SAJC 9758 2022 Prelim P1

1 [Turn Over 1 The equation of a curve C is given by ( ) ( )224 20x y x y+ + − = . (i) Show that the gradient of C at the point ( ),xy is given by d 5 3 d 3 5 y x y x x y +=− + . [3] (ii) Find the equation(s) of the tangent(s) to the curve C which are perpendicular to the line yx= . [6] 2 The curves 1C and 2C are defined by the equations 2 6 4y x= − and 26yx=− respectively. (i) On the same axes, sketch the graphs of the curves 1C and 2C , stating the equations of any asymptotes , the exact coordinates of the turning point(s) and any points where the curve crosses the x- and y-axes. [5] (ii) Solve the inequality 2 2 6 64 xx −− . [2] (iii) The transformations A and B are given as follows: A : Reflection about the y-axis; B : Translation of 4 units in the negative x-direction. The graphs 1C and 2C undergo in sequence, the transformations A and B. The resulting equations of the transformed graphs of 1C and 2C are ( )fyx= and ( )gyx= respectively. Deduce the solution set of the inequality ( ) ( )fg xx . [2]

2 [Turn Over 3 The function f is defined by 32f: x ax bx cx d+ + + , where x and a, b, c and d are constants. The graph of f intersects the y-axis at 3y=− and passes through the points ( )01,− and ( )2 ,0 . (i) Explain why f does not have an inverse. [1] (ii) Given also that the tangent to the graph of f at 1x= is a horizontal line, find f(x). [3] (iii) Sketch the graph of y = f(x), giving the coordinates of the turning points and the points which the graph intersects the axes. [2] (iv) Given that the function f has an inverse if its domain is restricted to xk , state the smallest possible value of k. [1] For the rest of the question, use the domain given and value of k found in part (iv). (v) Describe the relationship between the graphs of f ( )yx= and 1f ( )yx −= . [1] (vi) Show that the solution of the equation 1f ( ) f ( )xx −= satisfies the equation 33 11 6 0xx− − = . Hence, find the solution of the equation 1f ( ) f ( )xx −= . [3] (vii) It is g iven that g( ) ln( 5)xx=+ , where 5.x− A student attempts to find the composite function gf. The student’s solution is shown below: Comment on the validity of the student’s solution. [1]  ) ( ) gf 2 f 3 g( ) ln( 5) D D , gf ( ) l . ,n 5 , xx k x ax bx cx d x x k =+ ==  = + + +  +

3 4 (i) By sketching the graph of 1 21 xy x += − , find the range of values of x for which 1 021 x x + − . [4] (ii) Hence, without the use of a calculator, show that 0 2 1 3 3d ln 3 ln 52 1 2 4 x xx− + =−− . [4] 5 The curve C is defined by the equation ( ) 11 tan 22yx −= and the line L is defined by the equation 11 π 2 4 8yx = + − . It is given that the line L intersects the y-axis at the point Q and is a tangent to the curve C at the point P where 1 2x=− . (i) Find the y-coordinates of P and Q. [2] The region R is bounded by the line L, the curve C and the y-axis, for x < 0. (ii) Find the exact volume of the solid generated when R is rotated through 2π radians about the y-axis, giving your answer in the form ( )8 ab − where a and b are positive constants to be found. [6] 6 With reference to the origin O, the points A and B have position vectors a and b respectively, where a and b are perpendicular. A point P lies on AB between A and B such that : :1AP PB = − , 01 . (i) Show that ( ) ( ) (1 )cos 1AOP −= − + a ab . [4] (ii) Prove that ( ) ( ) ( ) 222 21 1 1− + − + = − +      a b a b a b . Hence, given also that OP bisects AOB , find the ratio of a b , leaving your answer in terms of  . [6]

4 7 The rate of temperature loss of an animal corpse can be estimated using Newton’s Law of Cooling, which states that the rate of change of temperature C , t hours after death of an animal is proportional to the difference between its body temperature C and the surrounding temperature 0 C  , where 0 . (i) Write down a differential equation for this situation. Solve this differential equation and show that the general solution of the above differential equation is given by 0 , where and are positive constants. ktAe A k −=+ [3] It is given that 0 24 = , the initial value of  is 36 and the initial rate of temperature loss is 2.5 C per hour. (ii) Calculate the exact values of A and k. [3] (iii) Hence, sketch the graph of  against t. [2] (iv) Explain why the rate of change of temperature of an animal corpse cannot be modelled by a constant rate of decrease of 1.5 C . [1] 8 (a) It is given that the equation 3230z az bz c+ + + = , where , and a b c are real numbers, has roots 5 11 i33− and 2− . Find the integer values of a, b and c. [4] (b) It is known that a complex number ii ii ee , eew   += − where 2n  − and   for any integer n. (i) Show that 2 πi 1e cot ( ) .2w  − =−  [3] (ii) Hence, find w and ( )arg w . [2]

5 9 The coach of Besto running club designed a training programme such that runners begin with a 400 m run on the first training session. On each subsequent session, the distance covered is 250 m more than the distance covered on the previous session. (i) Find the minimum num ber of sessions required for runners from Besto on the training programme to run at least 20 km in a training session. [3] For another group of runners in Besto, a circuit training exercise was designed to build up their stamina. In this exercise, this group of runners from Besto run from a starting point O to and from a series of points, 1 2 3, , ,P P P , increasingly far away in a straight line. In the exercise, they start at O and run stage 1 from O to 1P , and back to O, then stage 2 from O to 2P , and back to O, and so on. The distances between the points are such that 1 50 mOP = , 12 100 mPP = , 23 300 mPP = and 11 3n n n nPPPP+−= (see Fig. 1). (ii) Find an expression for the distance run by a runner from Besto who completes n stages of the circuit training exercise. [3] (iii) Hence, find the distance from O and the direction of travel, of a runner from Besto undergoing the circuit training exercise after he has run exactly 42 km. [3] Another running club, Choco, designed a different training programme. The runners in Choco began with running 400 m on the 1 st session. On each subsequent session, the distance covered was increased by 10% of the distance covered on the previous session. From the 11th session onwards and for all subsequent sessions, the distance covered was increased by r% of the distance covered on the previous session. (iv) Given that the runners from Choco club covered at least 20 km on the 70 th training session, find the range of values of r. [4] O 50 m 100 m 300 m 900 m Fig. 1

6 10 The diagram below shows the floorplan of a square lawn with a circular pond with its centre coinciding with the centre of the square lawn (see Fig. 2). The floorplan consists of a square lawn ABCD of side 26 m with a circular pond of radius 11 m built at the centre P of the square lawn. The owner intends to build a rectangular flower crate with base QTCS, with one corner of the base at C, and the opposite corner Q of the base, touching the circular pond, where angle RPQ =  radians, measured from RP and 0 4  . The base of the flower crate has area X m2. (i) By finding an expression of X in terms of  ,