Download Algebra

Achievement Standard 90638
Manipulate real and complex numbers, and solve equations 3.4
Achievement Questions
2
1. Evaluate 64 3
2. Simplify a)
64a
c3m  c 4
b)
c 2 m 5
64
3. Simplify as much as possible
a) 5  4m  4  2 8  m
4. Factorise completely
a) 12a  48a3
c) 5 x 2  29 x  6
5. Simplify
a2 1
a 2  2a  1
6. Simplify
3x  1 2 x  1

4
3
b)  2 x 1   2 x  1
2
2
c) 5  a   a 2  a 1
b) x 2  5 x  24
d) 4 x  8  xy  2 y
1
1

1 x 1 x
7. Simplify
8. Solve the quadratic equations:
a) x 2  x  30  0
b) 5 x 2  3 x  0
c) 4 x 2  49
d) 3 x 2  5 x  3  0
e)
 x  4
2
 3x  1
b  b2  4ac
9. Use the quadratic formula x 
to solve the equation
2a
5x2  2 x  1  0
10. Show that the equation x 2  x  5  0 has no real solutions.
11. The sum of the squares of two numbers is 369. If one number is three more than the
other, find the two numbers.
12. A right-angled triangle has sides  2 x 1 ,  x  6 and  x 1 . Find the value of x.
 2x 1
is the hypotenuse.
Algebra 2009
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 x  a  b
2
in the form a  x  b   c
13. By completing the square, write x 2  8 x  12 in the form
14. By completing the square, write 2 x 2  12 x  13
 x  3
15. Find the remainder when 2 x 3  x  8 is divided by
16. Show that
 x  4
is a factor of x 4  8 x 2  24 x  32
17. Use long division to divide x3  2 x2  3x  6 by
 x  2
18. Solve the cubic equations
a) x 3  27
b) 2 x3  16  0
d) x3  x 2  0
e) x3  2 x 2  7 x  4  0
19. Write as a single surd
32  2 8  6 2
20. Write as a single surd
25 p 2 q  3 p q
21. Expand and simplify
a) 2 3 3  2
b)

2


6 2 3

6 2 3
c) 3 x 3  12 x  0
f) x3  7 x  6  0

22. Rationalise the denominator of
9 2
3 2
a)
b)
2 3
3 2
23. Write
6 1
in the form a  b 6
6 2
24. Write in index form log5 125  3
25. Write in log form 34  81
26. Evaluate log 4 64
27. Find x if log 2 x  7
28. Write as a single logarithm 3log 2  2 log 3  2 log 6
29. Write as a single logarithm 2  log10 3
30. Simplify as much as possible 3log2 4  log2 16
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1
31. Simplify log    log x
x
32. Solve ln  2 x   ln  6  4
33. Solve log  x3  x   log x  log 2 x
34. Solve 53 x1  200
35. Solve e2 x 1  250
36. Solve:
a) ln  3x  4  2.6
b) log10  3x   1.4
d) log5 30 x  log5  x  3  2
c) log 2  4 x   7
e) log6  3x  5  1.1
37. Two complex numbers are: u  5  3i, v  4  2i .
Find:
a) u  v
b) 5v
c) u
f) uu
g) v 2
h) u  3v
d) uv
e)
u
v
38. Simplify i 7
2
in the form a  bi
3i
40. Show the following complex numbers on an Argand diagram.
a) 4  2i
b) 3  i
c) 3i
39. Write
41. If two complex numbers are given by u  3  i, v  2  4i , show the sum z  u  v
on an Argand diagram.
42. For the complex number z  12  5i, find z and arg  z  .
43. Convert w  3  2i to polar form.
44. Show 3cis   on an Argand diagram.
 
45. Show 2cis   on an Argand diagram
3
46. For the two complex numbers u  12cis 80 , v  4cis  25 find:
a) uv
Algebra 2009
b)
u
v
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 
47. Convert 20cis   to rectangular form.
6

  
48. a) Use de Moivre’s Theorem to simplify  3cis   
 2 

6


b) Write  3cis  in the form a  bi
3

5
49. A complex number is z  3  2i
a) Convert z to polar form
4
b) Use this result and de Moivre’s Theorem to find  3  2i  in the form rcis
50. Show the solutions of z 3  1 on an Argand diagram.
51. Simplify

