Archives

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 = –1... 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) 0... 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... 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 Solution: (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 and det(p) = ± 1 3) there are more than one... 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, II and III By the properties of determinants all the above I, II,... 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 + b + c Solution: (1) 0... View Article

If a, b and c are cube roots of unity, is

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

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... View Article

The value of the determinant is

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

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... 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... View Article

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... View Article

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... 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 C^TAC is

1) an identity matrix of order n 2) an identity matrix of order 1 3) a zero matrix of order 1 4) None of the above Solution: (3) a zero... 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 , – 1... View Article

If A + I = [latex]\begin{bmatrix} 3 &-2 \\ 4&1 \end{bmatrix}[/latex], then (A + I) (A – I) =

1) 2) 3) 4) Solution: (1)... 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 = PAP^T, then P^TQ²⁰⁰⁵P =

If P = and Q = PAPT, then PTQ2005P = 1) 2) 3) 4) Answer: (1) Solution: P = and A = and Q = PAPT = (P . PT). A PPT =... 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 + 3I... View Article

M	T	W	T	F	S	S
					1	2
3	4	5	6	7	8	9
10	11	12	13	14	15	16
17	18	19	20	21	22	23
24	25	26	27	28	29	30
31