right|[/latex] = k (1 – ɑ)2 (1 – β)2 (ɑ – β)2, then k =
1) αβ 2) 1/ αβ 3) 1 4) – 1 Solution: (3) 1
If m [–3 4] + n [4 –3] = [10 – 11] , then 3m + 7n is equal to
1) 3 2) 5 3) 10 4) 1 Solution: (4) 1 \(\begin{array}{l}\begin{array}{l} m\left[\begin{array}{ll} -3 & 4 \end{array}\right]+n\left[\begin{array}{ll} 4 & -3... View Article
If A is a skew-symmetric matrix of order n and C is a column matrix of the order of n × 1, Then CTAC is
1) an identity matrix of order n 2) an identity matrix of order 1 3) a zero matrix of order... View Article
If A = [latex]\begin{bmatrix} 1 &-1 \\ 2&-1 \end{bmatrix}[/latex] and B = [latex]\begin{bmatrix} 1 &a \\ 4&b \end{bmatrix}[/latex] and (A + b)2 = A2 + B2. Then a and b are respectively
1) 1 , – 1 2) 2 , –3 3) – 1, 1 4) 3, – 2 Solution: (1) 1... View Article
If A + I = [latex]\begin{bmatrix} 3 &-2 \\ 4&1 \end{bmatrix}[/latex], then (A + I) (A – I) =
1) \(\begin{array}{l}\begin{bmatrix} -5 &-4 \\ 8&-9 \end{bmatrix}\end{array} \) 2) \(\begin{array}{l}\begin{bmatrix} -5 &-4 \\ -8&9 \end{bmatrix}\end{array} \) 3) \(\begin{array}{l}\begin{bmatrix} 5 &4... View Article
If P = [latex]\begin{bmatrix} \sqrt{3}/2 &1/2 \\ -1/2 &\sqrt{3}/2 \end{bmatrix}[/latex] then A = [latex]\begin{bmatrix} 1 &1 \\ 0 &0 \end{bmatrix}[/latex] and Q = PAPT, then PTQ2005P =
If P = \(\begin{array}{l}\begin{bmatrix} \sqrt{3}/2 &1/2 \\ -1/2 &\sqrt{3}/2 \end{bmatrix}\end{array} \) then A = \(\begin{array}{l}\begin{bmatrix} 1 &1 \\ 0 &0... View Article
If A is matrix that A2 = A + I, where I is unit matrix then A5 is equal to
1) 5A + I 2) 5A + 2I 3) 5A + 3I 4) 5A + 4I Solution: (3) 5A +... View Article
If U = [latex]\begin{bmatrix} 2 &-3 &4 \end{bmatrix},X=\begin{bmatrix} 0 &2 &3 \end{bmatrix},V=\begin{bmatrix} 3\\ 2\\ 1 \end{bmatrix},Y=\begin{bmatrix} 2\\ 2\\ 4 \end{bmatrix}[/latex], then UV + XY =
1) [20] 2) 20 3) [-20] 4) – 20 Solution: (1) [20] \(\begin{array}{l}\begin{array}{l} U=\left[\begin{array}{lll} 2 & -3 & 4 \end{array}\right]... View Article
If A = [latex]\begin{bmatrix} 1 &1 \\ 1&1 \end{bmatrix}[/latex] then A100 =
1) 2100A 2) 299A 3) 100A 4) 299A Solution: (2) 299A \(\begin{array}{l}\begin{array}{l} A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1... View Article
If matrix A = [latex]\begin{bmatrix} 1 &-3 \\ 2&k \end{bmatrix}[/latex] and A2 – 4A + 10I = A, then k =
1) 0 2) – 4 3) 4 but not 1 4) 1 or 4 Solution: (3) 4 but not 1... View Article
If matrix A = [latex]\begin{bmatrix} 1/\sqrt{2} &1/\sqrt{2} \\ -1/\sqrt{2}& -1/\sqrt{2} \end{bmatrix}[/latex] is
If matrix A = \(\begin{array}{l}\begin{bmatrix} 1/\sqrt{2} &1/\sqrt{2} \\ -1/\sqrt{2}& -1/\sqrt{2} \end{bmatrix}\end{array} \) is 1) unitary 2) orthogonal 3) nilpotent 4)... View Article
If A (ɑ) = [latex]\begin{bmatrix} cos\alpha &sin\alpha \\ -sin\alpha& cos\alpha \end{bmatrix}[/latex], then the matrix A2 (ɑ) =
1) A(2α) 2) A(α) 3) A(3α) 4) A(4α) Solution: (1) A(2α) \(\begin{array}{l}\begin{array}{l} A(\alpha)=\left[\begin{array}{cc} \cos \alpha & \sin \alpha \\ -\sin... View Article
If A = [latex]\begin{bmatrix} 1 &2 \\ 2&4 \end{bmatrix}[/latex] and B = [latex]\begin{bmatrix} a &0 \\ 0&b \end{bmatrix}[/latex] a, b ∈ N. Then,
1) there exist more than one but number of B’s such that AB = BA 2) there exist exactly more... View Article
If A and B are square matrices of size n × n such that A2 – B2 = (A – B)(A + B), then which of the following will be always correct?
1) AB = BA 2) either A or B is a zero matrix 3) either A or B is an... View Article
