ACJC 2023 JC1 H1 Maths Revision Set A

0 ANGLO-CHINESE JUNIOR COLLEGE MATHEMATICS DEPARTMENT MATHEMATICS 8865 Higher 1 JC 1 REVISION EXERCISE SET A Mar 2023 Topics Start Page Number Complete solutions will be uploaded into Math Google Site one week before CA1. Exponential & Logarithmic Functions 1 Graphing Techniques 2 Equations & Inequalities 5 ACJC CA1 past year papers 8 Name: _________________________________ H1 Maths Group: 1MAX ___ Form Class: ___________

1 EXPONENTIAL & LOGARITHMIC FUNCTIONS 1 (a) Solve 9lg( 8) lg 1 lg 24 xx    − + = +       . [4] (b) Solve the equation ( ) ( ) 153log12log 55 =−−+ xx [3] 2 SRJC 2016 Promo Q3 (a) Given that ln p = k, find ln (pe25) in terms of k. [2] (b) Given that 3e x – 4e –x = 11, find the exact value of x. [4] 3 CJC 2012 Promo Q3 (a) Find the exact value(s) of the equation xxx −=− e3e5e2 3 . [4] (b) Given that 3log =xa , find )(log 2axa . [2] 4 RI 2012 Promo Q2 Solve the simultaneous equations y = (1 + e x ) (1 − e x ), x = ln (y + 1). [5] 5 YJC 2013 Prelim Q2 Solve the simultaneous equations 34 464 x y= and lg( ) 1 lg( 1)y x x− = − − . [4] 6 TJC 2016 Promo Q3 Given that 2 44log log 0q q p− − = and log 2p q = , show that 44p p= . [3] (i) On the same set of axes, sketch the graphs of 4xy= and 4yx= . [1] (ii) Hence find the solutions of p and corresponding values of q . [2] 7 JJC 2013 Promo Q4 In a research project, 5 000 ants are put in a man -made habitat for the population to grow. After t days, the population of the ants, P (in thousands) is given by 45eAtPB=− + , for some constants A and B. Given that 10 days later, the population has increased to 35 000. (i) Show that B = 50 and ln 3Ak= , where k is a constant to be found. [4] (ii) Find the population of the ants after 20 days. [2] (iii) Find the population of the ants after a very long time. [1] (iv) Sketch the graph of P against t. [1]

2 8 RVHS 2016 Promo Q11 The temperature, C, of the water in a flask, t minutes after the start of heating is modelled by the equation ( )ln 2 1b k t = + + , where b and k are constants. Initially the temperature of the water is 20 C . After 2 minutes of heating, the temperature is 65 C . (i) Find the values of b and k. [4] (ii) Find the time it takes for the water to reach a temperature of 50 C . [3] (iii) Sketch the graph of the temperature of the water against time for 08 t , labelling clearly the end points of the graph. [3] (iv) State, with a reason, whether this model is suitable in the long term. [2] Answers 1 (a) x = 18 (b) −2 2 (a) k + 25 (b) ln 4 3 (a) ln 3 2x= (b) 13 4 x = 0, y = 0 5 x = 3.5, y = 7.5 6 (ii) 2, 16pq== ; 4, 216pq== 7 (i) 1 10k =− (ii) 45 000 (iii) 50 000 8 (i) 45 ln 5k = (ii) 0.962 min (iv) No Suggested reason: Boiling point of water is 100 C but the graph is an increasing curve, so the model will be inaccurate once the temperature rises above 100 C as t→ . GRAPHING TECHNIQUES 1 DHS 2014 Promo Q3 Sketch the graph of 32 1,y x x x= − − + stating clearly the coordinates of any points of intersection with the axes and the coordinates of the turning points. [2] By sketching a suitable graph on the same diagram, find the number of real roots for the equation 322 2 2 2 ln 0.x x x x− − + + = [3] 2 ACJC 2009 CT1 Q6 [modified] Sketch the circle 012622 =++−+ yxyx , stating the coordinates of its centre and radius. [4]

