Download MATH1050 Additional practice problems

MATH1050
Additional practice problems - Section 5
1. Simplify the following expressions.
√
√
√
(a) ( 8 − 18)2
(b)
20
√
2− 5
2. Solve the following inequalities. Illustrate your solutions on the real number
line and state your answers in interval format.
2
(a) 2x − 1 < 5x + 7
(b) x2 ≥ 2x + 8
(c)
≤5
x−3
3. Solve the following equations.
(a) |3 − 4x| = 15
(b) 1 − y = |y − 2|
2x − 1 =3
(c) x+1 4. Solve the following inequalities. Illustrate your solutions on the real number
line and state your answers in interval
format.
2 − x
|x − 5|
≤3
(a) |3 − 2x| < 4
(b) (c) 3 <
x+1
x+2
5. Let u = 4i, w = 2 − 3i and z = 1 + 5i. Write the following in Cartesian form.
u
(a) 3w − z
(b) wz
(c) u
(d)
z
6. Let z = a + bi and w = c + di be complex numbers. Prove the following.
(a) |z| = |z|
(b) z + w = z + w
√
7. Let z = − 3 + i and let w = 2 − 2i.
(a) Write each of z and w in polar form.
(b) Use the polar forms of z and w to determine zw and
w
.
z
(c) Use De Moivre’s Theorem to determine z 5 and w6 (in polar form).
π
8. Let z = 6e− 3 i . Write z in Cartesian form.
9. (a) Determine the cube roots of 1 + i.
(b) Suppose w6 = −64i. Determine the six values of w.
10. Determine which of (z − 2), (z + 3) and (z + 4) is a factor of 2z 3 + 9z 2 − 6z − 40.
Hence write 2z 3 + 9z 2 − 6z − 40 as a product of linear factors.
11. Write each of the following polynomials P (z) as a product of linear factors, and
hence determine all solutions of P (z) = 0 (over C).
(a) P (z) = z 2 − 4z + 8
(b) P (z) = 16z 4 − 9
(c) P (z) = 4z 4 − 16z 2 − 9
(d) P (z) = z 4 − 2z 2 + 1
12. One solution of the equation 4z 3 − 41z 2 + 174z − 41 = 0. is z = 5 − 4i. Find
all the solutions of this equation (over C).