DHS EJC RVHS H3 Math 2024 Prelim Questions

1 (a) Given a differential equation of the form d gd xx yy =  , use the substitution x vy= to show that 11 d dg( ) vyv v y =− . [2] (b) Show that the differential equation ( ) 22 2 2 2dee d xx yy xxy y y x y             = + + , ,0xy  can be written in the form d gd xx yy =  . [2] (c) It is given that the curve of the differential equation (b) passes through the point ( )ln 9, 2− . Solve the differential equation given in (b), leaving your answer in the form h xy y =  . [7] 2 (a) Show that 1 1 n k k n n k −  − =   for any positive integers n, k where nk . [2] It is given that 12, , ... , na a a + and s is a positive integer where sn . Let S be 12 ... si i ia a a , where the sum is taken over all 1, ..., { 1,2,..., }si i n  such that 121 ... si i i n     . (b) Find, in terms of n and s, the number of terms in S. [1] (c) Let m be a fixed integer where 1 mn . Find, in terms of n and s, the number of terms in S such that  12, ,..., siim i . [1] (d) Let ( ) 1 12 nng a a a= . Prove that ( )( ) ( ) ( )121 1 1 1 n na a a g+ + +  + . [6]

2 3 In this question, we will examine different ways of dividing circles. (a) Consider the number of regions formed when a circle is cut n times. The cases where the maximum number nm is achieved for 2n= and 3 is illustrated below with the values of nm stated. Illustrate the case where the maximum number nm is achieved for 4.n= Hence deduce, with justification, an expression for 1nnmm −− in terms of n and solve for .nm Explain briefly why this represents the maximum number of regions. [7] (b) Suppose a circle is divided into 21n+ congruent sectors, with n of them randomly coloured black and the other 1n+ randomly coloured white. A smaller concentric circle is placed on the larger circle and also divided into 21n+ congruent sectors, with 1n+ of them randomly coloured black and the other n randomly coloured white. A possible case for 7n= is illustrated below. Prove that, for all n, we can always find 1n+ sectors with matched colours by suitably rotating the smaller circle if necessary. [5]

3 5 An ice-cream shop sells single scoop cones with k different flavours available. You may assume that the shop does not allow mixing of flavours in any single scoop. (a) An order of n single scoop cones is made where n k . Given that all k flavours are bought and the order includes an odd number of cones for each flavour, state a condition between n and k and find the number of such possible orders. [3] Choosing from the k possible flavours, a group of n kids orders one single scoop cone each. (b) By considering all possible orders made by the n kids, explain why ( ) 0 S, k nk r r k n r P = = where S(n, r) denotes the number of ways to partition n distinct objects