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 equa