additional mathematics - chapter 3

Success in SPM – Additional Mathematics – Form 4

1

Chapter 3: Quadratic Functions

1. Find the range of values of � for which the line � = � + 4 intersects the curve

� = " − 4 + 5 at two distinct points.


2

2. Given that � = ( " + 8 + 10 − (, find the range of values of ( for which � is always

positive.


3

3. Find the range of values of . of which 8 − 3 − " ≤ . for all real values of .


4

4. Find the range of values of � for which line � = � + 1 does not intersect the curve

�" = 8 .


5

5. Show that the equation 12 + 13 " + 122 + 33 + 12 + 23 = 0 has real roots for all real

values of 2.


6

6. Find the value of 2 for which the equation 11 − 223 " + 82 − 12 + 823 = 0 has two

equal roots.


7

7. Show that the line + � = 6 will intersect the curve " − 2 + 2�" = 3 at two distinct

points if 6" < 26 + 5.


8

8. Find the range of values of � for which the straight line � = � − 5 intersects the curve

� = " − 1 at two distinct points.


9

9. Find the value of 2 for which the line � = 6 is a tangent to the curve

� = " + 11 − 23 + 22.


10

10. Find the range of values of 6 for which the line + 2� = 6 meets the curve

1 + �3 + 8 = 0.


11

11. Find the range of the values of 2 for which equation " − 22 + 32 = 0 has real roots.


12

12. Calculate the value of . for which the line � = 2 + . is a tangent to the curve

� = 2 " − 4 + 5.


13

13. Given that the curve whose equation is � = 2 − 1 − 63" crosses the -axis at the points

11, 03 and 13, 03, find

i. The value of 2 and 6

ii. The maximum value of �


14

14. Find the range of values of for which 2 1 − 13 < 3 − .


15

15. Find the values of for which < " − 6.


16

16. Find the range of values of = for which 12 − 3=3 " + 14 − =3 + 2 = 0 has no real roots.


17

17. Find the value for . for which the line � = + . is a tangent to the curve

" + � + 2 = 0.


18

18. Calculate the smallest positive integer 2 for which the equation 3 " + 2 + 7 = 0 has

real roots.


19

19. Find the range of values of for which > " − 12.


20

20. Find the value of 2 for which the equation 12 + 23 " − 2122 + 13 + 42 = 0 has equal

roots.


21

21. Find the range of values of for which 12 − 33 < " − 2.

additional mathematics - chapter 3

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