2023 JC1 Promo Practice Paper B VJC

2023 PROMO PRACTICE PAPER B 1 VICTORIA JUNIOR COLLEGE 2023 PROMO PRACTICE PAPER B (Modified from 2019 VJC H2 Math PROMO) 1 (i) Sketch the curve with equatio n 3ln 3yx= + , giving the equation of the asymptote and the coordinates of any points of intersection with the axes. On the same diagram, sketch the curve with equation 5 3yx= . [3] (ii) Solve the inequality 5 33ln 3xx+  . [2] 2 Given that ( ) 22 d32 d yx y xy x−= , and that 1y= when 0x= , find the values of d d y x and 2 2 d d y x when 0x= . Hence write down the first two non- zero terms in the Maclaurin series for y. [4] 3 (i) Using integration, show tha t ( ) 22 1e sin d e 2sin cos5 xx xx x x C= −+∫ , where C is an arbitrary constant. [4] (ii) Hence find the gradient of the curve ( ) ( ) 22e 2sin 1 cos 1xy xx += +− + at π 12x= − , leaving your answer in an exact form. [2] 4 The curve C has equation ( ) ( ) 22 23 194 yx−− −= . (i) Sketch C, giving the equations of its asymptotes and the coordinates of any turning points. [4] The curve D has equatio n 2212 48 48 6 3 0y y ax ax a− ++ − −= , where a is a positive constant. (ii) Find the set of values of a for which C and D do not intersect. [4] 5 (i) Find the binomial expansion for 2 2 1 2 x x + − , up to and including the term in 4x . Give the coefficients as exact fractions in their simplest form. [3] (ii) Find the set of values of x for which this expansion is valid. [2] (iii) Using your answer in part (i), find the series expansion of ( ) 222 x − − , up to and including the term in 2x . [3] • Exam conditions (one sitting, 3 hours) • Manage your time well • Check against solutions and learn from your mistakes before the next practice

2023 PROMO PRACTICE PAPER B 2 6 The diagram below shows the graph of g( )yx= , where ( )g 2 ax bx xc += + . Determine the values of a, b and c. [3] It is also given that 1g( ) f 1 2xx = − . State a sequence of 2 transformations that will map the graph of g( )yx= to the graph of f( )yx= . Find f( )x . [5] 7 (a) Find 6s i n 2c o s 2dx xx∫ . [2] (b) (i) Find 2 3 d2 x xx + ⌠⌡ . [2] (ii) Show that ( ) ( ) 2 2 2 2 31 1 32 32 x Ax Bx xx xx + +++=−+ −+ , where A and B are constants to be found. [2] (iii) Using your answers to parts (i) and (ii), find ( ) ( ) 2 2 2 0 10 16 2 d 32 xx x xx −+ −+ ⌠  ⌡ . Give your answer in the form 1tan lna bc− − , where a, b and c are constants to be determined. [4] (c) JJC Prelim 9758/2018/02/Q4 By using the substitution 21 sin3x θ= , where π0 2θ≤< , find the exact value of 1 4 0 d13 x xx−∫ . [5]

2023 PROMO PRACTICE PAPER B 3 8 2018/NJC/Prelim/2/4 (a) It is given that ( ) 3 1i* 1i3 z −−= − , where *z is the conjugate of a complex number z. (i) Find the exact values of the modulus and argument of 1 z . [5] (ii) Hence determine the exact values of a and b (where ππ b−<≤ ) in the equation 2i 4 1e ab z + = . [3] (b) The complex variables u and v satisfy the equations i3uv−= and * (1 i) 7 4iuv +− =+ . Find the values of u and v, giving your answers in the form x + iy. [4] 9 A curve C has equation 2 1 2 xxy x α ++= + , where α is a real, non-zero constant. Show that if C has 2 stationary points, then 0α < or kα > , where k is a constant to be determined. [4] Sketch the curve C for 1α =− , giving the equations of asymptotes, the coordinates of stationary points and points of intersection with the axes. [4] 10 During test drives, a sensor is used to record the number of revolutions per minute made by a particular wheel of a vehicle. (a) In a test drive, a car is initially travelling at constant speed but starts to slow down due to engine malfunction. The total number of revolutions is recorded on every 1- minute interval after malfunction. In the first n minutes, the total number of revolutions recorded, nS , is given by ( )54 29nSnn= − . (i) Show that the number of revolutions recorded in each minute after the malfunction occurs, before the car comes to a complete stop, follows an arithmetic progression. [3] (ii) The diameter of each wheel of the car is measured to be 61 cm. Show that the car travels a distance of 21.7 km, correct to 3 significant figures, fr om the time the malfunction occurs until it comes to a complete stop. [3] (b) In another test drive, a truck was travelling at constant speed before it entered the rough terrain. Before entering the rough terrain, the wheel was rotating at 486 revolutions per minute (rpm). After entering the rough terrain, engine power increases the rate of rotation by 20 rpm almost immediately at the beginning of each minute. However, at the end of each minute, friction slows the truck down such that the rate of rotation is 2 3 of that recorded at the beginning of that minute. The rate of rotation of the wheel at the end of the the nth minute after entering the rough terrain is denoted by nv rpm. (i) Show that 2446 403 n nv = +  . [4] (ii) Explain why the wheel always rotates at a rate of more than 40 rpm. [2] (iii) Given that the rate of rotation of the wheel was less than 45 rpm at the end of m minutes, find the least integer value of m. [2]

2023 PROMO PRACTICE PAPER B 4 11 The diagram shows a string that is unwound from a circle while being held taut. The curve traced by the end point P of the string is called the involute of the circle. One of the major applications of involute of circle is in designing of gears for revolving parts where gear tooth follow the shape of involute. A circle has fixed radius a units and centre O and the initial position of P is at ( ),0a . The parameter θ, 0 2 πθ   , is the angle measured from the positive x -axis to OT in the anti-clockwise direction, where T is the point on the circle such that PT is tangential to the circle. Show that the involute has parametric equations ( )cos sinxa θθ θ= + , ( )sin cosya θθ θ= − , for π0 2θ . [3] The point W on the involute has parameter π 3θ = . (i) Show that the equation of the normal to the involute at W is 32y ax= − . [5] (ii) At W, x increases at a rate of 0.3 units per second. Given that z xy= , determine, in terms of a, the rate of change of z at W. [4] Involute of the circle