SAJC 2026 Prelim P1 QP

This document consists of 8 printed pages. ST ANDREW’S JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 MATHEMATICS 9758/01 Paper 1 31 August 2026 (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.

2 1 (a) By sketching suitable graphs, solve exactly the inequality sin cos 0xx− , where 0 x  . [2] (b) Hence find 0 sin cos dx x x  − , giving your answer in exact form. [3] 2 Find d1 d 1 e xx  + . [1] Hence evaluate ( ) ln3 20 e d 1e x x x x + , leaving your answer in exact form. [4] 3 The curve C has equation f ( )yx= where 36f ( ) e 2 xx +=− . C undergoes, in succession, the following transformations to obtain the curve 1C : 1. Translation of 2 units in the positive x-direction. 2. Scaling parallel to the x-axis with scale factor k. 3. Translation of 1 unit in the positive y-direction. The resulting curve 1C has equation g( )yx= . (a) Find g( )x in terms of k. [2] (b) Given that the graphs of g( )yx= and 1 ln( 1)2yx=+ are reflections of each other in the line yx= , find the value of k. [2] 4 A sequence of numbers 1u , 2u , 3u ,  nu has the sum 2155n+ − . (a) Express ru in the form (5 ), where and are constants.qrp p q [2] (b) Find 21 1 n r r u − =  . [3] (c) Show that ( ) 2 1 ln n r r u an bn = =+ , where the exact values of a and b are to be determined. [3]

3 [Turn over 5 The function f is defined by f : 3 2 ,x x x→− , 0 1.xx    (a) Show that f has an inverse. [2] (b) Find ( ) 1f x− and state the domain of 1f − . [4] It is given that ( ) 2 2, 0 2, g 4 , 2 4.1 xx x x xx −  = −  − (c) Sketch the graph of ( )g , 0 4.y x x=   [2] (d) Show that the composite function gf exists. [1] (e) Find the range of gf . [2] 6 With reference to the origin O, the points A and B have position vectors a and b respectively, where a and b are non-zero and non-parallel vectors. (a) It is given that the vector u = a + b and the vector v = a – 2b have equal magnitudes. Using suitable scalar products, show that |b| = 2 |a| cos  , where  is the angle between a and b. [3] It is given that a is a unit vector and that the angle between a and b is 30 . (b) Using the result in part (a), show that |b| = 3 . [1] Point P lies on the line segment AB such that AP: PB = 2: 1. (c) Find OP  in terms of a and b. [1] A point N lies on the line passing through O and B such that PN  is perpendicular to OB  . Find ON  in terms of b. [3] (d) Using a suitable vector product, find the exact shortest distance from the point P to the line OB. [3]

4 7 A large digital screen of height h metres and negligible thickness is mounted on a vertical wall. The bottom edge of the screen is d metres above the eye level of an observer. The observer stands at a distance of x metres from the wall, as shown in the diagram below. Let  be the angle subtended by the screen at the observer’s eye. (a) Show that 11tan tand h dxx −− +   =−        . [1] (b) Hence by using the formula tan tantan( ) , 1 tan tanABAB AB−−= + show that 22tan hxx dh d = ++ . [2] (c) The observer moves towards or away from the wall to find the "optimal" viewing position where  is maximised. Use differentiation to find in terms of d and h, the value of x which gives the maximum value of  . (You need not show that your answer gives a maximum. [4] (d) Suppose 3 mh= and 2 md = . The observer starts walking towards the wall at a constant speed of 0.5 m/s. Find the rate of change of  at the instant when the observer is standing 4 m from the wall . [3] h Screen d x

5 [Turn over 8 Do not use a calculator in answering this question. (a) Show that for any complex numbers z and w , ( )( ) ( ) 2*2 2 Rez w z w z w z w− − = − + . [1] (b) The equation 32 16 0z pz qz+ + + = , where p and q are real, has roots 2− and w , where w is a complex number. (i) Using the result in part (a), show that the equation can be expressed as ( ) ( )( ) 222 2 Re 0z z w z w+ − + = and deduce that 22w = . [1] (ii) Given that ( ) 3arg 4w = , find w and solve the equation. [2] (iii) Hence determine the values of p and q. [2] (c) The complex number v is given by 3ivw = , where  is a positive real constant. The points A and B represent the complex numbers w and v respectively on an Argand diagram, and O is the origin. (i) Using the w found in (b)(ii), describe a sequence of two geometric transformations that maps point A onto point B . [2] (ii) Point C represents the complex number u v w=+ . nn the Argand diagram in the Printed Answer Booklet, plot the points A, B and C. [2] (iii) Given that the area of quadrilateral OACB is 24 units2, determine the value of  . [1]

6 9 The diagram below shows the curve C with equation 21 xy x = + for 0x . The curve passes through the origin O and has a horizontal asymptote at 1y= . The region R is bounded by C, the x-axis and the vertical line xk= , where 0k  . (a) Find the volum e V of the solid formed when R is rotated through 2 radians about the x-axis, leaving your answer in terms of k and  . [3] (b) (i) Find the area of R in terms of k. [2] (ii) It is now given that 1k = . When the area R is divided into n equal parts, the area may be approximated by the sum of the areas of n rectangles of equal width from 0 to 1xx== using the right endpoint of each subinterval . Illustrate on the diagram, printed on your answer booklet, why the resulting approximation is an overestimate of the area R. [2] (iii) Hence, show that 221 1 21n r r n nr= − + . [2] x y O

7 [Turn over 10 A right pyramid VABCD with a square base ABCD where VA = VB = VC = VD = 5 units is placed on a cuboid ABCDEFGH where EF = EH = 4 units and EA = 2 units. O is the centre of the square base EFGH. The perpendicular unit vectors i, j, k are parallel to EF, EH, EA respectively. (a) Show that 0 0 2 17 OV  =  +  . [2] (b) State the geometrical meaning of 1 2 17 OV VA +   . [1] (c) Find the vector equation of the plane passing through VBC, in the form ax by cz d+ + = , where the values of a, b, c and d are to be given exactly. [3] (d) The point X lies on the diagonal AC such that AX : XC = 3 : 1. Find the vector equation of the line VX. Hence determine whether this line intersects the plane of the square base EFGH at a point within EFGH or outside EFGH. [5] 4 E A D V C B G F H 2 O k j i 4

8 11 When studying an endangered species of marsupials in Western Australia, scientists found that the birth rate of the marsupials is directly proportional to its population. The survival of the marsupials is threatened by the presence of predato