2023 JC1 Promo Practice Paper C VJC

2023 PROMO PRACTICE PAPER C 1 VICTORIA JUNIOR COLLEGE 2023 PROMO PRACTICE PAPER C (Modified from 2020 VJC H2 Math PROMO) 1 RVHS Promo 9758/2020/Q2(i) Show that 2 30xx++> for all real values of x . Without the use of a calculator, find the range of values of that satisfy [4] 2 YIJC Prelim 9758/2020/02/Q2(a) Given that 4i ,1iz λ λλ −= ∈−  and arg( ) πz = , find the value of z. [4] 3 NJC Prelim 9758/2018/01/Q7(b) In the diagram above, QR = 6, PS = 4, PR = 5, 2PSR π∠= and QRS θ∠= radians. (i) Show that ( ) 1 2 61 36cos 48sin .PQ θθ= −+ [2] (ii) Given that θ is a sufficiently small angle, show that 25PQ p q θθ≈+ + for some rational constants p and q to be determined exactly. [4] 4 It is given that d ed xy xyx −+= and 1y= when 0x= . (i) Find the Maclaurin series for y, up to and including the term in 3x . [5] (ii) Hence, or otherwise, state the first three non -zero terms of the Maclaurin series for d d y x . [1] 5 (a) Find x 2 2 26 1 .12 4 xx xx x ++ ≥−− − • 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 C 2 (i) 2 1 d25 xxx++ ⌠⌡ , [2] (ii) ( ) 3 ln dx xx ⌠ ⌡ . [2] (b) Find the exact value of 2 0 cos dx xx π ∫ . [3] 6 The curve C has equation 22 0x ay bx cy+ ++= where a, b and c are constants. (a) Given that C passes through the point ( )5, 3− and that the tangent to C at the point 13,22  − is parallel to the y-axis, find the values of a, b and c. [4] (b) Given instead that 1a=− , 10b=− and 6c=− , sketch C. [4] 7 The diagram below shows a drain in the form of a triangular prism 8 m long and 2 m deep. The opening of the drain is 1.5 m wide. During a heavy downpour, rainwater flows into the drain at a constant rate of 30.03 m per second. At the same time, the rainwater is drained away at a constant rate of 30.02 m per second. At time t seconds after the start, the depth of the rainwater in the drain is h m. (i) Show that the volume of rainwater in the drain, 3 mV , is given by 23Vh= . [3] (ii) Find the rate at which h is increasing when 7.2.V = [4] 8 An infinite geometric series has first term a and common ratio r , where 0r > . The third term is 36 and the sum to infinity is 243. (i) Find the value of a and r. [3] An arithmetic series has first term 1 and common difference d. The sum of the first 6 terms of the arithmetic series is equal to the sum of the first 3 terms of the geometric series. (ii) Find the value of d. [3] (iii) Find the least value of n for which the thn term of the arithmetic series is more than the sum of the first 2n terms of the geometric series. [3] Cross section of drain 1.5 m 2 m h m

2023 PROMO PRACTICE PAPER C 3 9 (a) The graph of f( )yx= is shown in the diagram. It passes through the origin and has a min