Sec 3 A Math EOY Mock Exam Paper

Page 1 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department Name: School: Target Grade: MOCK EOY PAPER 2025 SECONDARY 3 AMATH READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Find a nice comfortable spot without distraction. 2. Be fully focused for the whole duration of the test. 3. Speed is KING. Finish the paper as soon as possible then return-back to Check Your Answers. 4. As you are checking your answers, always find ways to VALIDATE your answer. 5. Avoid looking through line by line as usually you will not be able to see your Blind Spot. 6. If there is no alternative method, cover your answer and REDO the question. 7. Give non-exact answers to 3 significant figures, or 1 decimal place for angles in degree, or 2 decimal place for $$$, unless a different level of accuracy is specified in the question. Wish you guys all the best in this test. You can do it. I believe in you. Team Paradigm PARADIGM [Turn Over]

Page 2 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 1 [Simultaneous Equation] Find the coordinates of the points of intersection of the curve 𝑥ଶ−𝑥𝑦+𝑦ଶ = 16 and the line 2𝑥−3𝑦=4. [5] 2 [Complete The Square] A dolphin jumps out of the water during a dolphin show. The height, 𝑦 𝑚, of the dolphin is given by 𝑦 = −4.5𝑡ଶ+ 9𝑡 , where 𝑡 is the time in seconds after the dolphin jumps out of the water. (i) Express 𝑦 = −4.5𝑡ଶ+ 9𝑡 in the form 𝑦 = 𝑎(𝑡 − ℎ)ଶ+ 𝑘 . (ii) How long did it take the dolphin to reach the maximum height and what is the maximum height? (iii) Find the range of values of t where the height of the dolphin is at least 3.375 m. [2] [2] [2] [4] 3 [Nature of Roots] (a) Find the largest integer k for which the line 𝑦 = 2𝑥 + 𝑘 intersects the curve 𝑥ଶ+ 2𝑦ଶ = 8 at two distinct points. (b) Show and explain why there are no values of 𝑚 for which the curve 𝑦=(𝑚−5)𝑥ଶ+2𝑚𝑥+(𝑚+3) is always positive. [5] [4] 4 [Surds] A cylinder has volume ൫√45 + √3൯ cm3. If the base area of the cylinder is ൫√45 + √3൯ cm3, find the height of the cylinder. Give your answer in the form 𝑎+𝑏√15 , where a, b and are rational numbers. [3] 5 [Exponential] Solve the equation 2𝑒ଶ௫=7+15𝑒ଶ௫ giving your answer correct to 1 significant figure. [4] 6 [Exponential] Explain why7௡ାଵ−4(7௡)−ଵ ଻(7௡) is an even number for all positive values of 𝑛. [3] 7 [Exponential Word Problem] The mass, m grams, of a radioactive substance remaining, t days after being measured is given by 𝑚 = 10𝑒ି଴.଴ଵ௧+ 0.2 . (a) Find the initial mass. (b) Sketch the graph 𝑚 = 10𝑒ି଴.଴ଵ௧+ 0.2 𝑓𝑜𝑟 𝑡 ≥ 0. (c) Find the least number of days it takes before the amount of substance is reduced to 5% of its initial mass. (d) Explain why the mass of the radioactive substance can never be less than 0.2 g. [1] [2] [3] [2]

Page 3 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 8 [Logarithm] Solve the following equations. (i) logଷ(3𝑥+2)−2logଷ𝑥=2 , (ii) 5 logହ 𝑦−2log௬5=9 . [6] [5] 9 [Polynomial] The function 𝑓 = 𝑥ଷ− 𝑎ଶ𝑥 + 𝑏ଶ, where a and b are positive constants, leaves a remainder of 1 when divided by 𝑥 + 3 . The function 𝑔(𝑥)=3𝑎𝑥ଷ−2𝑥ଶ+𝑏𝑥−6 is exactly divisible by 𝑥−1. (a) Find the value of a and of b. [5] 10 [Partial Fraction] Express ଻௫మିଶଷ௫ା଺ (௫ିଶ)(௫మିସ) in partial fractions. [5] 11 [Binomial Theorem] (a) Write down the first three terms in the expansion, in ascending powers of x, of (2 + 𝑎𝑥)଼, where a is a constant. Give the terms in their simplest form. (b) In the expansion of (1 + 2𝑥 − 4𝑥ଶ)(2 + 𝑎𝑥)଼ the coefficient of 𝑥ଶ is 22 times the coefficient of x. Given that a is positive, calculate the value of a. (c) Explain why there is no even power of x in the expansion of ቀ௫య ଶ + ଶ ଷ௫ቁ15. 12 [Coordinate Geometry] 𝐴𝐵𝐶𝐷 is a trapezium such that the coordinates of 𝐴 and 𝐶 are (6, 1) and (−5, 4) respectively. Angle 𝐴𝐷𝐶 and angle 𝐵𝐶𝐷 are right angles and the gradient of 𝐴𝐵 is −ସ ଷ . (i) Show that the coordinates of 𝐵 are (0, 9). (ii) Show that the coordinates of 𝐷 are (2, −3). (iii) Find the area of triangle 𝐴𝐵𝐷 . (iv) Find angle 𝐴𝐵𝐶 . [2] [4] [2] [3]

