page 05 MARKS 6. On a suitable domain, a curve is defined parametrically by x t = + 2 1 and ( ) ln y t = + 3 2 . Find the equation of the tangent to the curve where t = −1 3 . 7. Matrices C and D are given by: C − ⎛ ⎞ ⎜ ⎟ = − ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ 2 1 2 1 1 0 1 0 1 and D k ⎛ ⎞ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 1 2 3 0 2 1 1 1 , where k ∈\. (a) Obtain C D − ′ 2 where C′ is the transpose of C. (b) (i) Find and simplify an expression for the determinant of D. (ii) State the value of k such that 1 D− does not exist. 8. Using the substitution sin u θ = , or otherwise, evaluate sin cos θ θ dθ π π∫ 2 4 6 2 . 9. Prove directly that: (a) the sum of any three consecutive integers is divisible by 3; (b) any odd integer can be expressed as the sum of two consecutive integers. [Turn over 5 2 2 1 4 2 1