Define a binary operation '*' on the set A = {0, 1, 2, 3, 4, 5}, given by a*b=(ab) mod 6. Show that for *,1 and 5 are only invertible elements with \(1^{ -1 }=1\) and \(5^{ -1 }=5\).
[Here (a, b) mod 6, we mean the remainder after dividing ab by 6 ]

Write the value of  \(tan^{-1}\left[2sin\left(2cos^{-1}{\sqrt{3}\over2}\right)\right]\)

Using elementary transformations, find the inverse of the matrix

\(\left[ \begin{matrix} 1 & 3 & -2 \\ -3 & 0 & -1 \\ 2 & 1 & 0 \end{matrix} \right] \).

If \(A=\begin{bmatrix} 4 & 1 \\ 5 & 8 \end{bmatrix}\), show that A + A^T is a symmetric matrix, where A^T denotes the transpose of matrix A.

Find the inverse of the matrix A by elementary operations and verify that \({ A }^{ -1 }A=I\), when:
\(A=\left[ \begin{matrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{matrix} \right] .\)

Solve for x,\(\begin{vmatrix} 3x-8 & 3 & 3 \\ 3 & 3x-8 & 3 \\ 3 & 3 & 3x-8 \end{vmatrix}=0\)

Mrs. Rodger got a weekly raise of $145. If she gets paid every other week, write an integer describing how the raise will affect her paycheck.

Find the vector and Cartesian equations of the line passing through the point P (1,2,3) and parallel to the planes \(\vec { r } .(\hat { i } -\hat { j } +2\hat { k } )=5\) and \(\vec { r } .(3\hat { i } +\hat { j } +\hat { k } )=6\).

Find the co-ordinates of the point, where the line \(\frac { x-2 }{ 3 } =\frac { y+1 }{ 4 } =\frac { z-2 }{ 12 } \) intersects the pIane x - y + z - 5 = 0.
Also find the angle between the line and the plane.

(Manufacturing Problem) A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

An experiment succeeds twice as often as it fails. Find the probability that in the next six trails, there will be at least 4 successes.

A husband and a wife appear in an interview for two vacancies for the same post. The probability of husbands selection is \(1\over7\) and that of wife's selection is \(1\over5\). What is the probability that
(i) both will be selected
(ii) only one of them will be selected  
(iii) none will be selected?

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of diamond cards drawn. Also, find the mean and the variance of the distribution.