HCI 9758 2023 Prelim P1

1 The variables x and y are related by the differential equation 2 2 2 d 2ed xy x −= . Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote. [4] 2 In the triangle PQR, angle π 3PQR= radians and angle π 3PRQ θ= − radians, where θ is small enough for 3θ and higher powers of θ to be negligible. Show that 21PR PQ αθ βθ≈+ + , where α and β are exact real constants to be determined. [5] 3 A curve C has equation 24 5xy y+= . (a) Find d d y x in terms of x and y. [2] (b) Find the exact distance between the two tangents to C that are parallel to the x- axis. [3] 4 Without the use of a calculator, show that 2 22 2 d ln 3 π24 x xp qxx− − = +−+∫ , where p and q are exact real constants to be determined. [6] P Q R

2 © HCI 2023 5 A curve G with equation f (3 2)yx a= −+ , where a is a positive real constant, is transformed onto the curve with equation f( )yx= by a sequence of transformations. (a) Describe fully, in terms of a , the sequence of transformations. [3] A point on G with coordinates ( ,0)b is mapped onto the point with coordinates (, )cd on the curve with equation 1 f( )y x= . (b) Find c and d in terms of a and b . [4] 6 (a) By writing 1 (2 1)(2 3)rr++ in partial fractions , find an expression for 1 1 (2 1)(2 3)r n rr= ++∑ in terms of n . [4] (b) Explain why ( ) 1 11 3 (2 1)(2 3) r r n rr= −+ ++∑ is a convergent series, and state its sum to infinity. [4] 7 It is given that πtan for ,4f( ) πsin for 3 ,2 x axaax x ax aa   −<≤ =  <≤  and that f( ) f( 4 )x xa= − for all real values of x , where a is a positive real constant. (a) Sketch the graph of f( )yx= for 47ax a− ≤≤ . [3] (b) By using the sketch in part (a), write down an equation relating f( )x and f( ) x− . [1] (c) Show that 4 2 f ( ) d (1 ln )π a a kaxx m − = +∫ , where k and m are constants to be determined. [5]

3 © HCI 2023 8 Relative to the origin O , two points A and B have position vectors given by 2= −a ij and α=bk respectively, where α is a negative real constant. (a) Find the angle between OA and OB . [1] Let OBA θ= , where θ is in degrees. It is given that 1sin 2θ = . (b) Find the two possible values of ,θ and justify which of the two values is the correct value of θ. [2] (c) By considering a suitable scalar product, find the exact value of α . [3] A point C has position vector c such that = −cba . (d) State the shape of the quadrilateral OABC , justifying your answer. Find the exact area of the quadrilateral OABC . [3] 9 The function f is defined by 2f : ln ( 1) 2xx ++ , , 0xx∈≥ . (a) Show that f has an inverse. [2] (b) Find 1f () x− and state the domain of 1f− . [3] The function g is defined by 1g: 1 xx x − + , , 1xx∈ ≠− . (c) Given that 1g− exists, show that 11gf−− exists, and find the range of 11gf−− . [3] (d) Find the exact value of x such that 11 1gf () 2x−− = . [3] 10 A complex number z is given by i π e nzr= , where 01 r<< and n is a positive integer with 3n≥ . The numbers 21, , ,..., nzz z can be represented by the points 012, , , ... , nPPP P respectively in an Argand diagram. The ( 1)-sidedn+ polygon formed by using 012, , ,..., nPPP P is called the ( 1)-n polygon+ generated by z . An example of a (3 1)-polygon+ generated by z is shown in the following Argand diagram (not drawn to scale).

4 © HCI 2023 Let 1 (1 3 i)4z= + for parts (a) to (c). (a) Express z in the form ie,r θ where 0r > and 0 πθ<≤ . [2] (b) Hence write down 2z and 3z in similar form. [2] (c) Find the exact area of triangle 01OP P , where O is the origin. Hence find the exact area of the (3 1)- polygon+ generated by z . [4] Let i π1 e2 nz= for part (d). (d) Find the area of an ( 1)-n polygon+ generated by z in terms of n , leaving your answer in the form π(1 ) sin ,nab n− where a and b are real numbers to be determined. [3] 11 [The volume of a right circular cone of radius r and height h is 21 π3 rh .] A student makes a printer nozzle for his self-built 3D printer. The printer nozzle consists of a hollow inverted right circular cone of negligible thickness with radius r mm where 02 r<< , height h mm and slanted edge 2 mm joined to a hollow cylinder of negligible thickness with radius r mm and height r mm. A height of 1 4 h mm is cut off from the vertex of the cone for the nozzle opening (see diagram). The volume of the printer nozzle is V mm3 . (a) Show that 3 22 21π π4 64Vr r r= +− . [3] The student wants V to be a maximum. (b) It is given that 1rr= gives the maximum value of V . Show that 1r satisfies the equation 424537 18736 3136 0rr− += . [4] r mm mm h mm 2 mm mm

5 © HCI 2023 (c) Show that one of the positive roots of the equation in part (b) does not give a stationary value of V . Suggest a reason why this value is a solution to the equation in part (b) even though it does not give a stationary value of V . [3] (d) Given that 1r is the largest positive root of the equation in part (b), state the value of 1r and find the corresponding value of h. (You need not show that your answer gives a maximum.) [1] (e) With reference to the value of 1r found in part (d), comment on the practicality of having a maximum volume for the printer nozzle. [1] 12 A data processing centre processes data (in suitable units) every day. Let nu , where n is an integer, 1n≥ , represents the amount of data processed each day starting from 1st September. It is given that 1 (1 )nnu pu+ = + , where p is a positive real constant. (a) Explain why the sequence {}nu is a geometric progression. [1] On 1st September, 2m units of data, where m is an integer, 6m≥ , were processed. (b) If the total amount of data processed in the three days from 1st September to 3rd September is 127 (2 )36 m units, find the value of p . [3] (c) Explain if there is a limit to the total amount of data the centre can handle in the long run. [1] Let 1 4p= for the rest of the question. It is desired that data processors at the centre operate at below their maximum capacities to avoid processor downtime due to overheating. In a revised operating procedure, it is proposed that rv (where r is a positive integer, 1r≥ ), the amount of data processed each day starting from 1st October, follows a sequence given by 1 for 1 4, 25 for 5. r r r urv vr− ≤≤= −≥ (d) Show that 6[5(2 ) 4]m rvk r −= −+ for 4r≥ , where k is a constant to be determined. [4]

6 © HCI 2023 (e) Find the total amount of data processed up to the th15 day, starting from 1st October. Give your answer in the form (2 )mst + , where s and t are constants to be determined. [3] (f) Assume that the revised operating procedure is adopted. Explain, in context, if it is meaningful to compute the total amount of data processed in the long run. [1]