1.If \[D_{1}\] and \[D_{2}\] are two \[3 \times3\] diagonal matrices, then
a) \[D_{1}D_{2}\] is diagonal matrix
b) \[D_{1}D_{2}=D_{2}D_{1}\]
c) \[D_1^2+D_2^2\] is a diagonal matrix
d) All of the above
Explanation:
2. Let \[A=\begin{bmatrix}1 & 0 \\1 & 1 \end{bmatrix}\] then
a) \[A^{-n}=\begin{bmatrix}1 & 0 \\-n & 1 \end{bmatrix}\forall n \epsilon N\]
b) \[\lim_{n \rightarrow \infty}\frac{1}{n^{2}}A^{-n}=\begin{bmatrix}0 & 0 \\0 & 0 \end{bmatrix}\]
c) \[\lim_{n \rightarrow \infty}\frac{1}{n}A^{-n}=\begin{bmatrix}0 & 0 \\-1 & 0 \end{bmatrix}\]
d) All of the above
Explanation:
3. Let A, B and C be \[2 \times2\] matrices with entries from the set of real numbers. Define '*' as follows
\[A*B=\frac{1}{2}\left(AB+BA\right)\] , then
a) A * B = B * A
b) \[A * A = A^{2}\]
c) A * (B + C) = A * B + A * C
d) All of the above
Explanation: A * B = B * A
4. With A, B, C as in Q.No.3, define 'o' as follows:
\[AoB=\frac{1}{2}\left(AB-BA\right)\] , then
a) AoA = O
b) AoI = O
c) Ao(B + C) = AoB + AoC
d) All of the above
Explanation: AoA = O
5. Suppose a, b > 0 and let \[\triangle\left(x\right)=\begin{bmatrix}x & a & b \\a & x & b \\a & b & x\end{bmatrix},x \epsilon R\]
then \[\frac{\triangle'\left(0\right)}{\triangle\left(0\right)}\] is equal to
a) \[\frac{1}{a+b}\]
b) \[1-\left(\frac{1}{a}+\frac{1}{b}\right)\]
c) 0
d) \[\frac{1}{a+b}-\frac{1}{a}-\frac{1}{b}\]
Explanation:
6. Let \[\triangle=\begin{bmatrix}ln \left(2/15\right) & ln\left(4\right) & ln\left(4\right) \\ ln\left(9\right) & ln\left(3/10\right) & ln\left(9\right) \\ ln\left(25\right) & ln\left(25\right) & ln\left(5/6\right)\end{bmatrix}\]
then \[\triangle\] is equal to
a) \[\left[ln\left(30\right)\right]^{3}\]
b) \[\left[ln\left(60\right)\right]^{3}\]
c) \[\left[ln\left(30\right)\right]^{2}\]
d) \[\left[ln\left(60\right)\right]^{2}\]
Explanation: Let a = ln(2), b = ln(3), c = ln(5), then
7. Let \[\triangle\] ABC be an acute angled triangle
such that
\[\frac{\tan A}{\tan B}-\frac{\tan B}{\tan A}+\frac{\tan B}{\tan C}-\frac{\tan C}{\tan B}+\frac{\tan C}{\tan A}-\frac{\tan A}{\tan C}=0\]
then
a) \[\triangle ABC\] is an equilateral triangle
b) \[\triangle ABC\] is an isosceles triangle
c) one of the angles of \[\triangle ABC\] is \[\pi/3\]
d) one of the angle of \[\triangle ABC\] must be \[\pi/4\]
Explanation: Let x = tan A, y = tan B, z = tan C, then
8. Let a, b, c be distinct positive real numbers
such that \[S=\sum\frac{1}{a\left(a-b\right)\left(a-c\right)}\] and \[T=\frac{1}{abc}\]
then
a) S=T
b) S+T=0
c) S = (a + b + c)T
d) S = (bc + ca + ab)T
Explanation:
9. Let x, y, z be three real distinct real
numbers, and let
\[S=\begin{bmatrix}x+y^{2} & y+z^{2} & z+x^{2} \\x^{2}+y^{3} & y^{2}+z^{3} & z^{2}+x^{3} \\x^{3}+y^{4} & y^{3}+z^{4} & z^{3}+x^{4}\end{bmatrix}\]
and \[T=\begin{bmatrix}1 & 1 & 1 \\x & y & z \\x^{2} & y^{2} & z^{2}\end{bmatrix}\]
then
a) \[\frac{\triangle}{\triangle_{1}}=xyz \]
b) \[\frac{\triangle_{1}}{\triangle}=\frac{1}{1+x^{2}y^{2}z^{2}}\]
c) \[\frac{\triangle_{1}}{\triangle}=\frac{1}{1+xyz}\]
d) \[\frac{\triangle_{1}}{\triangle}=xyz +x^{2}y^{2}z^{2}\]
Explanation:
10. Let \[A=\left(a_{ij}\right)_{3\times 3}\] be such that det(A) = 5.
Suppose \[b_{ij}=2^{i+j}a_{ij}\left(1\leq i, j\leq3\right)\] and let \[B=\left(b_{ij}\right)_{3 \times 3}\] , then
det(B) is equal to
a) \[2^{16}\]
b) \[2^{15}\]
c) \[2^{14}\]
d) \[2^{12}\]
Explanation:
