RI 2026 H3 Math Prelim

2026 RI H3 Math Prelims Questions 1 Let 1 20 d , 0.1 n n xI x n x= + (a) Find the exact values of 0I and 1I . [2] (b) Show that 1 1 nI n + for all nonnegative integers n. [1] (c) Show that for all integers 2n , 2 1 .1nnII n−+= − [2] (d) (i) Hence show that for all positive integers m, ( ) ( )21 1 1ln 2( 1) . 22 km m m k I k+ = −− = +  [2] (ii) Deduce the exact value of ( ) 1 1 1 . k k k − = −  [2] (e) Determine the exact value of ( ) 1 1 1 .21 k k k − = − − [3] 2 (a) Let ( ) 0 1 f24n kn kkSx == , where ( ) 4f sinxx= . (i) Show that there exists a function g such that ( ) ( ) ( )1f g g 2 4x x x=− . [2] (ii) Hence find ()nSx in terms of g. [3] (iii) Show that ()nSx converges for all real x and find lim ( )nn Sx→ . [2] (b) Let ( ) ( )0 1 h2 4 n k n k k Cx = = − , where ( ) 4h cosxx= . Show that ()nCx converges for all real x and find lim ( )nn Cx→ . [4]

2026 RI H3 Math Prelims Questions 3 (a) Let x, y, z be nonnegative real numbers. (i) Using the AM-GM inequality, show that 3 3 3 242 .7x y y z z x x yz++  [2] (ii) Hence show that 3 3 3 ( ).x y y z z x xyz x y z+ +  + + [2] (b) Let a, b, c, d be nonnegative real numbers. Show that 4 4 4 4 ( ).a b b c c d d a abcd a b c d+ + +  + + + [4] (c) Let u, v, w be positive real numbers such that uvw = 1. Prove that 2 2 2 .u v w u v w+ +  + + [3] 4 (a) Consider the differential equation (E) 2d( 2) 2 2 ( 1)dyx x x y xx+ = + + − + , where 2x− . (i) Show that all the solution curves of (E) pass through the fixed point ( )2, 2−− . [1] (ii) Use the substitution u y x=− to find the general solution of the differential equation (E). [5] (iii) State the solution curve that is a line and show that all other solution curves will approach it as x tends to infinity. [2] (b) Let n be a positive integer. Consider the differential equation (F) ( ) 2d( 2) 2 2 ( 1)dyx n x x y xx+ = + + − + , where 2x− . (i) Show that all the solution curves of (F) pass through a fixed point. [1] (ii) Solve (F). [3]

2026 RI H3 Math Prelims Questions 5 (a) A log of wood of unit length is to be cut into three pieces. Two cuts will be made at random along the log to create these three pieces. Let the distances from the left end of the log to the two cuts be x and y. (i) Suppose 0 1.xy   Write down 3 inequalities which the 3 lengths of log must satisfy if they were to form the sides of a triangle. [2] (ii) Draw a diagram to illustrate the region in the xy-plane such that the inequalities in (a)(i) hold. [2] (iii) Hence determine the probability that when 2 cuts are made at random points along a log, the lengths of the three pieces obtained form the sides of a triangle. [1] (b) Let a, b, c, d and n be positive integers such that ,,a b c n . (i) Using a combinatorial argument, show that 2 1 .23na an= +   =       [2] (ii) Use the bijection principle to show that there are 2 a  positive integer solution triplets (b, c, d) to the equation 1b c d a+ + = + . [3] (iii) Hence show that the probability 2 2 1P( ) . 6na b c n− + = [2] (iv) Hence determine the probability that 3 randomly chosen positive integers can form the lengths of the sides of a triangle. [2]

2026 RI H3 Math Prelims Questions Use the information in the mathematical text to answer this Question. You should read the whole mathematical text before you start answering the questions. The Mandelbrot Set – A Fractal In the Complex Plane The Mandelbrot set is one of the most famous fractals in mathematics, named after the mathematician Benoit Mandelbrot who studied it in the 1980s. Despite its simple definition, it produces extraordinary complexity and has become an icon of chaos theory an d complex dynamics. For a given complex number c, define a sequence 0 1 2, , ,...z z z by 2 01 0, .nnz z z c += = + The number c belongs to the Mandelbrot set M if this sequence remains bounded as n→ . For example, take c = 0. The sequence becomes 0, 0, 0, …, which is clearly bounded as it is constant and thus 0 M . If we take c = −1, the sequence becomes 0, −1, 0, −1, … which is periodic with period 2 and thus bounded and thus 1 M− . What makes the Mandelbrot set fascinating is its infinite complexity. Zooming in on its boundary reveals smaller copies of the whole set, spirals and intricate patterns that repeat at ever -smaller scales. This property, known as self-similarity, is characteristic of fractals. To generate an image of the Mandelbrot set like Figure 1 below, a computer cannot iterate infinitely to test whether c is in M. Instead, the following property is used : if 2nz  for some n, then the sequence will not be bounded. Thus, the computer will compute the sequence until either: 2nz  , and thus conclude c is not in M, or it reaches a large number of iterations (say, 100) with 2nz  and it assumes that c is in M. Figure 1: The Mandelbrot Set

2026 RI H3 Math Prelims Questions 6 In this question you will investigate the Mandelbrot set, M. (a) (i) Show that c = i belongs to M. [2] (ii) Explain clearly why c = 1 does not belong to M. [2] (iii) With reference to Figure 1, state a symmetry you observe and prove it algebraically. [3] (b) Let x and y be real numbers. Using the triangle inequality or otherwise, show that x y x y−  − . [1] (c) Let c  be such that 2c  . Show that for all positive integers n, (i) .nzc  [3] (ii) ( )1 1.nnz c z+ − [1] (iii) Hence explain why c is not in the Mandelbrot set. [1] (d) Let cM . We will show in this part of the question that for all nonnegative integers n, 2nz  . (i) Show that the equation 2x x c=+ has only one positive real solution  satisfying 1 2. [3] Let nn z=− for all nonnegative integers n. (ii) Show that 1 2nn +  for all nonnegative integers n. [2] (iii) Deduce that 0n  for all nonnegative integers n and hence show that for all cM , 2nz  . [2]