5 i 3

5 i 3



52. Write  2cis  as a complex number in rectangular form, a  bi
3

4
 
 3 
53. If u  5cis   and v  3cis   , find uv, leaving the answer in the form rcis
4
 2 
Merit questions
1. Solve for x: 3x  5  x  1
2. Solve the equation
3. Solve
x
2x 1  x  4  1
2
1
x
4. Show that the function f ( x)  x3  x 2  2 x  4 has no turning points.
5. If 2i is a root of 2 x3  ax 2  8x  12  0 , what is the value of a.
6. Write down and simplify the cubic equation which has solutions 2,
2i,  2i .
7. The cubic equation p( x)  x3  3x 2  4 x  12  0 has one real solution.
a) Calculate p(1), p(2), p(3) to find the real solution.
b) Use long division to find the other solutions.
8. Find all the solutions of x3  5 x 2  17 x  13  0
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9. Find the square roots of 12  5i by solving z 2  12  5i
10. Find the fourth roots of 16. Show these solutions on an Argand diagram.
11. Show that if w  1  i, then w4  4
12. Solve the equation z 4  6  8i by converting to polar form and using de Moivre’s
Theorem.
13. Find all the solutions of the equation x3  x 2  3x  5  0 given that one solution
is 1  2i .
14. One root of the equation z 3  7 z 2  16 z  k  0 is 3  i .
Find the value of k and the other two solutions.
15. Solve the equation 40e6 x  200e 2 x
16. Solve x 2  2 x  6  0 , giving the solutions in the form x  a  b c i
17. Solve the following equation for x in terms of p.
 4 
ln  x  2   ln 
  ln 2 x
 p 1 
18. Solve the following equation for x in terms of k.
xk  xk  4
19. Solve z 3  27 . Leave the answers in polar form.
20. Solve z 3  27i . Answer in polar form.
21. Solve for x, giving the answer in terms of a:
a 2 x 2  5ax  4  0
22. Solve for x, giving the answer in terms of a:
3 xa 2 xa  0
23. Solve for x, giving the answer in terms of a:
a x 1  2 x  1  0
24. Solve for x, giving the answer in terms of a:
43 xa  2x a
25. Solve for x, giving the answer in terms of a:
32 x 1  2 x  a
Excellence questions
x
1. Simplify as much as possible
1
1
1
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x3   x  2  x  2 x
2. Show that
3. Show that
1
 cos  i sin  
2
 cos 2  i sin 2
4. Two complex numbers are given by u  cos  i sin  , and v  sin   i cos 
Find the product uv in its simplest form.
5. Show that if z  cis , then z n  z  n  2cos n .
Complete working must be shown.
6. Find the complex number z  a  bi which satisfies the equation
z  2  3i   11 16i  i
7. Prove that
cos  i sin 
2
  cos   i sin  
cos  i sin 
8. If z  2  i , show that z 3  5 z 2  9 z  5  0
9. A train moving at r km/hr can cover a given distance in h hours. By how many km/hr
must its speed be increased in order to cover the same distance in one hour less time?
10. A pilot flies a distance of 600km. The pilot could fly the same distance in 30 minutes
less time by increasing the plane’s average speed by 40 km/hr. Find the actual average
speed. You must write an equation and solve it.
11. The diagonal of a rectangle is 85cm. If the short side is increased by 11cm and the
long side is decreased by 7cm, the diagonal remains the same length. Find the
dimensions of the original rectangle.
12. The amount of money in a fund between years 4 and 20 follows the formula
y  60log10  x  a   b , where y is the amount in the fund in thousands of dollars, x is
years and a, b are constants.
After 4 years, the fund holds $20000 and after 13 years the fund holds $80000
Find the values of a and b.
13. If w is a complex root of 1, show that;
a) ww  1
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b) w2  w
c)
1 1
  1
w w
d) w  w  1  1
14. Expand and simplify  3x  1
5
 2 x  3
10
15. Write down the first three terms in the expansion of
16. Find the term in x 6 in the expansion of  3x 2  1
8
12
2

17. Find the constant term in the expansion of  x 2  
x

Sketch the locus for each of the following equations on an Argand diagram.
18. z  5
19. z  4  2
20. z  i  3
21. z  3  4i  5
22. z  z  i
23. z  1  z  i
24. Show the region which satisfies the inequation z  2  4
25. Find all the solutions of
Algebra 2009
z
2
 1  4 , where z is a complex number.
2
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Answers: Achievement
1. 16
2. a) 8a 32
b) c m1
3. a) 22m  36 b) 8x c) a 3  6a 2  4a  5
4. 12a 1  2a 1  2a  b)  x  8 x  3 c)  5x 1 x  6
d)  x  2 4  y 
a 1
x7
2x
6.
7.
a 1
1  x2
12
8. a) 6, -5
b) 0, -0.6
c) 3.5, -3.5
d) 2.14, -0.468 e) 9.405, 1.595
2
9. 0.29, -0.69
10. b  4ac  19  0 so no solutions
5.
11. 15, 12 or -12, -15
14. 2  x  3  5
2
17. x 2  3
18. a) 3
19. 2 2
13.  x  4   4
2
12. 9
16. f (4)  0 , so is a factor
15. –59
b) 2 c) 0, 2, -2
d) 0, 0, -1
20. 2 p q
21. a) 6  4 3
3 6
b) 3 6  6
2
25. log3 81  4
26. 3
22. a)
23. 4 
3
6
2
27. 128
24. 53 = 125
29. log10 300
30. 2
31. 0
33. 1, 1
34. 0.764
35. 3.26
36. a) 3.15
b) 8.37
c) 32 d) 15
5– 1
1
2
3
4
1– 1
2
3
4
5
37. a) 1  5i
b) 20 10i
f) 34
38.  i
g) 12  16i
39. 0.6  0.2i
40. a)
e) 4.059
e)
h) 7  3i
Im
2
1
4– 2
1
2
3
1
1– 5
2
3
4
5
6
1
2
3
4
2
14  22i
20
Im
40. b)
3
– 5– 4– 3– 2– 1
– 1
1
– 1
– 1
28. log 2
32. 4.55
d) 26  2i
5
4
5– 1
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
1– 3
2
1
2
c) 5  3i
1– 5
2
1
2
3
4
f) –1, -2, 3
e) 1, 1, -4
b) –6
1
2
3
4
1
2 Real
– 2
5 Real
– 3
41.
Im
40. c)
5
4
3
4
3
2
1
Im
2
1
– 5– 4– 3– 2– 1
– 1
1 2 3 4 5 Real
42. 13, 157.4º
Algebra 2009
– 5– 4– 3– 2–– 11
– 2
1 2 3 4 5 6 Real
43. 13cis(33.7)
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Im
4
3
2
(44)
1
1
2
(45)
1
– 4– 3– 2– 1
– 1
1
2
3
4 Real
46. a) 48cis(105º)
b) 3cis(55º)
47. 17.32 + 10i
48. a) 729cis(3  ) b) 121.5 - 210.4i 49. a) 13cis(146.3) b) 169cis  134.8
50.
Im
1
– 2
1 Real
– 1
– 1
 7 
53. 15cis 

