JPJC 9758 2024 Promo Question (JCMTC)

Name:____________________________________ Class:_____________ JURONG PIONEER JUNIOR COLLEGE JC1 Year End Examination 2024 MATHEMATICS 9758/01 Higher 2 27 Sept 2024 Paper 1 3 hours Additional materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST This document consists of 6 printed pages and 2 blank pages. [Turn over Answer all questions. Write your answers on the Printed Answer Booklet. Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers co rrect to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given by [ ] at the end of each question or part question .

2 1 Differentiate each of the following expressions with respect to x. (a) ( ) 12cos 3 x− , [2] (b) 3 21ln x x  + . [3] 2 Water is poured at a rate of 0.1 m 3 per minute into a container in the form of an open cone. The semi-vertical angle of the cone is 30 o . At time t minutes after the start, the radius of the water surface is r m (see diagram). Find the rate of increase of the depth of water when the volume of the water is 3 m 3 . [5] [The volume of a cone of base radius r and height h is given by 21 3V r h = ] 3 A curve C has equation 2f ( ) ax bx cx= + + , where a, b and c are constants. It is given that C passes through the points ( )1, 4− and ( )2, 11− and the curve of 1 f ( )y x= has a vertical asymptote with equation 1x= . Find the equation of C. [4] 4 By writing ( )88x A x B= − + , where A and B are constants, find 1 2 0 d4 8 5 x xxx−+ , giving your answer in the form 1ln tanp q r s −+ , where p, q, r and s are values to be determined. [5] r m 30

3 5 The curve C has equation 2 1 x ax by x ++= + , where a and b are constants. It is given that C has a minimum point at ( )2,0 . (i) Show that 4a=− and 4b= . [3] (ii) Sketch the graph of C, indicating clearly the equations of any asymptotes and the coordinates of any turning points and axial intercepts. [3] (iii) By adding an appropriate graph to your sketch in (ii), deduce the number of real roots of the equation ( ) 32 1 1 0x ax b x+ + + + = . [2] 6 The diagram shows an ellipse, centred at the origin with semi -minor axis a and semi-major axis 2a, where 0a . The rectangle PQRS is inscribed in the ellipse such that its four vertices P, Q, R and S lie on the ellipse. The coordinates of P is ( ),xy , where x and y are positive. (i) Write down the cartesian equation of the ellipse in terms of a. [1] (ii) By considering the coordinates of P, show that A, the area of rectangle PQRS, is 228x a x − . [2] (iii) Using differentiation, find, in terms of a, the value of x when A is a maximum. You do not need to prove that A is a maximum. [3] (iv) Find the value of a if the maximum value of A is 100. [2] y x Q P(x, y) S R [Turn over

4 7 (i) Without using a calculator, solve the inequality 2 53 22 x xx − −+− . [4] (ii) Deduce the range of values of x for which 2 53 22 x xx −− −− . [3] 8 (a) The diagram below shows the graph of ( )fyx= . The curve passes through the points ( ),0Aa− and ( )0,Ba − and has a minimum point at ( )2 , 2C a a − , where 0a . The lines 2xa=− and ya=− are asymptotes of the curve. On separate diagrams, sketch the graphs of (i) ( )fy x a a= − + , [3] (ii) ( ) 1 fy x= , [3] indicating clearly in each case, where appropriate, the equations of any asymptote(s), axial intercepts and coordinates of the points corresponding to A, B and C. (b) The curve whose equation is sinyx= , undergoes in succession, the following transformations: A: A translation by π units in the negative x-direction. B: A scaling parallel to the x-axis by a factor of 3. C: A translation by 2 units in the positive y-direction. Find the equation of the resulting curve. [3] y x O B C A

5 9 (a) Use the substitution sinx = to find 2 2 2 d 1 x x x− . [4] (b) (i) Find d sin(ln )d xx . [1] (ii) Hence, using integration by parts, find sin(ln ) dxx . [3] 10 A curve C has parametric equations 2x t t=+ , 2y t t=− , t . (i) Sketch the graph of C, indicating the coordinates of the points where the curve crosses either axis. [2] (ii) Find the coordinates of the point on C where the tangent to C is parallel to the line 5 4 20yx=− . [3] (iii) Find the equation of the tangent to C at the point ( )0, 2 and show algebraically that this tangent does not cut C again. [5] 11 Referred to the origin O, points A and B have position vectors a and b respectively, where a and b are non-zero and non-parallel vectors. Point C is the midpoint of OA and point D lies on BC such that BD: DC = 2: 3. (i) Find the position vectors OC and OD , giving your answers in terms of a and b. [2] (ii) Find the area of triangle OCD in the form of abk  , where k is a constant to be found. [3] (iii) Given that 2, 2==ab and angle AOB is 4  radian, using scalar product, show that CD is perpendicular to OA. [3] (iv) Given instead that a is a unit vector, give a geometrical interpretation of ab . [1] [Turn over

6 12 The line l contains the point P with coordinates (1, 5, 5)−− and is parallel to 36+−i j k . The plane 1 has cartesian equation 2 5 3 4x y z− − = . (i) Find the acute angle between l and 1 , giving your answer to the nearest degree. [3] (ii) Find the coordinates of Q, the point of intersection between l and 1 . [3] The plane 2 contains l and is perpendicular to 1 . (iii) Find a cartesian equation of 2 . [3] (iv) Find a vector equation of the line where 1 meets 2 . [2] 13 The function f is defined by 2 4f : 1 , for , 1x x x x−  − . (i) Sketch the graph of ( )fyx= , stating the equation of the asymptote and the coordinates of the point where the curve crosses the axes. [2] (ii) Show that f has an inverse. Hence, find ( ) 1f x− and state its domain. [4] (iii) By considering the domains of ( ) 1ffyx −= and ( ) 1ffyx −= , find the range of values of x such that ( ) ( ) 11ff f f xx−− = . [2] Another function g is defined by 2g: e 2, for .xxx − −  (iv) Explain why fg exists. Find fg and its range. [3]

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