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Investigation

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TUTORIAL 1 NUMBER AND SET

1.1 Exponential and Logarithm Equations

1. Evaluate each expression (a) [pic] (b) [pic] (c) [pic]

2. Solve the following exponential equations.

(a) [pic] (b) [pic]

(c) [pic]

3. Solve each of the following equations.

(a) [pic] (b) [pic]

(c) [pic]

4. Solve the logarithmic equations for [pic]

(a) [pic] (b) [pic]

(c) [pic]

5. Solve the following equations: (i) [pic] (ii) 32x+ 2 – 28 (3x) + 3 = 0 (iii) [pic] (iv) [pic] (v) [pic]

6. Solve the following equations: (i) [pic] (ii) log2 x + 3logx 2 = 4 (iii) [pic] (iv) [pic] (v) [pic]

7. Solve the following simultaneous equations: (i) [pic] [pic]

(ii) [pic] [pic]

(iii) [pic] [pic]

8. Find the smallest positive integer n such that (i) [pic] (ii) [pic] (iii) [pic]

9. (a) If[pic], prove that[pic]. (b) Given [pic]solve for x.

1.2 Complex Numbers

10. Evaluate the expression and write the result in the form [pic]

(a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

(g) [pic] (h) [pic]

11. Given that [pic], find the values of p and q.

12. Find [pic] in the form [pic], where [pic]

13. If [pic] and [pic], find

(a) [pic] [pic] (b) [pic] [pic]

14. Solve the following equation for the values of x and y:

[pic]

15. Express the following complex numbers in the form a + bi. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic] (vi) [pic] (vii) [pic] (viii) [pic] (ix) [pic] (x) [pic] (xi) [pic] (xii) [pic]

16. Express z = [pic] in the form x + yi, where x and y are real numbers. Draw an Argand diagram to show the location of z. Hence, find the modulus and argument of z.

17. Find the value of x and y, [pic], for the following equations. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic]
18. Given that[pic], find[pic] and[pic] where[pic] is the conjugate of z. Hence, find the modulus and argument of [pic] and[pic].

19. (i) Find the square root of [pic] in the form x + iy, where [pic]. (ii) If [pic]find z.

1.3 Set

20. Given that A = {1, 2, 3, 4, 5}, and B = {1, 3, 6, 9, 10}. Find [pic], [pic], [pic] and [pic].

21. Given [pic] = {1, 2, 3, 4, 5, 6, 7, 8}, A = {3, 4, 5}, B = {2, 4, 6, 8} and C = {1, 3, 5, 7}. Write down each of the following sets:

(a) [pic] (b) [pic]

(c) [pic] (d) [pic]

22. Given [pic] = [−5, 10), A = (1, 8], B = [−3, 5), C = [−2, 7). Find each of the following set.

(a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic]

23. Using the algebraic laws of sets, show that for any set A and set B,

[pic]

24. Given that [pic]and [pic] Find [pic][pic][pic] and [pic]

25. Given [pic][pic][pic]and[pic]. Write down each of the following sets: (i) [pic], (ii) [pic], (iii) [pic], (iv) [pic], (v) [pic], (vi) [pic].

26. Given that [pic][pic] [pic]and[pic] Find (i) [pic], (ii) [pic], (iii) [pic].
27. Given [pic][pic][pic]and [pic] Find each of the following set. (i) [pic], (ii) [pic], (iii) [pic], (iv) [pic], (v) [pic], (vi) [pic].

ANSWERS

1. (a) -9 (b) [pic] (c) 4
2. (a) [pic] (b) 0 or 2 (c) -2 or 1
3. (a) 1.3979 (b) 2.1492 (c) -2.9469
4. (a) [pic] (b) [pic] (c) 3 or [pic]
5. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

6. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

7. (i) [pic] (ii) [pic] (iii) [pic]

8. (i) [pic] (ii) [pic] (iii) [pic]

9. (b) [pic]

10. (a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic] (f) [pic] (g) [pic] (h) [pic]
11. [pic], [pic]
12. [pic]
13. (a) [pic], [pic] (b) [pic], [pic]
14. [pic]
15. (i) 10 (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic] (vi) 100 (vii) [pic] (viii) [pic] (ix) [pic] (x) [pic] (xi) [pic] (xii) [pic]

16. [pic]

17. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic]

18. [pic] [pic]; [pic]

19. (i) [pic] (ii) [pic]

20. [pic] = {1, 2, 3, 4, 5, 6, 9, 10}, [pic] = {1, 3}, [pic] = {2, 4, 5}, [pic] = {6, 9, 10}

21. (a) {1, 3, 5, 7}, (b) {1, 7} (c) {1, 2, 3, 5, 6, 7, 8} (d) {1, 3, 4, 5, 7}

22. (a) [-3,8] (b) [5,8] (c) [5,7) (d) [-2,1] (e) [5,7)

24. [pic] [pic]

25. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic] (vi) [pic]

26. (i) [pic] (ii) [pic] (iii) [pic]

27. (i) [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic] (vi) [pic]

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