Evil A Math Paper

Evil A Math (4049) Paper Specimen 1 Paper 1 Prepared by Thang Pang Ern MA THEMA TICS 4049/01 Paper 1 Specimen 1 2 Hours 15 Minutes Additional Materials: Writing paper READ THESE INSTRUCTIONS FIRST Write your name and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of the marks for this paper is 90. This document consists of 4 printed pages. 1

Question 1 By considering the expression sin2 θ + sin2 π 2 − θ

, evaluate S, where S = sin2 π 100

+ sin2 2π 100

+ sin2 3π 100

+ ... + sin2 99π 100

. [3] Question 2 (a) Show that log2(log4 x) =log8(log16 y) simplifies to 2(log4 x)3 = log4 y. [2] (b) For integers x and y, solve the system of equations, justifying your answer. log2(log4 x) =log8(log16 y) 4y − 3x = 1 [6] Question 3 Consider f (x) =14sin2 x + 5 and g(x) =α cos x − 3, where α is a positive integer and 0 ≤ x ≤ π. (a) Explain why regardless the value of α, there exist at least one solution to f (x) − g(x) =0. [5] (b) For α = 1, sketch the graphs of f (x) and g(x) on the same diagram and state the number of solutions tof (x)−g(x) = 0. [5] Question 4 (a) The mapping of polynomials of degree 3 to degree 2, can be expressed mathematically asT : P3 → P2 is defined by T

a0 + a1x + a2x2 + a3x3 = a1 + 2a2x + 3a3x2 and is said to be a linear transformation. Infer what kind of operator T is. [1] (b) (i) p(x) is a polynomial of degree n + 1, where the coefficient of xi is bi. A student applies T repeatedly until he obtains a polynomial q(x), which is of degree 3. Find the remainder when q(x) is divided by x + 2. [6] (ii) Another student applies T n +1 times and obtains a polynomial. State what it is and simplify your answer.[1] Question 5 (a) Derive the equation of the normal line to the curve y = f (x) at an arbitrary point. [3] (b) Suppose there exist three distinct points on a parabola given by the equation y = x2 such that their normal lines intersect at a single point. By considering two cases, prove that the sum of their x-coordinates is 0. [8] Question 6 n circles are plotted on a Cartesian plane, where n is some positive integer. The equation of the nth circle, for n ≥ 2, is given by Cn : (x − n + 1)2 + (y − n + 1)2 = n2. C1 has its centre at the origin and it has a radius of 1. Prove algebraically that the minimum value of n, where n ≥ 2, such that Cn does not intersect C1 is 6. [4] 2

Question 7 In a rectangle ABCD, E is a point on AD and F is a point on CD such that the line through the midpoint of EF , where M is the midpoint of EF and the centre of the rectangle is perpendicular to AC. It is given that AB = 100, BC = 60 and AE = 40. Consider a Cartesian plane and let D be the origin. (a) Find the equation of the line segment AC. [2] (b) What are the possible coordinates of E and F? You may use unknowns where appropriate. [1] (c) Hence, deduce the exact coordinates of M. [3] (d) Hence, find the area of triangle BEF . [2] Question 8 Idempotent functions are functions such that f ( f (x)) = f (x) for any value of x. The term ‘idempotence’ is of great importance in several areas of Mathematics. For example, in Linear Algebra, an idempotent matrix, A, is a square matrix such that A2 = A. One example of an idempotent function is the absolute value function, also known as the modulus function. It is denoted by f (x) =|x|. (a) (i) Prove that the modulus function is an idempotent function. [2] (ii) Give another example of an idempotent function. [1] (b) (i) Explain why y =

|x|3 − 2x2 10 is non-negative and symmetrical about the y-axis. [2] (ii) Find the coefficient of the constant term in the expansion of y =

|x|3 − 2x2 10 . [3] Question 9 Consider the function fk(x), where fk(x) =ln [(x + 1) (x + 2) . . .(x + k)] and k is a positive integer. (a) (i) Prove that ln(xy) =ln x + ln y and state the conditions on x and y for this to be true. [2] (ii) Prove that f ′2(x) = 2x + 3(x + 1)(x + 2) and explain why the graph of f ′2(x) in this case does not consider all values of x. [3] (b) (i) For the case wherek = 3, an amateur Calculus student decides to approximate the area under the curve between x = 3 and x = 4 using two rectangles. As such, each rectangle has a width of 1/2. Given that the height of the two rectangles are f3(7/2) and f3(4) and they are adjacent, find an approximation for I = Z 4 3 ln[(x + 1)(x + 2)(x + 3)] +2 dx. [4] (ii) State a way to improve the approximation for I in (i). [1] (c) When f ′k(x) is expressed as a single fraction, the coefficient of the constant term in the numerator is A. Consider another function gk(x), where gk(x) =Ak!. By using (ai) and the fact that 1 2 = 14 + 14 = 18 + 18 + . . .+ 18| {z } 4 times = 116 + 116 + . . .+ 116| {z } 8 times = . . .= 12n + 12n + . . .+ 12n| {z } n−1 times + . . . , show that as k increases, gk(x) tends to infinity regardless the value of x. [5] 3

Question 10 The catenary curve is a special geometrical figure and it has a variety of applications, such as in Physics and architecture. It has a U-like shape and resembles a parabolic arch, but it is not a parabola. Figure 1 The catenary curve can be modelled by a hyperbolic cosine function which is denoted by the general formula y = acosh x a

. The class of hyperbolic trigonometric functions are closely linked to exponential and logarithmic functions. That is, cosh x = 1 + e−2x 2e−x . (a) (i) Show, using the second derivative test, that the function y = cosh x has a minimum point and find its coordi- nates. [5] (ii) An even function h(x) has the property that h(−x) =h(x). Prove that any hyperbolic cosine function of the form y = acosh x a

is an even function. [2] (b) The equation of the cables’ profiles on the Golden Gate Bridge in San Francisco, United States of America, can be modelled using the hyperbolic cosine function. Figure 2 Denote the top of the poles connecting the cables as P1 and P2, where P2 is on the right of P1. Assume that the base of the catenary curve lies on the origin of a Cartesian plane. The distanceP1P2 is 1280 m and the height of the curve is 152 m. The equation of the Golden Gate Bridge catenary curve is given by y = 1280cosh x m

− 1280. (i) Compute the value of m. [5] (ii) Find the area bounded by one arm of the catenary curve. [3] 4

Evil A Math (4049) Paper Specimen 1 Paper 1 (Solutions) Prepared by Thang Pang Ern MA THEMA TICS 4049/01 Paper 1 Specimen 1 2 Hours 15 Minutes Additional Materials: Writing paper READ THESE INSTRUCTIONS FIRST Write your name and index number on all the work you hand in. Write in dark blue or black pen on both sides of the paper. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the