2020 CJC Promos (QP + Suggested Answer)

BP- 61 2020 JCI H2 Math Promo - CJC 1 Without using a calculator, solve Hence solve ., 3 I<-x-2 J'x 13l l2le-" -z 9758/01/J1PROMOt2020 61

BP.62 3 ) The curve C has equation y =!-2:.' 2x+3 (i) Express the equation of C in the form y = a*: b :, u,here a and b are constants to be' 2x+3' determined. 12) (ii) Describe a sequence of tlree transformations which transform the graph of y =-J- onlo x+3 the graph of C. t3l 97s8/01/J l PROMO/2020 62 [Turn Over

BP- 63 4 3 By writing *-rin partial fractions, fi"d Z#=, giving your answer inthe form M -f (n), where M is a real constant to be determined. Exnlain *fw i-tA 4r"; is a convergent series, and state the value ofthe sum to intinity tsl 12) 97s$t01tJ1PROMOt2020 63

BP-64 4 5 Find the derivative of the following expressions with respect to x, leaving your answers in terms of x only. (b) tan-l("") , l2l (c) *sec2x. t3l 97s8/01/J 1 PROMO/2020 64 [Turn Over

BP- 65 6 5 The diagram shows a circle with centre O and diameter PQ . The point R lies on the circumference of the circle. Taking the centre O as the origin, the position vectols of the points P , Q and R are p, q and r respectively. .R P (i) By first writingdown E and@. interms of p and r, prove that the lines PR and QR are perpendicular, showing your working clearly. t4] It is giventhat qx r : (q-r)rs where s is the position vector of the point,S (ii) Byconsidering ffix@, showthat -.lli =kjfr, kelR , k+0. t3l 9758/01/J 1PROMOt2020 o 65

BP- 66 7 6 The diagrarn shows the graph of y=f(r). The curve passes through the origin, has a minimum point at (-4,0) and a maximum point at (1,4) . fne lines y = 0 and x =2 are asymptotes of the curve. v v=f(r) (-z,q) (-+,0) o !=0 On separate diagrams, sketch the graphs of t2)(i) Y=f '(r) , (ii) y =f (3x+2), and t3l 1 (iii) ,=fu, t3l indicating clearly the equations of any asymptotes, the coordinates of any tuming points and any points where the curve crosses the x- and y -axes whenever possible. e7s8/01/J l PROMO/2020 Y I I I I I I I I I I I I I I I I I I I I ,l _1_Lx bb [Turn Over

8P.67 8 7 A curve lras equation y = rl2-- ry . (i) Without using a calculator, find the equations of the tangent and normal to the curve at the point P where x=-1. l7l (ii) Hence find the exact area of the region bounded by the tangent and normal to the curve at thepoint P andthe y-axis. t2) 9758tO1tJ1PROMO|2A20 67

BP-68 9 8 The curve C has equation 3xz -2x -1y -_ -;;- for x e IR., x + -1. (i) Find, using an algebraic method, the range of values that y cannot take. t3] (ii) Sketch C, stating clearly the equations of asymptotes, the coordinates of the axial intercepts and the coordinates of the turning points. t4l (iiD Verify that (-1, -8) lies onthe graph -y =m(x+l)-S, where me IR.. Hence find the range of values of m forwhich the equation lx'z -z!-l = m(x+l)-8 has 2 distinct real roots.[2] x+l \ 9758/01/J 1 PROMO/2020 6B [Turn Over

BP- 69 9 10 A sequence up u2) u3, ...is such that ur*r=3un+Pn, where P is a constant and ne Z' . The terms of the sequence are defined by their previous terms, for example, u, = 3u, + P . (i) Given that u, = I and uz = 7 , find P and u, . lt is known that the n th temr of this sequence is given by u.=a(3')+bn+c, where a, b and c are constants. (ii) Find a, b and c. (iii) Find f u, in terms of n. r=l 9758/01/J 1PROMOl2020 12) t4l t4l 69

BP-70 11 10 A ftinction f is said to be a self-inverse function if f (x) = f-'(r) . The function f is defined by cx -2' c where a, b, c and d are non-zero constants. (i) By finding f-' (r) , show that a = 2 for f to be a self-inverse function. For the rest ofthequestion, use a = 2, b =3 and c = 5. (ir) Find. f'?(x) . (iii) Evaluate f" (4) . The function g is defined by g (') = 2x2 -3' x e IR' (iv) Explain why composite function fg does not exist. (v) Find the exact solutions of gf (x) = S. 9758/01/Jl PROMO/2020 l2l tll 121 12) t3l [Turn Over