3 3 VJC 2009 Prelim Q5 (i) Sketch, for 0x , the graphs of 1 4 −= xy and 43112 2 −+= xxy on the same axes. [2] (ii) The graphs intersect on the y-axis. Find, correct to 2 decimal places, the x-coordinate of the point of intersection for which 0x . [1] 4 MJC 2008 Prelim Q4(i) [modified] Sketch, for 0,x the graphs of 213yx=− and 2 36y x= on the same axes, showing clearly any axial intercepts. State the exact coordinates of any points of intersection for which 0.x [4] 5 MI 2012 Promo Q3 [modified] Sketch, on separate diagrams, the graphs of (i) y = 2ex + 2, [2] (ii) 2x2 + 3x + 2y2 + 2y + 1 = 0, [4] (iii) 4 21 xy x −= + , [3] labelling clearly the equations of any asymptotes and the coordinates of any intersections with the coordinate axes. State the equations of any axes of symmetry. 6 JJC 2009 Promo Q4 The diagram given shows the graph of the equation ax cy xb += − . The graph intersects the x-axis at 3 ,02 − and the lines x = 1 and y = 2 are asymptotes. Find the values of a, b and c. [3] 7 TJC 2014 Promo Q4 A curve C has equation ( )ln 3 ,y kx=− where k is a positive constant. State in terms of k, the equation of the asymptote of C and the coordinates of the point where C crosses the x-axis. [2] Sketch the graph of ln(2 3)yx=− , showing clearly its asymptote and the point of intersection with the x-axis. By adding a suitable graph to the same diagram, solve the equation ( )e ln 2 3 1.x x−= [5] x y x=1 y =2 3 2−

4 8 CJC 2012 Promo Q2 (Explore using GC) A curve has equation 1 1 1 23 −−++= xxy . (i) State the equations of the asymptotes of the curve. [2] (ii) Sketch the curve, indicating clearly the equations of the asymptotes, and the coordinates of any turning points and any points of intersection with the axes. [4] 9 ACJC 2009 CT1 Q3 Sketch the graph of y = 22 14 − + x x , showing clearly the asymptotes and the axial intercepts. [3] Given that the equation x = )22(log)14(log 22 −−+ xx can be written in the form 22 14 − + x x = f(x), determine the function f(x). [3] The equation x = )22(log)14(log 22 −−+ xx can be solved graphically by drawing an additional curve in the same diagram as the graph of y = 22 14 − + x x . Sketch the additional curve and solve the equation x = )22(log)14(log 22 −−+ xx . [2] 10 TJC 2009 Prelim Q11 [modified] (a) A rectangular hyperbola H is given by the equation 2 2 xay xa −= + where a is a positive real number. Sketch H, stating clearly the equations of any asymptotes and the coordinates of any intersections with the coordinate axes in terms of a. [3] Verify that the point 34 , 27 a  lies on H. [2] (b) The diagram shows the graph of ( )fyx= . Express ( ) 2 21x x k+ + + in completed square form. [1] Hence explain why the equation ( ) ( ) 2f 2 1x x x k= + + + will always have two real solutions for 2k  . [2] f ( )yx= y x 1x=− 2y= 0

5 x y y = 3 1.2 6 – 1.59 6 x = 1 x = –1 (0.172, 5.91) (5.83, 3.09) Answers 1 (0,1), (−1,0), (1,0) . Coordinates of turning point (−0.333, 1.19) and (1,0) , 3 2 Radius of circle = 3 Coordinates of centre are (3, 1)− 3 (ii) x = −2.67 4 intersection points: (2, 9), (3, 4) for x ≥ 0 5 (i) asymptote: y = 2, (0,4) (ii) (−1,0), (−0.5,0) (iii) asymptote: y = −1/2, x = −1/2, (0,4), (4,0) 6 a = 2, b = 1, c = 3 7 asymptote: x = 3/k. Crosses x-axis at (4/k, 0). x = 2.07 8 9 intercepts: 11,0 , 0,42    −−       asymptotes: x = 1, y = 2 sketch f ( ) 2 xx = x = 2.10 (3 sf) 10 (a) asymptotes: x = −2a, y = 2 axes intercepts: 10, , ,022 a   −       EQUATIONS & INEQUALITIES 1 Two cubes of side x cm and y cm respectively, have a total surface area of 60 cm 2. The sum of the length of all the edges of the two cubes is 48 cm. Find the exact values of x and y, if y > x. 2 YJC 2013 Promo Q2(i) It i s giv en that 2f ( )x ax bx c= + + , where a, b and c are constants. Given that the curve with equation f ( )yx= passes through the points with coordinates (0, 1.2), (2, 34.4) and ( − 3, − 11.1), find the values of a, b and c. [3]

6 3 JJC 2013 Promo Q7b A factory produces 3 brands of drinks, A, B and C. The total price of 1 litre of A, 1 litre of B and 2 litres of