BE SURE TO ANSWER, I NEED CORRECT ANSWERS

1. a) A point travels 8 units from A1 to A2, then turns left and travels 4 units upwards to A3, as shown. At each stage, after the first one, the point moves left by half the distance of the previous stage. This pattern continues indefinitely. (i) Find the length of the line segment [A10A11]. (ii) Find the total length of all line segments, if the pattern continues indefinitely. (iii) Determine the orientation (vertical or horizontal) of the line segment [A2017, A2018]. (b) Find the sum of all natural numbers not exceeding 2018 which are divisible by11 (c) Prove that the number of diagonals in an n-sided polygon is n(n 3) , where n 3

2) A circle c passes through the points A(0, 0), B(6, 0) and C(6, 8). (i) Find the equation of the circle c. Write the length of the radius and the coordinates of the centre.

(ii) Find the measure of the angle BCA, correct to the nearest degree. (iii) Find the equation of the tangent l to the circle at C. (iv) Find the coordinates of the points where l cuts the x-axis and the y-axis and hence, find the area of the triangle formed by l and the two axes. (b) Assume the earth is a sphere with radius 6371 km. On the diagram, circle c1 represents the equator and c2 represents the circle that is at latitude 45 Both circles are on parallel planes. The point O is the centre of the earth and the point P lies at latitude 45 north of the equator. Find the length (circumference) of c2 to the nearest km

3. (a) Find the equation of the line l through the point (3, 4), which divides the line segment from (6, 2) to (3, 4) internally in the ratio 1 : 2. (b) Let ABCD be a parallelogram whose vertices have coordinates A(2, 1), B(4, 0), C(5, 3) and D(3, 2). (i) Find the area of the parallelogram ABCD. (ii) Let P be any point on the line segment [CD]. Prove that the area of the triangle ABP is half the area of the parallelogram

4. (a) How many arrangements are there of the letters LEAVING: (i) if there are no restrictions, (ii) if AE are side by side, in that order, (iii) if A and E are separated, (iv) if consonants and vowels alternate? If three letters are chosen at random from the word LEAVING without replacement, what are the chances that the first letter is N, second is I and third

5. (a) A coin is tossed five times. (i) What is the probability of getting three heads and two tails? (ii) Find the probability that the third head occurs on the fifth (last) toss. (b) Suppose that in a certain diagnostic screening programme the probability of a misdiagnosis is 30. In a sample of 100 cases independent of each other, what is the probability that 5 cases were misdiagnosed? (c) A jury of 12 people is selected from a group of 10 men and 10 women. Find the probability that the jury selected has more women than men.