Sec 3 E 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 E MATH 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 [Algebra] (a) Simplify 2𝑦2−3𝑦−5 𝑦2−1 . (b) 𝑏 = √ 4+5𝑎 3𝑎−11 Make a the subject of the formula. (c) (i) Express 𝑥2 + 6𝑥 − 11 in the form (𝑥 + 𝑏)2 + 𝑐, where b and c are constants. (ii) Hence, solve the equation 𝑥2 + 6𝑥 − 11 = 0. [3] [3] [2] [2] 2 [Quadratic Equation] A road up a mountain is 18 km long. Paul followed this road at an average speed of x km/h. (i) Write down an expression, in terms of x, for the number of hours he took to walk up the mountain. (ii) Paul came down the mountain by a different road. The length of this road was 25 km. His average speed coming down the mountain was 2 km/h greater than his average speed going up the mountain. Write down an expression, in terms of x, for the number of hours he took to walk down. (iii) It took Paul 1.5 hours less to come down than to go up. Write down an equation in x, and show that it simplifies to 3𝑥2 + 20𝑥 − 72 = 0. (iv) Solve the equation 3𝑥2 + 20𝑥 − 72 = 0, giving both answers to 2 decimal places. (v) Calculate, correct to the nearest minute, the total time Paul took to go up and come down the mountain. [1] [1] [3] [3] [2] 3 [Indices] (a) Simplify (5𝑥)3 × (9𝑦4)−1 2, leaving your answer in positive index notation. (b) Solve 62𝑦×36 6 1 3 = 216𝑦. [3] [3] 4 [Standard Form] (a) Sound travels at 1234.8 km/h. Express the speed of sound in metres per second, giving your answer in standard form. (b) Mercury is 6.58 × 107 kilometres from the Sun and the Pluto is 5.914 billion kilometres from the Sun. How much further from the Sun is Pluto than Mercury? [2] [2]

Page 3 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 5 [Finance] Peter invested a sum of money in an account paying compound interest at 1.5% per year. After 8 years, the money had earned total interest of $3162.32. Calculate the sum of money Peter invested in the account. [3] 6 [Quadratic Functions – Sketching] (a) Factorise −𝑥2 + 5𝑥 + 6. (b) Sketch the graph of 𝑦 = −𝑥2 + 5𝑥 + 6, indicate clearly the x-intercepts and y-intercept. (c) Write down the equation of the line of symmetry. (d) Write down the equation of a horizontal line that intersects the graph of 𝑦 = −𝑥2 + 5𝑥 + 6 at only one point. [2] [2] [1] [1] 7 [Graph Functions] Select a possible equation from the box to represent each of the sketch graphs below. 𝑥𝑦 = 2 𝑦 = 𝑥3 − 2 𝑦 = 2 − 𝑥3 𝑦 = 2 − 𝑥2 𝑦 = 𝑥2 − 2 𝑥2𝑦 = 2 [1] [1] [1] 8 [Speed Time Graph] The diagram shows the speed – time graph of a moving object. (a) Calculate the acceleration of the object at time 𝑡 = 22 𝑠. (b) Find the speed when t = 4 s. (c) Find the value of k if the total distance travelled is 1.35 km. [1] [2] [2] (a) (b) (c)

Page 4 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 9 [Coordinate Geometry] A is the point (– 2, 8) and B is the point (x, – 4). (a) Given that the gradient of line 𝐴𝐵 𝑖𝑠 − 1.5, find the value of 𝑥. (b) Find the length of 𝐴𝐵. The equation of line 𝐶𝐷 𝑖𝑠 6𝑦 + 9𝑥 = 5. (c) The point (4𝑏, −𝑏) lies on the line 𝐶𝐷. Find the value of 𝑏. (d) Explain why line 𝐶𝐷 does not intersect line 𝐴𝐵. [2] [2] [1] [2] 10 [Coordinate Geometry] Complete the table of values for 𝑦 = 𝑥3 5 − 𝑥 + 4 below. (a) Using a scale of 2 cm to represent 1 unit on both axes, plot the points given in the table and join them with a smooth curve for −3.5 ≤ 𝑥 ≤ 3. (b) By drawing a tangent, find the gradient of the curve at 𝑥 = 2. (c) Use your graph to find the solutions of the equation 𝑥3 5 − 1 2 𝑥 = 0 for −4 ≤ 𝑥 ≤ 3. (d) The solutions in part (d) above are also the solutions for the equation 2𝑥3 + 𝐴𝑥2 + 𝐵𝑥 = 0. Find the value of A and of B. [1] [3] [2] [2] [2] 11 [Congruency and Similarity] In the diagram below, 𝐵 is on 𝐴𝐸 such that 𝐴𝐵 = 6 cm and 𝐵𝐸 = 14 cm. 𝐴𝐷 = 15 cm and 𝐶 is on 𝐴𝐷 such that 𝐵𝐶 = 9 𝑐𝑚, 𝐷𝐸 = 22.5 cm and ∠𝐴𝐵𝐶 = ∠𝐴𝐷𝐸. (a) Prove that ∆𝐴𝐶𝐵 is similar to ∆𝐴𝐸𝐷. (b) Show that CD = 7 cm. [2] [2] x −3.5 −3 −2 −1 0 1 2 3 y −1.1 1.6 4.8 4 3.2 3.6 6.4

