Section A
Q1. If a subspace W of \mathbbR4 is generated by the vectors (3, 8, -3, -5), (1, -2, 5, -3) and (2, 3, 1, -4), then find a basis and dimension of W. Extend that basis to get a basis of \mathbbR4. Find a row echelon matrix which is row equivalent to Amreek has n number of children by his first wife. Shaina has (n + 1) children by her first husband. They marry and have children of their own also. The whole family now has 12 children. It is assumed that children of Amreek from his first wife do not fight among themselves, and likewise, children of Shaina by her first husband do not fight among themselves. Find the maximum possible number of fights between children that can take place. Prove that: Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9.
(b)
Find a row echelon matrix which is row equivalent to
A = \beginbmatrix 0 & 0 & -2 & 0 & 1 \\ 2 & 4 & 1 & 4 & 3 \\ 1 & 2 & -3 & 1 & 2 \\ 4 & 8 & 2 & 2 & 5 \endbmatrix and find the rank of A.
(c) Amreek has n number of children by his first wife. Shaina has (n + 1) children by her first husband. They marry and have children of their own also. The whole family now has 12 children. It is assumed that children of Amreek from his first wife do not fight among themselves, and likewise, children of Shaina by her first husband do not fight among themselves. Find the maximum possible number of fights between children that can take place.
(d) Prove that: \Gamma(n + 1/2) = (√(π) \Gamma(2n + 1))/(22n \Gamma(n + 1))
(e) Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9.
