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           Q2 The curve C has equation   2 21 19 x y   . (i) Sketch the curve C, showing clearly the equations of asymptotes, axial intercepts and coordinates of turning points, if any. (ii) Given that k is a positive constant and C intersects the curve with equation 2 2 2 1xy k  at exactly two distinct points, state the range of values of k. Answer(s): (ii) 2 4k 

Page 8 of 50 Q3 The curve C has equation 2 2 3 2 x xy x    . (i) Find the equations of the asymptotes of C. (ii) Draw a sketch of C, which should include the asymptotes, and state the coordinates of the points of intersection of C with the x-axis. (iii) On the same diagram draw a sketch of the curve   2 4 4y x  , showing clearly the asymptotes. (iv) Hence show that the equation 4 3 26 3 60 56 0x x x x     has exactly two real roots. Answer(s): (i) 2, 4x y x   (ii) 1, 0 , 3, 0 Q4 A curve C has parametric equations 2 2 3,x t t y t t    where 2 2t   . (i) Sketch C, indicating the end-points and the behaviour near the origin. (ii) Explain why the curve does not exist when 1 4x . (iii) Find the gradient of the tangent to C at (2, 2) and the angle that this tangent makes with respect to the horizontal. Answer(s): (iii) 1.67; 59.0

Page 9 of 50 Q5 Determine, with working, whether the curves 6 25 8 12 xy x   and 2 2 125 16x y  are tangent to each other at 2.5x . Answer(s): Yes, the curves touch each other at 2.5, 1.25  and both curves give the same gradient of 2 at 2.5, 1.25  .

Page 10 of 50 Topic 4: Transformations Q1 A curve with equation 2 24 2 24 0x y y    is transformed by a translation of 6 units in the positive y-direction, followed by a stretch parallel to the x-axis with scale factor 2. Find the equation of the new curve in the form     2 2 2x h y k r    . Answer(s):   22 7 25x y   Q2 The diagram below shows the graph of  fy x . Sketch, separately, the graphs of (i)  fy x , (ii)  fy x , (iii)  3f 2 1y x  , (iv)   1 fy x , (v)  fy ' x . For part (iii), describe a sequence of transformations on  fy x to obtain  3f 2 1y x  .