Evil E Math Paper

Evil E Math (4048) Paper Specimen 1 Paper 1 Prepared by Thang Pang Ern MA THEMA TICS 4048/01 Paper 1 Specimen 1 2 Hours Additional Materials: 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 80. This document consists of 5 printed pages. 1

Question 1 Consider the equation (x + 3)(x + 4)(x + 6)(x + 7) =1120. (a) By factorising the right side of the equation to be of the form abcd, where a, b, c, d are distinct and suitably chosen integers, show that the sum of all x satisfying the equation is −10. [3] (b) Use the substitution t = x + 5 and deduce that the same conclusion is yielded as in (a). [4] Question 2 On a Cartesian plane, the distance between P(a − 1,3) and Q(a, a + 1) is d1 = p a(3 − a) units. The distance between P and the image of Q when the latter is reflected about the origin is d2. Find the minimum value of d1 + d2 explicitly. [4] Question 3 James presses the button of a drinking fountain and water flows out. The graph of the path taken by the water is plotted on a Cartesian plane, with the basin being the x-axis. The mouth of the fountain has coordinates (a, b), the turning point of the path has coordinates (9,16) and the point of contact of the water and the basin is (i, j), where a + b = i + j and b > j. (a) Show that there exists only one pair of (a, b) which fits the description above. [3] (b) Hence, sketch the path taken by the water for a ≤ x ≤ i, labelling all the necessary coordinates. [2] Question 4 The Brahmagupta-Fibonacci Identity expresses the product of two sums of two squares as a sum of two squares in two different ways. The identity states that for real numbers a, b, c, d, (a2 + b2)(c2 + d2) = (ac + bd)2 + (ad − bc)2. (a) Verify that the identity is true by expanding the right side of the equation, then factorising it. [2] For the rest of the question, let a, b, c, d be integers. Some integers can be written as the sum of two squares. For example, 13 can be written as such because 13 = 22 + 32. (b) Show that 6 cannot be written as the sum of two squares. [2] (c) (i) By setting u = n2, solve the equation n4 + 13n2 + 36 = 3074. [3] (ii) Hence, without using a calculator, show that 3074 can be expressed as 3074 = 552 + 72 = 432 + 352. [1] 2

Question 5 Figure 1 shows △ABC, where AB = c, AC = b and BC = a. D is a point on BC such that BD = u, CD = v and AD = t. E is the foot of perpendicular from A to BC such that AE = h and DE = x. This question involves two proofs of a fundamental result in Geometry known as Stewart’s Theorem. It states that t2 = b2u + c2va − uv. Figure 1 (a) In △ACD, ∠ADC = θ. By using the cosine rule on △ACD and △ABD, prove Stewart’s Theorem. [3] (b) (i) Use Pythagoras’ Theorem to show that b2u = h2u + uv2 − 2uvx + ux2 and c2v = h2v + u2v + 2uvx + vx2. [2] (ii) Hence, prove Stewart’s Theorem. [2] (c) Two circles, C1 and C2, where the radius of C1 is greater than that of C2, are tangent to each other internally at a point T . Let the chord AB of C1 be tangent to C2 at P. The tangent at T meets the extension of AB at O. It is given that AB = 59(3)−1/2, AT = 32 and BT = 27. (i) Sketch a diagram, which does not need to be drawn to scale, to illustrate the above information. [3] (ii) Show that △OAT is similar to △OT B. [3] (iii) Hence, show that ATBT = APBP. [2] (iv) Hence, find the exact value of PT . [1] 3

Question 6 The Transportation Association of Canada wanted to study the effects of wearing a helmet on motorcycle accidents. The data in Table 1 was collected by Canadian hospitals between 2002 and 2014. Each motorcyclist who was in a traffic accident and arrived at a hospital had the information below recorded. Wearing helmet? Yes No Arrived in ambulance 18% 9% Had concussion 9% 3& Needed blood transfusion 0.8% 0.4% Table 1 Source: Journal of Internal Medicine (2015) Three students, Andrew, Joseph and Flippy, make the following statements. Andrew: Motorcyclists who came to a Canadian hospital and were wearing helmets in traffic accidents were twice as likely to arrive in an ambulance than motorcyclists who were not wearing helmets. Joseph: Motorcyclists who came to a Canadian hospital and were wearing helmets in traffic accidents were three times as likely to have a concussion than motorcyclists who were not wearing helmets. Flippy: 0.8% of all Canadian motorcycle riders who wear helmets will need a blood transfusion. Only two statements are true. Explain the misleading detail in the table which resulted in the student who made the false claim to have such an opinion. [2] Question 7 Let ξ , S, I, E, R be the following sets: ξ = {all triangles with perimeter less than or equal to 100} S = {all scalene triangles with perimeter less than or equal to 100} I = {all isosceles triangles with perimeter less than or equal to 100} E = {all equilateral triangles with perimeter less than or equal to 100} R = {all right-angled triangles with perimeter less than or equal to 100} so S ∪ I ∪ E ∪ R = ξ . Let Ti denote an arbitrary triangle. (a) Construct two Venn diagrams to illustrate the following sets for the cases where ξ comprises T of integer side lengths and when it comprises T of irrational side lengths, showing the relationship between 2 or more sets where necessary. [3] (b) If ξ comprises T of integer side lengths, (i) explain algebraically why R ∩ I = ∅; [3] (ii) evaluate n(E); [1] (iii) and now, I is the set comprising all isosceles triangles with a perimeter equal to 100. The triangle inequality states that the sum of any two sides of a triangle must be greater than the third side. Using this information, construct an equation and an inequality and hence, evaluate n(I). [5] (c) If ξ comprises T of irrational side lengths, (i) suppose T1 ∈ R ∩ I and T2 ∈ R ∩ S. Given an example, each, of T1 and T2; [2] (ii) using the sine rule, explain why R ∩ E = ∅. [3] 4

Question 8 Consider the 2 × 2 identity matrix I and another 2 × 2 matrix J = 0 1 1 0 ! . Point P on a Cartesian plane has coordinates (p, q) and can be described by the position vector p = p q ! , where p, q > 0. (a) (i) Find the matrices represented by (I + J)n for n = 1,2,3,4. [3] (ii) Hence, state the column vector represented by (I + J)np for any positive integer n. Explain the geometrical significance of this transformation. [3] (iii) Alice and Riley are standing on the coordinates represented by the position vectors (I + J)2022p and (I + J)2023p respectively. They are facing north. From (ii), calculate the bearing of Alice from Riley. [2] (b) Figure 2 shows four large squares, each comprising shaded and unshaded ones (except for the first term which only comprises a shaded square). It is given that any positive odd integer can be written as 2 k + 1, where k is a non-negative integer. The number of shaded and unshaded squares for the nth term are denoted by un and vn respectively. Define Ln to be 0 un vn 0 ! . Figure 2 (i) Using Figure 2, state the values of un and vn for n = 1,2,3,4. [2] (ii) It is known that un + vn is a perfect square. State the general term of un + vn. [1] (iii) By considering un + vn and un −