SAJC 9758 2025 Prelim P1

[Turn Over ST ANDREW’S JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 MATHEMATICS 9758/01 Paper 1 1 September 2025 (Monday) Preliminary Examination 3 hours Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) ______________________________________________________________________________ READ THESE INSTRUCTIONS FIRST Answer all questions. Total marks : 100 Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give 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. You are expected to use an approved graphing calculator. Unsupported 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 7 printed pages and 1 blank page.

2 [Turn Over 1 A function f is defined by 32f( )x ax bx cx d= + + + where a, b, c and d are constants. The graph of f( )yx= has a turning point at 1x=− and passes through the points (2,10) and ( 2, 2)−− . Given also that 2 2 f ( ) d 16xx − = , find the values of a, b, c and d. [4] 2 A closed cylinder is designed to have a fixed external surface area of p cm2 such that its volume is maximum. Find the radius of the cylinder in terms of p. [5] 3 The lines 1l and 2l have equations 1 2 12 : 2 1 , 02 02 : 5 2 , 83 l l   −       =+                 = + −      −    r r where  and  are parameters. (i) Show that 1l and 2l are skew lines. [2] The point P has coordinates ( )1 ,5,0 and the plane 1 has equation 20yz− − = . (ii) Find the coordinates of 'P , the reflection of P in 1 . [4] (iii) Find the cartesian equation of the plane 2 containing P and 1l . [3]

3 [Turn Over 4 (a) The function f is defined by 22 , 0 , f ( ) ( ), 2 , a x x a x a x a a x a  −   =  −   and f(x) = f(x + 2a) for all real values of x, where a is a positive constant. (i) Sketch the graph of y = f (x) for 23a x a−   . [3] (ii) Find 101f 2 a  in terms of a. [2] (b) The function g is defined by g : ln ,where , 0 e.x x x x    The diagram below shows the graph of the quadratic function h with domain defined where 04 x . The graph has a maximum point at x = 1 and has end - points e0, 2   and (4,0). (i) Determine whether hg exist. Justify your answer. [2] (ii) State the other value of x for which h(x) = e 2 . Given that h -1 exists, deduce the restricted domain of h such that domain of h-1 is e0, 2    . [2] (iii) Using the restricted domain of h found in (ii), find the range of gh. [2] y = h(x) O 1 4

4 [Turn Over 5. An arithmetic progression has first term a and common difference d, where a and d are non-zero. The 3 rd, 9 th and 11 th terms of the arithmetic progression are three distinct consecutive terms of a geometric progression. (a) Find d in terms of a. [3] (b) Let the sum of the first n terms of the arithmetic progression be nS where 1 143S = . Find the largest possible value of nS . [3] The first term of the geometric progression is b, where b is positive. (c) Determine whether the geometric series is convergent. [2] (d) Show that the sum of all terms after the thn term of the geometric progression is at most 1 2 b , where n is a positive integer. [2] 6. The graph of f( )yx= cuts the x-axis at the point (6,0) and has a maximum point and a minimum point at (0,0) and (3, 5)− respectively. The graph of g( )yx= cuts the x-axis at the point (2,0), and has a minimum point and a maximum point at (0,0) and (1,5) respectively. The diagrams below show the graphs of f( )yx= and g( )yx= . (i) Describe fully a sequence of two transformations which transform s the graph of f( )yx= onto the graph of g( )yx= . [2] It is given that g( )x can be expressed in the form f ( )a bx c d++ where , , and a b c d are constants. (ii) State the values of , , and a b c d . [2] (iii) The derivative of the function f( )yx= with respect to x is denoted by f ( )x . It is given that f (6) 4 = . Find the exact value of g (2) . [3] y x 𝑂 y x 𝑂

5 [Turn Over 7. (a) (i) On a single diagram, sketch the graphs of 3 1 xy x= − and 3yx= , stating the equation(s) of any asymptote(s) and the coordinates of points of intersection. [4] (ii) Hence find the area bounded by the curve 3 1 xy x= − and the lines 3yx= and 8y= . [2] (b) A curve has the equation 2 1yx=+ where x is non-negative. (i) With the aid of a sketch of the above curve, explain why the value of 2 1 2 2 2 2 0 1 1 4 ( 1)lim 1 1 1 ... 1 is 1 d n nS x x n n n n→  −= + + + + + + + +   . [4] (ii) Hence find the value of S. [1] 8 A curve has equation ( ) 2 2e .xyxy−= (i) Show that de de xy xy y x y y x x y x −−= −+ . [3] (ii) Given that the curve cuts the positive y-axis at point A, find the equation of the normal to the curve at A. [2] (iii) The normal to the curve at A meets the curve at another point B. Find the coordinates of B. [3] 9. (a) Use the substitution 3secx = to find 22 1 d 9 x xx − . [5] (b) The region R lies in the first quadrant and is bounded by the curve 2 22 1 9 y xx = − and the lines y = 0.1 and x = 4. Find the exact volume of the solid generated when R is rotated through 2 radians about the x-axis. [4]

6 [Turn Over 10. Do not use a calculator in answering this question. (a) Find the complex numbers z and w which satisfy the following simultaneous equations. 50 i 4 4 7i zw zw += − =− + Give your answers in the form of iab+ , where a and b are real constants. [5] (b) O The point A on the Argand diagram represents the complex number u. (i) On the copy of the Argand diagram in the Printed Answer booklet, plot the point B to represent the complex number u− . [1] The points C and D represent the complex numbers vu− and ( )*vu− respectively, where v is an unknown complex number. It is also given that 90CDA= . (ii) By using the Argand diagram or otherwise, state the value of ( )Im v and justify your answer. [2] Im Re A

7 [Turn Over 11. The points , and 'A B A lie on a circle with center O such that and OA OB==ab . It is also given that the line segment 'AA is a straight line that passes through O (see diagram). (i) Using a suitable scalar product, prove that ' 90ABA = . [4] (ii) Express the area of 'ABA in the form k ab , where k is a real constant. [2] 12. Welding fumes contain a dangerous amount of particulates. To protect workers in a workshop, an extraction device remove particulates in the air continuously. Let V mg represent the mass of particulates in the air in the workshop at time t min after the extraction device is activated. Particulates are produced at a constant rate of 0.32 mg/min. The rate at which particulates are removed is proportional to the mass of particulates in the air. When the mass of particulates in the air in the workshop is 18.75 mg, the mass of particulates in the air increases at a rate of 0.12 mg/min. (i) Show that ( )d4 30d 375 V Vt =− . [2] (ii) Solve the differential equation, given tha