Further Mathematics

1. 1.

   Column vectors are first reflected in y = x by S = [[0,1],[1,0]], then stretched parallel to x by D = [[2,0],[0,1]]. Find the combined matrix and image of (3,-1).

   [3 marks] · no calculator
2. 2.

   A triangle of area 7 is transformed by [[2,1],[1,-1]]. Find its image area and state whether orientation is reversed.

   [3 marks] · no calculator
3. 3.

   Find the eigenvalues and corresponding eigenvector directions of A = [[3,1],[1,3]].

   [3 marks] · no calculator
4. 4.

   For A = [[k,2],[3,6]], find k for singularity. At that k, describe the image of the whole plane and the vectors sent to zero.

   [3 marks] · no calculator
5. 5.

   For A = [[3,1],[1,3]], derive A^n for positive integers n using its eigenvectors.

   [3 marks] · no calculator
6. 6.

   Let B = [[2,1],[0,2]]. Show that B is not diagonalisable and derive B^n for n >= 1.

   [3 marks] · no calculator

Original, unofficial practice questions from SubjectScout — not official past-paper questions.