The upward speed v(t)of a rocket at time t is approximated by v(t) = at² + bt + c, 0 ≤ t ≤ 100 where a, b and c are constants. It has been found that the speed at times t = 3, t = 6, and t = 9 seconds are respectively, 64, 133, and 208 miles per second respectively. Find the speed at time t = 15 seconds. (Use Gaussian elimination method.)

Investigate for what values of λ and μ the system of linear equations x  +  2y  +  z  =  7 ,   x  +  y  +  λz   =  μ ,   x  +  3y  −  5z   =  5 has
(i) no solution 
(ii) a unique solution 
(iii) an infinite number of solutions

Test for consistency and if possible, solve the following systems of equations by rank method.
2x - y + z = 2, 6x - 3y + 3z = 6, 4x - 2y + 2z = 4

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)

Solve the following systems of linear equations by Cramer’s rule:
\(\frac { 3 }{ x } -\frac { 4 }{ y } -\frac { 2 }{ z } \) -1 = 0, \(\frac { 1 }{ x } +\frac { 2 }{ y } +\frac { 1 }{ z } \) - 2 = 0, \(\frac { 2 }{ x } -\frac { 5 }{ y } -\frac { 4 }{ z } \) + 1 = 0