Core Foundation Mathematics

EXAMINATION PAPER Examination Session: Year: Exam Code: Sum1 2021 F_FMA_S_RS2_EXAM_B Module Title: Core Foundation Mathematics Time Allowed: It is advised that you spend 1.5 hours to complete this paper. You do not need to complete the examination in one sitting and may take breaks if necessary. Additional Material provided: n/a Materials Permitted: n/a Calculators Permitted: n/a Models Permitted: n/a Students permitted to use dictionaries: n/a Instructions to Candidates: SUBMISSION DEADLINE: Mon June 28th 1pm You may use textbooks, lecture notes, DUO, and other web educational resources, but the examination attempt must be your own work. We advise you very strongly not to try to do any fresh research during the exam. You should not spend significantly longer than the scheduled duration of the exam in preparing and completing your answers. The work must be all your own and you must not have consulted any other person to assist you in any way when preparing and completing your answers. If you have any problems, please do not contact your subject tutors. You must contact Revision: Instructions There are 9 questions. Answer all questions. The total marks available for this exam is 60. This accounts for 60% of your overall marks for this module. Ensure all working is shown. Answers with no working may gain no credit. Any necessary formulae are given on the formula booklet provided on DUO. Question 1 (a) Find the smallest integer value of 𝑛 such that 1 βˆ’ 0.7𝑛 > 0.998 (4) (b) Solve the equation: log 3 (π‘₯ + 1) = 1 + 2 log 3 (π‘₯ βˆ’ 1) (4) Question 2 π‘₯ The first three terms in the expansion of (3 βˆ’ 2)𝑛 are 81 + 𝑏π‘₯ + 𝑐π‘₯ 2 . Find the values of each of the constants 𝑛 , 𝑏 π‘Žπ‘›π‘‘ 𝑐. (4) Question 3 The line 𝑦 + π‘˜π‘₯ + 4π‘˜ = 0 intersects the parabola 𝑦 = 2π‘₯ 2 + 4π‘₯ βˆ’ 8 , where π‘˜ is a constant. Find the condition on π‘˜ such that the line will be a tangent to the parabola. (5) Question 4 𝑓(π‘₯) = (π‘₯ βˆ’ 1)(π‘₯ βˆ’ 3)(π‘₯ + 1) (a) State the coordinates of the point at which the graph 𝑦 = 𝑓(π‘₯) intersects the 𝑦axis. (1) (b) The graph of 𝑦 = 𝑓(π‘₯ + 𝑏) passes through the origin. Find the possible values of 𝑏. Justify your answer clearly. (4) Question 5 𝑓(π‘₯) = 𝑝π‘₯ 3 + 6π‘₯ 2 + 12π‘₯ + π‘ž (a) Given that the remainder when 𝑓(π‘₯) is divided by (π‘₯ βˆ’ 1) is equal to the remainder when 𝑓(π‘₯) is divided by (2π‘₯ + 1). Find the value of 𝑝. (4) (b) Given also that π‘ž = 3, find the value of the remainder. (2) Question 6 A new employee’s salary in a company is Β£19000 in their first year of employment. Their salary rises by Β£1500 per year at the end of each of the first six years of employment, after which it remains constant. (a) Find the employee’s salary at the start of the seventh year of employment. (2) (b) Calculate the total salary received over the first 10 years of employment. (5) Question 7 The points A(π‘˜, βˆ’2), B (7,2) and C(6,4) , where π‘˜ is a constant, are the vertices of βˆ† ABC. ∠ ABC is a right angle. (a) Calculate the value of π‘˜. (5) (b) Find an equation of the straight line L passing through A and B, giving your answer in the form π‘Žπ‘₯ + 𝑏𝑦 + 𝑐 = 0, where π‘Ž, 𝑏 and 𝑐 are integers. (3) The line L crosses the π‘₯-axis at D and y-axis at E. (c) Find the coordinates of D and E. (2) Question 8 (a) Given that 𝑦 = 5π‘₯ 2 + π‘Žπ‘₯ + 𝑏 has a turning point at (𝑏, π‘Ž) where π‘Ž β‰  0. Find π‘Ž and 𝑏. (6) (b) Hence find the range of values of π‘₯ for which the function is decreasing. (2) Question 9 k Given that k is a positive constant and 1 5 2 x + 3 dx = 4 Find the values of π‘˜. (7) END OF EXAMINATION
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