2024 JC2 H2 Math prelim - SAJC

1 [ Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION HIGHER 2 Candidate Name CLASS 2 3 H2 MATHEMATICS 9758/01 Paper 1 26 AUGUST 2024 (Monday) Candidates answer on the Question Paper. 3h Additional Materials: MF 26 N umber of pieces of additional writing paper : _____________________ (N.A. if none) Q 1 2 3 4 5 6 7 8 9 10 11 TOTAL M 4 4 7 5 8 8 11 12 12 13 16 100 READ THESE INSTRUCTIONS FIRST W rite your name, civics group, index number and calculator models on the cover page. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. A nswer all the questions. Total marks : 100 W rite your answers in the spaces provided in the question paper. G ive non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an approved graphing calculator is expected, where appropriate. U nsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. Y ou are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 26 printed pages and 2 blank pages including this page.

2 [Turn Over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 1 Show that 2 24xx++ is always positive for all real values of x. Hence solve the inequality 2 2 24 032 xx xx ++ <+− . [4] 2 A function is defined as f ( ) 3 sin cosθ θθ= + . (i) Show that f ( ) 3 sin cosθ θθ= + can be written in the form sin( )R θα+ where exact values of R and α are to be found. [1] (ii) Hence, state a sequence of transformations that will transform the graph with equation siny θ= on to the graph with equation 3sin cos 5y θθ= ++ . [3] 3 (i) Show that 2121 1 ( 1) ! ! ( 1) ! ( 1) ! rr r rr r −−−+ =− ++ , where r +∈  . [1] (ii) Hence find an expression for 2 2 1 ( 1) ! n r rr r= −− +∑ . [4] (iii) Show that 2 2 1 ( 1) !r rr r ∞ = −− +∑ is convergent. [2] 4 A curve has equation 2335 2x xy y− += . Find d d y x in terms of x and y. Hence, deduce the number of tangent(s) to the curve that is/are parallel to the y-axis. [5] 5 (a) Using the substitution exu= , find d2 eexx x−−∫ . [3] (b) (i) Find cos sin 3 dx xx∫ . [2] (ii) Without the use of a calculator, evaluate 3 6 π 4 π cos sin 3 dx xx∫ . [3]

3 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN 6 Referred to the origin O , the points A and B are such that OA= a    and .OB= b   (a) The point C lies on OB such that OC kOB=     , where k is a constant. X is on AC such that AX : XC = 1 : 2 and Y is on AB produced such that AY : BY = 5 : 4. (i) Find OX  and OY   in terms of a, b and k. [2] (ii) Given that O, X and Y are collinear, find k. [3] (b) It is given that 12 and cos .4AOB= ∠= −ab Give the geometrical meaning of () m+×a bb , for some real value of m, and evaluate the value of () m+×a bb , leaving your answer in terms of |a|. [3] 7 An inverted right pyramid is inscribed in a sphere of fixed radius R and center O, where the vertices A, B, C, D and T are touching the surface of the sphere as shown in Figure 1 . The pyramid has a square base ABCD and height, ST, where ST = h units. S is the point where the diagonals AC and BD of the square intersect. Figure 1 Figure 2 [It is given that the volume of a pyramid is 1 base area height3×× .] (i) Show that the volume of the pyramid, V is ( ) 22 23 h Rh h− . [3] (ii) Given that h varies, show that the maximum volume of pyramid, V, is obtained when the length of the square base is equal to the height of the pyramid. [5] A T D C B S O D T A C B S H

4 DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN (iii) A container in the form of an inverted right pyramid, shown in Figure 2 , with maximum volume V as described in (ii) is made. Water is poured into this container at a rate of 10 units3 per second. Find the rate of increase of the water level, H, when the height of the water is 5 units. [3] 8 Do not use a calculator in answering this question. (i) Given that 1iz= + , find 23 4, and zz z in cartesian form. Given also that 432 2 8 80z az z z+ − + −= , where a is real, find the value of a. [4] (ii) Using the value of a in (i), express 432 2 88z az z z+ − +− as the product of two quadratic factors. [3] (iii) Hence, deduce the two quadratic factors of 2 341 i 2 8i 8aw w w w−+ + − . [2] (iv) A second complex number is given by i 22e .v π = Find arg ( ) 3 zv . [3] 9 The curve 1C has equation 23 23 1 xxy x −+= − . The curve 2C has the equation 2e5xy= −− . (a) Sketch 1C , stating the equation(s) of any asymptote and the coordinates of any turning point(s) and point(s) where the curve crosses the axes. [4] (b) 1C and C2 intersect at a point P (a, b) where x < 0. Find the values of a and b, giving your answer correct to 4 decimal places. [1] The region R is bounded by 1C , C2 , x-axis, y-axis and the line x = k where k < a. (c) Given that the area of region R is at least 10 units2, show that k satisfies the inequality 21 e5 02 k kp+ +≤ where p is a constant to be determined. Hence find the maximum value of k. [4] (d) Now, given that k = – 4, f ind the volume of the solid formed when R is rotated completely about the x-axis, correct to 2 decimal places. [3] 10 Scientists are investigating the growth in length of a particular species of fish. (i) The scientists used the von Bertalanffy growth model to predict the length L cm of the fish at a particular time, t years after birth. It is said that the rate of growth in length of the fish is directly proportional to the difference between its theoretical maximum length of the fish, 60 cm and its length at the time t years. The constant of proportionality, k, is known as the growth coefficient which is always positive . At birth, it is known that the rate of growth when the fish is 10 cm long is 5 cm per year. (a) Write down a differential equation for this situation. Solve this differential equation to get L as an exact function of t. [7] (b) Find the length of the fish 5 years after it is born. [1]

5 [Turn over DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN (ii) Based on the data collected, one of the scientists proposed a second model to predict the length L cm of the fish at a particular time, t years after birth. The proposed model is as follows: d 50 d L tL= . (a) Given that the length of the fish is 10 cm long at birth, find L in terms of t. [4] (b) Comment on the validity of the second model in modelling the length of fish, L cm, at a particular time, t years after birth