 4 
52. 8  8 3i
51. 8
Answers: Merit
3
1
2
1
2
3
1. 2, 3
2. 12
3. 4
2
2
4. f ( x)  3x  2 x  2 b  4ac  20 < 0, so no solutions, so no turning points.
5. 3
6. x3  2 x 2  2 x  4
7. a) –10, -8, 0 so 3 is real solution
b) 2i, -2i
8. 1,  2  3i,  2  3i
9. 3.61cis 11.3 , 3.61cis  168.7
10. –2, 2, 2i, -2i
Im
3
2
1
– 3 – 2 – 1
– 1
1
2
3 Real
– 2
– 3
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11. w2  2i, w4  w2  w2  2i  2i  4i 2  4
12. 1.78cis  103.3 , 1.78cis  13.3 , 1.78cis  76.7 , 1.78cis 166.7
13. 1  2i, 1  2i,  1
14. k  10, 3 + i, 1
4
1 p
17. x 
16. 1  5 i
 
  
19. 3, 3cis   , 3cis 

3
 3 
4 1
13a
21. ,
22.
5
a a
Answers: Excellence
15. 0.402
64  k 2
16
 
 5
20. 3cis   , 3cis 
6
 6
a2  4
23. 2
a 4
18. x 
3a
24.
5

  
 , 3cis 


 2 
25.
ln
 
2a
3
ln  4.5 
1. x  y
2.
LHS  x
3
x
3
2
 x x
2
x
3
2
1
2
2 x
2 x
2 x
3. Multiply out bottom line and simplify. Multiply top and bottom by conjugate and
simplify.
4. i
5.
1
n
z n  z  n   cos   i sin   
n
 cos  i sin  
  cos   i sin   
n
 cos n  i sin n 
1

 cos n  i sin n   cos n  i sin n 
  cos n  i sin n  
 cos n  i sin n 
 2 cos n
1
6. a  5, b  2
7. multiply LHS by complex conjugate of the denominator.
8. substitute and simplify.
2  i  52  i  92  i  5
  2  11i   5  3  4i   18  9i  5
3
2
 2  11i  15  20i  18  9i  5
0
r
h 1
10. 200 km/hr.
9.
Algebra 2009
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11. 40cm, 75cm
12. a  3, b  20
4–
2
– 4
2
4
1
3
4
5
6
2
1
3
5
6
2
1– 6
3
5
2
4
6
8
10
1
2
3
4
5
1
3
 2 
13. Write w as cis 

i and simplify
 or
2
2
 3 
14. 243x5  405 x 4  270 x3  90 x 2  15 x  1
15. 1024 x10  15360 x9  103680 x8
16. 1512x 6
17. 126720
y
18.
–0 2
4
6
– 10
2
4
6
8
y
6
5
4
3
2
1
19.
4
2
– 6
– 5
– 4
– 3
– –2
– 11 1 2 3 4 5 6 x
– 2
– 3
– 4
– 5
– 6
2
4
6
– 2
– 4
y
y
20.
10
0.5
1
2
4
6
8
– 10
0.5
1
2
4
6
8
2– 10
4
6
8
10
2
4
6
8
6
10
21.
4
8
2
6
– 6– 4– 2
– 2
2
4
6 x
4
2
– 4
– 6
– 10– 8– 6– 4– 2
– 2
y
22.
10 x
8
4 x
y
1
23.
0.5
6– 6
2
4
2
4
2– 10
4
2
4
6
8
– 10– 8– 6– 4– 2
2
2 4 6 8 10 x
– 0.5
– 1
10
8
6
4
2
– 10
– 8– 6– 4–– 22
– 4
– 6
– 8
– 10
2 4 6 8 10 x
y
24.
6
4
2
– 10 – 8 – 6 – 4 – 2
– 2
25.
2
3,  3, i,  i
4 x
– 4
– 6
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