Page 5 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 12 [Congruency and Similarity – Similar Figures] Two geometrically similar containers have volume 250 ml and 54 ml respectively. Find the ratio of the base area of the bigger container to the base area of the smaller container. [2] 13 [Trigonometry – Triangle Formula] In the diagram, 𝐴𝐵 = 7 𝑐𝑚 and 𝐵𝐶 = 8 𝑐𝑚. The area of triangle 𝐴𝐵𝐶 is 24.249 𝑐𝑚2. Given that ∠𝐴𝐵𝐶 is an obtuse angle, find the length of 𝐴𝐶. [5] 14 [Trigonometry – Area of Triangle] The area of triangle 𝐴𝐵𝐶 is 36.7 cm2 The length of 𝐴𝐵 is 14.2 cm and 𝐵𝐶 is 6.1 cm. Find the possible values of angle 𝐴𝐵𝐶. [2] 15 [Trigonometry] In the diagram above, triangle 𝐴𝐵𝐶 forms a garden on level ground. 𝐵𝐶 = 32 m, angle 𝐴𝐶𝐵 = 100° and angle 𝐵𝐴𝐶 = 53° . (a) Show that 𝐴𝐶 = 18.19 m, correct to 4 significant figures. (b) Triangle 𝐵𝐷𝐸 is a roof designed for the garden. Two vertical poles, 𝐴𝐷 = 2 m and 𝐶𝐸 = 5 m, were built to hold up the roof. Find (i) the length of 𝐷𝐸, (ii) angle 𝐷𝐵𝐸. [2] [2] [5]

Page 6 Paradigm Specialising in O Level Mathematics Paradigm Secondary Math Department 16 [Circle] In the diagram, 𝑋𝐴𝑌 is a tangent to the circle 𝐴𝐵𝐶𝐷 at 𝐴. 𝑂 is the centre of the circle 𝐶𝐷𝐸𝐹 𝑎𝑛𝑑 𝐵𝐶𝐹 is a straight-line. It is given that ∠𝐵𝐶𝐴 = ∠𝑋𝐴𝐵 = 58° , ∠𝐴𝐵𝐷 = 32° 𝑎𝑛𝑑 ∠𝐶𝐹𝐸 = 120° (a) Find angle ACD. Give a reason for each step of your working. (b) Explain why BD is a diameter of circle ABCD. (c) Find angle DAY. Give a reason for each step of your working. (d) Given that 𝐹𝐶 = 𝐹𝐸, show that triangle CDE is equilateral. Give a reason for each step of your working. (e) (i) Prove that triangle OCE is congruent to triangle FCE. (ii) Hence, what is the special name given to quadrilateral COEF? [1] [1] [1] [3] [3] [1] 17 [Arc Sector Segment] (a) 𝐴𝐵𝐶𝐷 is a major segment of a circle, centre 𝑂. 𝐵𝐷 is 32 cm, 𝐴𝐶 is 48 cm and angle 𝐷𝐵𝐴 is 𝜋 2 . (b) The sector 𝑂𝐴𝐷𝐶 was cut from the above to form a cone where the 𝑂𝐴 is glued to 𝑂𝐶. (i) Calculate the circumference of the base circle of the cone. (ii) Hence, calculate the vertical height of the cone. [3] [4] [2] [3]

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