Test 5

Pure Mathematics – Matrix and Determinants

1.

p.1

King’s College 2007 – 2008 F.6 Pure Mathematics Revision Test 5 Time Allowed: 35 minutes Total Mark: 30 Consider the system of linear equations

⎧x + y + z = a (E ) : ⎪⎨kx + y + kz = b ⎪k 2 x + y + kz = c ⎩ (a) Find all real values of k such that for any given values of a, b and c, (E) has a unique solution, and hence solve (E). (b) For k = 0, find the relation between a, b and c such that (E) is consistent, and hence solve (E). 2.

In each of the following, find the equation of the image of the curve (Γ), under the transformation specified. (a) (Γ): 5 x 2 − 2 3 xy + 7 y 2 − 4 = 0

A rotation through an angle

5π about the origin. 6

(b) (Γ): y 2 − 2 3 xy − x 2 − 2 = 0 A reflection about the line y = (c) (Γ): x 2 + y 2 − 4 x = 0

3x .

A reflection about the line y = x, followed by a shear which maps (1, 1) to (-1, 3).

--End of Test --