SPM · Matematik (Mathematics) · Chapter 1

Quadratic Functions and Equations in One Variable

Quadratic expressions, graphs, roots and real-life problems (Form 4, Chapter 1)

100 marks · suggested time 60 minTime 0:000 / 43 answered

Worked example

Solve x² − 5x + 6 = 0. Factorise: (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Enter numbers or fractions (units are optional), and write coordinates as (x, y). Give rounded answers to the accuracy the question asks for. Written answers are marked by you, against the mark scheme, after you submit.

Section A · Basics 20 marks

1. Which of these is a quadratic expression?[2]

A. 3x + 2B. 2x² − 5x + 1C. x³ + 1D. 1/x + 4

2. f(x) = x² − 3x + 2. Find f(4).[2]

3. Find the positive root of x² − 9 = 0.[2]

4. The axis of symmetry of f(x) = x² − 6x + 5 is x = k. Find k.[2]

5. Find the minimum value of f(x) = (x − 2)² + 5.[2]

6. The graph of f(x) = −2x² + 3 has:[2]

A. a minimum pointB. a maximum pointC. no turning pointD. two turning points

7. Find the y-intercept of f(x) = 2x² − 7x − 4.[2]

8. Expand (x + 3)(x − 5). Find the coefficient of x.[2]

9. Find the smaller root of x² − 5x + 6 = 0.[2]

10. Find the larger root of 2x² − 7x + 3 = 0.[2]

Section B · Graphs and equations 35 marks

f(x) = x² − 4x − 12.

11. Find the smaller root of f(x) = 0.[2]

12. Find the larger root of f(x) = 0.[2]

13. The axis of symmetry is x = k. Find k.[2]

14. Find the minimum value of f(x).[2]

15. Find the y-intercept.[1]

Solve 3x² = 5x + 2.

16. Find the larger root.[2]

17. Find the smaller root, as a fraction.[2]

A rectangle has length (x + 4) cm, width x cm and area 45 cm².

18. Find x.[3]

19. Find the perimeter of the rectangle.[2]

20. Explain why x = −9 is rejected.[2]

The graph of f(x) = x² + bx + c passes through (1, 0) and (3, 0).

21. Find b.[2]

22. Find c.[2]

23. The product of two consecutive positive integers is 132. Find the smaller integer.[3]

A ball is thrown upwards. Its height is h = 20t − 5t² metres after t seconds.

24. Find the height after 1 second.[2]

25. After how many seconds does it return to the ground?[2]

26. Find its maximum height.[2]

27. Describe how the graph of f(x) = ax² changes when a changes from 1 to 3, and then to −1.[2]

Section C · Problem solving 45 marks

A rectangular garden 20 m by 15 m has a path of uniform width x m around the outside. The total area of garden and path is 500 m².

28. Show that 2x² + 35x − 100 = 0.[4]

29. Find x.[3]

30. Find the area of the path.[2]

A company’s profit is P(x) = −x² + 40x − 300 (RM thousand), where x is hundreds of units sold.

31. Find the smaller break-even value of x.[2]

32. Find the larger break-even value of x.[2]

33. Find x for maximum profit.[2]

34. Find the maximum profit, in RM thousand.[2]

35. Explain what P(x) < 0 means when x > 30, and suggest one reason it could happen.[3]

A right-angled triangle has sides x cm, (x + 7) cm and hypotenuse (x + 8) cm.

36. Find x.[4]

37. Find the perimeter.[2]

38. The sum of a positive number and its reciprocal is 10/3. Find the larger possible number.[3]

f(x) = −x² + 2x + 8.

39. Find the larger root of f(x) = 0.[2]

40. Find the maximum value of f(x).[2]

41. Describe the sketch of y = f(x): shape, intercepts and turning point.[3]

42. Aiman solves x² = 4x by dividing both sides by x and gets only x = 4. Explain his mistake and give the full solution.[4]

43. Form a quadratic equation in general form whose roots are 3 and −5. Show your working.[5]

43 questions still blank. Blank answers score 0.