The Minimalist Problem Set (Final version)

Page 1 of 50 H2 Mathematics The Minimalist Problem Set By Wee WS This minimalist problem set of 12 pure mathematics topics is designed to support every H2 Mathematics student in:  Developing critical thinking, problem-solving, and the practical application of acquired knowledge.  Realising the best results within a short, intensive period of focused practice before the A-level examination. A bonus topic permutations & combinations has also been included. 30 August 2023

Page 2 of 50 Topic 1a: Inequalities and equations Q1 (i) Show algebraically that 2 4 0x x   for all real values of x. (ii) Hence, without using a calculator, solve the inequality 1 2 2 x x x   . Answer(s): (ii) 2x or 0x Q2 (i) Sketch in a single diagram, the graphs of 1 1 xy x   and   33y x  . (ii) Hence solve the inequality   31 31 x xx    . (iii) Using the answer obtained in part (ii), deduce the range of values of x that satisfies   3e 1 e 3e 1 x x x    . Answer(s): (ii) 3.84, 1x x  (iii) , 0x x 

Page 3 of 50 Q3 Find the exact solutions of the equation 2 1 1 3 2x x    . Answer(s): 13, 17 32x  Q4 Solve the inequality 1sin 3 2x , where 2 2x    . Answer(s): 7 2 18x    or 5 18 18x  

Page 4 of 50 Topic 1b: Discriminants Q1 The curve C has equation 2 2 , 11 x x qy x x     , where q is a constant. Given that C has two stationary points, show that 3q . Q2 Use an algebraic method to show that the curve 2 2 1 1 x xy x    , 1x does not lie between 0 and 3, not including 0 and 3.

Page 5 of 50 Topic 2: System of linear equations Q1 Let f x be a cubic polynomial. It is given that the graph of fy x passes through the origin and has a stationary point at the point 6, 3  . The tangent at the point where 4x is parallel to the line 6 1y x  . Find f x . Answer(s):  3 21 1 3f 72 4 2x x x x   Q2 A collection of 41 shapes consisting of identical triangles, squares and pentagons have a total of 169 sides. The base and height of each triangle have the same length as the side of each square. Given that the total area of the triangles is twice the total area of the squares, find the number of triangles, squares and pentagons respectively. Answer(s): 16 triangles, 4 squares, 21 pentagons

Page 6 of 50 Q3 Find positive integers x, y, s, t such that the given chemical reaction is balanced, in terms of the number of molecules of the elements Na, P, Ca and F. 3 2 3 2Na P CaF NaF Ca Px y s t   Answer(s): 2, 3, 6, 1x y s t   

Page 7 of 50 Topic 3: Graphs Q1 The curve C has equation 3axy x b   where a, b are positive integers such that 3ab . (i) State the equations of the asymptotes of C, in terms of a and b. (ii) Sketch the graph of C, indicating all asymptotes and axial intercepts. Answer(s): (i) ,y a x b  (ii) 3 30, , , 0b a     