8. (a) Prove by induction that, for any positive integer n, = + 1)2. n2(n (5) (b) Using the formulae for and , show that = + 2)(n2 + 7). n(n (5) (c) Hence evaluate . (2) 9. M = . (a) Describe fully the geometrical transformation represented by the matrix M. (2) The transformation represented by M maps the point with coordinates (p, onto the point A q) B with coordinates (3√2, 4√2). (b) Find the value of and the value of p q. (4) (c) Find, in its simplest surd form, the length where is the origin. OA, O (2) (d) Find M2. (2) The point is mapped onto the point by the transformation represented by M2. B C (e) Find the coordinates of C. (2) TOTAL FOR PAPER: 75 MARKS END JUNE 2014 MARK SCHEME Question Scheme Marks Number 1.(a) M1 M1 A1, A1 (4) (b) ` M1 dM1 dM1A1 OR M1
8. (a) Prove by induction that, for any positive integer n, = n2(n + 1)2. (5) (b) Using the formulae for and , show that = n(n + 2)(n2 + 7). (5) (c) Hence evaluate . (2) 9. M = . (a) Describe fully the geometrical transformation represented by the matrix M. (2) The transformation represented by M maps the point A with coordinates (p, q) onto the point B with coordinates (3√2, 4√2). (b) Find the value of p and the value of q. (4) (c) Find, in its simplest surd form, the length OA, where O is the origin. (2) (d) Find M2. (2) The point B is mapped onto the point C by the transformation represented by M2. (e) Find the coordinates of C. (2) TOTAL FOR PAPER: 75 MARKS END JUNE 2014 MARK SCHEME Question Number Scheme Marks 1.(a) M1 M1 A1, A1 (4) (b) ` M1 dM1 dM1A1 OR M1
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