Page 4 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 13 [Circles] A circle, centre 𝐶, has a chord 𝐴𝐵 where A is the point (7, 27) and 𝐵 is the point (11, 15). The normal at 𝐴 passes through the point 𝐷 (3, 3). (a) Showing all your working, find the equation of the circle. (b) Show that 𝐷 lies on the circle. [7] [2] 14 [Linear Law] 𝑡 1.0 15 2.0 2.5 3.0 3.5 𝑠 4.20 4.65 5.09 5.50 5.89 6.25 An experiment is conducted to find the distance, 𝑠 metres, travelled by an object over time, 𝑡 seconds. The table below shows the corresponding values of 𝑡 and 𝑠. It is known that s and t are related by the equation 𝑠 = 𝑎√𝑡+ ௕ √௧, where a and b are constants. (a) On the grid below, plot s√𝑡 against 𝑡 and draw a straight line. (b) Use your graph to estimate 𝑎 and 𝑏. (c) By drawing a suitable straight line on your graph, find the value of t that satisfies the equation 2𝑠√𝑡+2𝑡−10=𝑎𝑡+𝑏. [2] [4] [3] 15 [Trigonometry Simplifying & Quadrant] (a) Without using a calculator, find the exact value of cotቀଶగ ଷቁ. (b) Given that sin 𝐴 = −ହ ଵଷ and tan 𝐴 > 0, find without using a calculator, the numerical value of (i) sec𝐴, (ii) tan(−𝐴). [1] [1] [1] 16 [Trigonometry Graph] The function f is defined by f(x) = −3 cos (0.5x). (i) Write down the period and amplitude of f(x). (ii) Sketch the graph of f(x) for 0°≤𝑥≤360°. [2] [2] 17 [Trigonometry Solving] Solve the equation 2tanଶ𝑦+5sec𝑦−1=0 for 0≤𝑦≤2𝜋. [3]

Page 5 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department Answer Key 1 𝑥ଶ−𝑥𝑦+𝑦ଶ =16 ----- (1) 2𝑥−3𝑦=4 𝑥 =ସାଷ௬ ଶ --------- (2) Substitute (2) into (1) ቀସାଷ௬ ଶ ቁ ଶ − 𝑦ቀସାଷ௬ ଶ ቁ+ 𝑦ଶ = 16 7𝑦ଶ+ 16 − 48 = 0 (7𝑦 − 12)(𝑦 + 4) = 0 𝑦 = 1ହ ଻ or −4 𝑥 = 4ସ ଻ or −4 The coordinates of the points of intersections are ቀ4ସ ଻, 1ହ ଻ ቁ and (−4, −4). Ans: ቀ4ସ ଻ ,1ହ ଻ቁ and (−4 ,−4 ) 2 (i) 𝑦=−4.5𝑡ଶ+9𝑡 =−4.5(𝑡ଶ+2𝑡) = −4.5(𝑡ଶ+ 2𝑡 + 1 − 1) = −4.5(𝑡 − 1)ଶ+ 4.5 (ii) Time taken to reach maximum height = 1 seconds Maximum height = 4.5 m (iii) −4.5𝑡ଶ+ 9𝑡 ≥ 3.375 4.5𝑡ଶ− 9𝑡 + 3.375 ≤ 0 4𝑡ଶ− 8𝑡 + 3 ≤ 0 (2𝑡 − 3)(2𝑡 − 1) ≤ 0 ଵ ଶ≤ 𝑡 ≤ 1ଵ ଶ Ans: (i)−4.5(𝑡−1)ଶ+4.5 (ii) 4.5 m (iii) ଵ ଶ≤𝑡 ≤1ଵ ଶ 3 (a) 𝑥ଶ+2(2𝑥+𝑘)ଶ =8 9𝑥ଶ+8𝑘𝑥+2𝑘ଶ−8=0 Line and Curve intersect at 2 distinct points. (8𝑘)ଶ− 4(9)(2𝑘ଶ− 8)> 0 64𝑘ଶ− 72𝑘ଶ+ 288 > 0 8𝑘ଶ− 288 < 0 𝑘ଶ− 36 < 0 (𝑘 − 6)(𝑘 + 6) < 0 −6 < 𝑘 < 6 Largest integer k is 5. (b) Since 𝑦 > 0, (2𝑚)ଶ− 4(𝑚 − 5)(𝑚 + 3) < 0 4𝑚ଶ− 4𝑚ଶ+ 8𝑚 + 60 < 0 8𝑚 + 60 < 0 𝑚 < −ଵହ ଶ But 𝑦 > 0 means the 𝑥ଶ coefficient must be positive, that is 𝑚−5>0 ⟹𝑚>5. Since m must be 𝑚<−ଵହ ଶ and 𝑚>5 for 𝑦>0, ∴ there are no values of m for which 𝑦>0 (is always positive)

Page 6 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 4 ൫√45−√3൯×ℎ=√45+√3 ൫3√5−√3൯×ℎ=3√5+√3 ℎ =ଷ√ହା√ଷ ଷ√ହି√ଷ =ଷ√ହା√ଷ ଷ√ହି√ଷ×ଷ√ହା√ଷ ଷ√ହା√ଷ =ସହା଺√ଵହାଷ ସହିଷ =ସ଼ା଺√ଵହ ସଶ =ቀ଼ ଻+ଵ ଻√15ቁ cm Ans: ቀ଼ ଻ +ଵ ଻ √15ቁ𝑐𝑚 5 2𝑒ଶ௫ =7+15𝑒ିଶ௫ Let 𝑦=𝑒ଶ௫ 2𝑦 = 7 +ଵହ ௬ 2𝑦ଶ− 7𝑦 − 15 = 0 (2𝑦 + 3)(𝑦 − 5)= 0 𝑦 = 5 or 𝑦 = −ଷ ଶ 𝑒ଶ௫= 5 or 𝑒ଶ௫