SAJC 9758 2023 Prelim P1

Question 1 2 3 4 5 6 7 8 9 10 11 TOTAL Marks 5 7 7 10 5 12 7 8 12 13 14 100 This document consists of 30 printed pages and 2 blank page including this page. ST ANDREW’S JUNIOR COLLEGE PRELIMINARY EXAMINATION MATHEMATICS HIGHER 2 9758/01 Tuesday 29 August 2023 3 hr Candidates answer on the Question Paper. Additional Materials: List of Formulae (MF26) NAME:_________________________________________( _____ ) C.G.: __________ TUTOR’S NAME: _________________________________________ SCIENTIFIC / GRAPHIC CALCULATOR MODEL: _______________________ NUMBER OF ADDITIONAL PIECES OF WRITING PAPER :________________ READ THESE INSTRUCTIONS FIRST Write 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. Answer all the questions. Total marks : 100 Write your answers in the spaces provided in the question paper. Give non-exact numerical answers correct to 3 signi ficant 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. 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 are required to present the mathematical steps using mathem atical notations and not calculator commands. You 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.

2 [Turn Over 1 A function f is defined by 32f ( )x ax bx cx d   . The graph of f ( )yx passes through the points ( 1, 15) and (2, 3). The graph has a turning point at 1x  , and 2 0 f ( ) d = 5.xx∫ Find the values of a, b, c and d. [5] 2 (a) The sum, ,nS of the first n terms of a sequence is given by 2 nSnn= + Show that the sequence follows an arithmetic progression and state the value of the common difference. [3] (b) An arithmetic series has first term b and common difference d, where b and d are non-zero. A geometric series has first term a and common ratio r, where a is non- zero and r is positive. It is given that the 3rd, 5th and 7th terms of a geometric series are equal to the 7th, 13th and 25th terms of an arithmetic series respectively. Find the value of r and determine if the geometric series is convergent. [4] 3 Figure 1 shows the net of a square-based pyramid. The net consists of a square of side length x cm and four isosceles triangles, each with base x cm and fixed sides k cm. The net is folded to form a right pyramid which has a square base of side length x cm, as shown in Figure 2. k cm Figure 1 k cm Figure 2 k cm k cm x cm x cm k cm x cm

3 [Turn Over (i) Let the volume of the pyramid be V cm3. Show that V satisfies the equation 4222 92 xxVk  = −  . [2] (ii) Find, in terms of k, the value of x that will maximise the volume of the pyramid. [5] 4 The function f and g are defined by 1 (2 ) 1 for2f( ) 34 fo , 2, ,r3 2,5 x xx x xxx ∈< ∈≥  −=  − −   g ( ): f( ) ,forxx x x xa ∈ < . (i) Sketch the graph of f( )yx= , indicating clearly the axial intercepts and equations of asymptotes. [3] (ii) State the maximum value of a such that g has an inverse. [1] (iii) For the value of a found in (ii), find 1g ()x− and state its domain. [3] (iv) Given that the composite function 2f exists, find 2f . [3] [You do not need to simplify the expressions for 2f .]

4 [Turn Over 5 The diagram below shows the region R bounded by the circle with equation ( ) 2 249xy−+= , the line with equation 5 35yx= −+ and the x-axis. The line and the circle meets at the point A with coordinates (2, 5) . Show that the volume V, formed when R is rotated through 2π radians about the y-axis is given by ( ) ( ) 2 22 5 0 35 25 8 9 d .5 y y yyπ  − − −− −  ∫ Hence evaluate V, leaving your answer correct to 3 decimal places. [5] 6 The curve C1 has equation ()xx ay xa −= + and a is a positive real constant. (i) Find algebraically, in terms of a, the range of values that y can take. [3] It is now given that a = 1 and that curve C2 has equation 5 10 23y x= −+ + . (ii) Find the exact x-coordinates of the intersection points of C1 and C2. [2] (iii) Sketch C1 and C2 on the same graph, giving the coordinates of any points where C1 and C2 intersect and the equations of any asymptotes. [4] (iv) Find the exact area of the region bounded by the two curves C1 and C2, simplifying your answer in the form ( ) 2ln 2 ln 5 unitsabc ++ , where a, b, c are constants to be determined. [3] y x A R 4 3 1 O

5 [Turn Over 7 With reference to the origin O, the points A and B have position vectors a and b respectively. The point P lies on AB such that : 1:4AP AB = . The point Q lies on the line passing through the points O and B such that AQ is perpendicular to OP. (i) Explain why Q has position vector in the form λb where λ is a real constant. [1] (ii) Given that a and b are unit vectors and AOB θ∠= , show that 3 cos .3cos 1 θλ θ += + [4] (iii) Given that 0 2 πθ< < , explain with justification why Q does not lie between O and B. [2] 8 The complex number w is given by 1 i zw z −= + , where izxy= + where , xy ∈  and i.z≠− (i) If i,z= find w in the form e,ir θ where 0 and .r > −π<θ≤π Hence find 14 ,w leaving your answer in cartesian form. [4] (ii) If cos isinz θθ= + , show that w can be expressed as 1 cot 2 k θ+  , where k is a complex number to be found in exponential form. [4]

6 [Turn Over 9. A tentage for an outdoor activity is built by attaching a piece of polyester tent fabric to the tops of four vertical poles AE, BF, DG & CH, where E, F, G and H are positioned at ground level on the x-y plane. Coordinates of A, B, C and D are as shown in the diagram below, with lengths measured in meters. The length of the pole DG is k meters. (i) Find the cartesian equation of the plane ABDC. [3] (ii) Find the acute angle that plane ABDC makes with the horizontal. [2] (iii) Find the value of k. [2] (iv) Show that the shape of the polyester tent fabric ABDC is a trapezium. Hence, or otherwise, find its exact area in the form of 3 2a units2, where a is an integer to be determined. [5]

7 [Turn Over 10 Hyperglycaemia, also known as high blood sugar, is a medical condition that can affect a significant portion of the population. One potential treatment involves regulating blood sugar levels by administering insulin. During a clinical trial, insulin is added to the bloodstream of a patient, and the body's insulin regulation removes excess glucose from the bloodstream at a rate directly proportional to the amount of glucose present in the bloodstream. The rate of increase of glucose level in the bloodstream is inversely proportional to the amount of glucose present in the bloodstream of hyperglycaemic patients. The amount of glucose present in the bloodstream of the patient at time t (in minutes) is denoted by Q mmol/L (millimoles per litre). I t is given that the amount of glucose in the bloodstream remains constant when Q = 4. (i) Show that the rate of change of the amount of glucose present in the bloodstream at time t satisfies the differential equation 2d