NJC 9758 2024 Promo

* © NJC 2024 [Turn_over NATIONAL JUNIOR COLLEGE SENIOR HIGH 1 Higher 2 NAME SUBJECT CLASS 1ma2 REGISTRATION NUMBER MATHEMATICS 9758 Term 4 Promotional Examination 25 September 2024 3 hours Candidates answer on the Answer Booklet. Additional Materials: List of Formulae (MF27) Printed Answer Booklet READ THESE INSTRUCTIONS FIRST Write your name, class and registration number on the work you hand in. Write in dark blue or black pen. Answer all the questions. Give non-exact numerical answers correct 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. You are reminded of the need for clear presentation in your answers. Up to 2 marks may be deducted for improper presentation. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages and 2 blank pages.

2 © NJC 2024 1 Without using a calculator, solve the inequality 2 3 12 x xx + +− − . [4] 2 The region bounded by siny xx= , the x-axis, and the lines 0x= and πx= is rotated through 2π radians about the x-axis. Find the exact volume of the solid generated. [5] 3 (i) Given that k is a positive constant, s ketch the curve E with equation ( ) 2y x x k=− , labelling the coordinates of the points where the curve cuts the axes. [2] (ii) Hence, find, in terms of k, the area of the region (s) bounded by E and the x-axis for 02 x k . [3] 4 (a) The diagram shows the curve f ( )yx= that passes through the origin and (1,0). It has a minimum point 1 2 , 1 2  −   and asymptotes 1x=− and 3.y = ( )fyx= 3y= O ( )1, 0 1x=− 1 2 , 1 2  −   x y Sketch the curve ( ) 1 ,fy x= showing clearly the coordinates of any stationary points, axial intercepts and equations of all asymptotes. [3]

3 © NJC 2024 [Turn_over (b) A curve with equation h( )yx= undergoes the following sequence of transformations. Step 1: A translation of 2 units in the positive direction of the x-axis. Step 2: A scaling parallel to the x-axis by factor of 3. Step 3: A reflection in the y-axis. The resultant curve has equation 2 6 45 xy x=− + . Find h( )x . [3] 5 The line 1l passes through the point ( )7,6,5A and the point ( )5, 2, 5B − − − while the line 2l has equation 10 85 2 yzx k −−− = = , where k is a constant. The lines 1l and 2l intersect at point P. (i) Show that 2k= and find the coordinates of P. [5] (ii) Find the position vector of the point on 2l which is closest to A. [3] (iii) Hence, find a vector equation of the line which is a reflection of 1l in 2l . [2] 6 A curve C has equation 2 2 2 3 2 2 0x y xy y x+ + − − = . (i) Find d d y x in terms of x and y. [3] (ii) Find the gradients of the tangents at 0.x= [2] (iii) Using an algebraic method, show that there is no part of the curve for which 4.x− [3] 7 (a) (i) Given that =p q 0 , what can be deduced about the vectors p and q? [2] (ii) With respect to the origin on a xy-plane, a fixed point G has position vector g and a variable point R has position vector r. Describe the locus of R if ( ) 0.= −r r g [2] (b) With respect to the origin, plane p has an equation given by ( ) ( )2 2 2 3 2 , ,.    − − = + + + + + +r i j k i j k i j k Points Q and R have position vectors 3 9 5++i j k and 4 11 7++i j k respectively. Find the length of the projection of QR ⎯⎯ → onto p. [4]

4 © NJC 2024 8 The diagram below shows that the graph of g ( )yx = has two x-intercepts at ( )1, 0− and ( )1, 0 as well as two stationary points whose x-coordinates are 1− and 1 2 respectively. ( )gyx = 1 2 ( )1 ,0− O ( )1,0 x y (i) A stationary point on the curve o f g( )yx= has a x-coordinate that is positive. State the value of this x-coordinate and determine the nature of this stationary point on the curve g( )yx= . [2] It is given that ( ) 243g x ax bx cx dx e = + + + + . (ii) Show that 0a b c d e− + − + = . [1] Given further that the gradient of the tangent at the point of inflexion on g( )yx= is 27 ,8− find the values of a, b, c, d and e. [4] 9 Referred to the origin O, fixed points A and B have position vectors a and b respectively, where O, A and B do not lie on the same line. Point C with position vector c is a varied point such that 2a b c 0+ + = , where  is a parameter. (i) Show that Area of 3 Area of ABC ABO   + = . [3] (ii) Deduce the value of  given that A, B and C are collinear. [1] (iii) Given instead that A, B and C are not collinear and the line OC intersects AB at the point D, find the ratio :AD BD . [2] (iv) Further given that 90OAC OBC = =  , find  in terms of a and b . [4]

5 © NJC 2024 [Turn_over 10 (a) The functions f and g are defined by ( ) 2 22f : , 2 xx xx − +− , 1,xx g: e 1 , xx − − .x (i) Determine if the composite function fg exists. [2] (ii) Find the exact range of gf. [3] (iii) The domain of f is now restricted to 0 1.x Find the value of ( ) 11g f 2 .−− [3] (b) The function h is defined by ( )h : 3 sin cos ,x x x + 5,, 2x x p −   where p is a real constant. (i) Find the largest exact value of p such that 1h− exists. [3] (ii) With the value of p obtained in part (b)(i), find the solution of ( ) ( ) 11hh h h ,xx−− = leaving your answer to 3 decimal places. [2] 11 (a) (i) Express ( )( ) 2 13 2 3 1 4 x xx − −+ in partial fractions. [3] (ii) Hence, find the exact value of ( )( ) 1 2 21 2 13 2 d 3 1 4 x x xx− − −+   , leaving your answer in the form ln πpq+ , where p and q are rational numbers to be determined. [4] (b) You are given that ( ) 22 222 daxIx ax −= +   , where a is a positive constant. Use the substitution tanx a t= , where ππ 22 t−   , to find an expression for I in terms of a, leaving your answer in a non-trigonometric form. [5]

6 © NJC 2024 12 D dsd R S O y x The diagram above shows Curve D which has equation ( ) 2 2 136yx− − = . Points R and S are stationary points on D. Another curve E has equation 22 1yx−= . (i) By considering a single transformation that transforms the graph of E onto D or otherwise, find the equations of the asymptotes of D and the coordinates of S. [2] (ii) Verify that point ( ),sec 3 6tan p pP + lies on D, where