C2 Test 1B: Remainder Factor Theorem, the CircleChurston Ferrers Grammar School1C2 Test 1B: Remainder Factor Theorem, the CircleTest is 35 minutes long1.The circle C has centre (5, 13) and touches the x-axis.(a)Find an equation of C in terms of x and y.(2) (b)Find an equation of the tangent to C at the point (10, 1), giving your answer in the form ay + bx + c = 0, where a, b and c are integers.(5)(Total 7 marks) 2.f(x) = 2x3 - x2 + px + 6,where p is a constant.Given that (x 1) is a factor of f(x), find(a)the value of p,(2) (b)the remainder when f(x) is divided by (2x + 1).(2)(Total 4 marks) 3.A circle C1 has equationx2 + y2 12x + 4y + 20 = 0.(a)Find the coordinates of the centre of C1.(2) (b)Find the radius of C1.(2) The circle C1 cuts the x-axis at the points A and B.(c)Find an equation of the circle C2 with diameter AB.(6)(Total 10 marks) 4.f(n) = n3 + pn2 + 11n + 9, where p is a constant.(a)Given that f(n) has a remainder of 3 when it is divided by (n + 2), prove that p = 6 .(2)(b)Show that f(n) can be written in the form (n + 2)(n + q)(n + r) + 3, where q and r are integers to be found.(3)(c)Hence show that f(n) is divisible by 3 for all positive integer values of n.(2)(Total 7 marks)