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Brock University
Physics 1P21/1P91
Mathematics Review Exercises
(answers at end)
1. Express each number in scientific notation without using a calculator.
(a) 437.1
(b) 563, 000
(c) 0.0023
(d) 0.0000468
2. Multiply the numbers without using a calculator.
(a) (5 × 103 ) × (7 × 106 )
(b) (3.2 × 104 ) × (1.5 × 102 )
(c) (4 × 10−2 ) × (9 × 10−4 )
(d) (2.1 × 104 ) × (8 × 10−5 )
3. Divide the numbers without using a calculator.
(a) (7 × 103 ) ÷ (5 × 106 )
(b) (6 × 106 ) ÷ (3 × 104 )
(c) (4 × 10−4 ) ÷ (8 × 10−6 )
(d) (2.5 × 104 ) ÷ (10−2 )
4. Multiply the fractions without using a calculator.
2 4
×
3 5
3 ∆t
(b) ×
4
2v
a c
(c) ×
b d
(a)
5. Divide the fractions without using a calculator.
2 4
÷
3 5
3 ∆t
(b) ÷
4
2v
a c
(c) ÷
b d
a
(d) cb
d
(a)
6. Add or subtract the fractions without using a calculator.
2 4
+
3 5
3 5
(b) −
4 6
a c
(c) +
b d
(a)
7. Simplify each expression by expanding.
(a) (x + 2)(x − 3)
(b) (x − vt)(x + vt)
(c) 2(x1 + x2 )(m + 3)
8. Solve each equation.
(a) 2x + 5 = −7
(b) 3 − 2t = 4t
(c) 2(2t + 3) = 3(t − 4) + 1
9. Solve each equation.
4
2t
=
3
9
7
2
(b)
=
4t
5
4
5t
(c)
=
3t
7
(a)
10. Solve each quadratic equation.
(a) (t − 2)(t + 3) = 0
(b) t2 − 5t + 6 = 0
(c) 2t2 + t − 3 = t2 + 5t − 6
(d) 2t2 − 5t + 3 = 0
(e) 3t2 − 7t + 5 = 0
(f) 4t2 − 28t + 49 = 0
11. Solve each equation to obtain a formula for t.
(a) x = vt
b
(b) at =
t
a t
(c) − = 0
t
b
1
(d) xf = xi + vi t + at2
2
12. Solve each system of equations.
(a) 5x + 3y = 1, 4x − 2y = 14
(b) 2x + 3y = 7, −4x − 6y = 1
(c) 3x − 2y = 5, −6x + 4y = −10
13. Sketch a graph to represent the solution of each system of equations in the previous
question.
14. Determine the area and perimeter of a rectangle that has dimensions 4 m and 3 m.
15. Determine the area and perimeter of a triangle that has sides of lengths 3 cm, 4 cm,
and 5 cm.
16. Determine the area and circumference of a circle that has radius 5.3 mm.
17. Plot each point on a rectangular coördinate system: (1, 3), (2, −1), (−3, 2), and
(−1, −2).
18. Evaluate each quantity.
√
(a) 32 + 42
√
√
(b) 2 3 + 3 2
√
√ √
√ (c)
5− 3
5+ 3
19. A right triangle has sides with lengths 2.7 cm and 4.1 cm. Determine the length of the
hypotenuse and the measures of the angles.
20. A right triangle has a side with length 5.6 km and a hypotenuse with length 10.2 km.
Determine the length of the other side of the triangle and the measures of the angles.
21. A right triangle has a hypotenuse with length 3.7 mm and angles with measures 40◦
and 50◦ . Determine the lengths of the other two sides of the triangle.
22. A right triangle has a hypotenuse with length 14 m and angles with measures 25◦ and
65◦ . Determine the lengths of the other two sides of the triangle.
23. A triangle has sides of lengths 5.00 cm, 6.00 cm, and 7.00 cm. Determine the measures
of the angles of the triangle, in degrees correct to one decimal place.
24. When a surveyor on level ground sights the top of a tall building, his telescope makes
an angle of 51◦ relative to the horizontal. When the surveyor moves 20.0 m further
away, the angle is 42◦ . Determine the height of the building.
25. A triangle has sides of lengths 2.00 m, 3.00 m, and x. The corresponding sides of a
similar triangle have lengths 4.20 m, y, and 9.03 m. Determine the values of x and y.
Table of Greek letters
There are very many concepts in physics, and not enough letters in the English language
to comfortably represent them. As a result, often we use Greek letters to represent physics
concepts. It’s helpful if you get to know the Greek letters; here’s a table of them (both upper
case and lower case) to help you do this.
Alpha
A α
Beta
B β
Gamma Γ γ
Delta
∆ δ
Epsilon E ε
Zeta
Z ζ
Eta
H η
Theta
Θ θ
Iota
I ι
Kappa
K κ
Lambda Λ λ
Mu
M µ
Nu
N ν
Xi
Ξ ξ
Omicron O o
Pi
Π π
Rho
P ρ
Sigma
Σ σ
Tau
T τ
Upsilon Y υ
Phi
Φ φ
Chi
X χ
Psi
Ψ ψ
Omega
Ω ω
Answers.
1. (a) 4.371 × 102 (b) 5.63 × 105 (c) 2.3 × 10−3 (d) 4.68 × 10−5
2. (a) 3.5 × 1010 (b) 4.8 × 106 (c) 3.6 × 10−5 (d) 1.68
3. (a) 1.4 × 10−3 (b) 200 (c) 50 (d) 2.5 × 106
4. (a)
8
15
5. (a)
5
6
6. (a)
22
15
(b)
(b)
3∆t
8v
3v
2∆t
(b) −
(c)
ad
bc
(c)
1
12
ac
bd
(c)
(d)
ad
bc
ad + bc
bd
7. (a) x2 − x − 6 (b) x2 − v 2 t2 (c) 2mx1 + 2mx2 + 6x1 + 6x2
8. (a) −6 (b)
2
9. (a)
3
1
2
35
(b)
8
(c) −17
r
(c)
28
15
10. (a) −3 and 2 (b) 2 and 3 (c) 1 and 3 (d) 1 and
x
11. (a) t =
v
r
(b) t = ±
b
a
3
2
(e) no solutions (f)
√
−vi ±
(c) t = ± ab (d) t =
7
2
p
vi2 + 2a(xf − xi )
a
12. (a) x = 2 and y = −3 (b) no solution (c) there are an infinite number of solutions,
because each equation represents the same line
13. (a) The two lines intersect at a single point, which represents the solution to the system
of equations.
y
4
3
4x − 2y = 14
2
1
−4
−3
−2
−1
1
2
3
4
x
−1
−2
(2, −3)
−3
−4
5x + 3y = 1
(b) Parallel lines means no solution.
y
4
3
2
1
−4
−3
−2
−1
1
−1
2
3
4
x
2x + 3y = 7
−2
−3
−4x − 6y = 1
−4
(c) The graph of each equation is the same line, so each point on the line is a solution
of the system of two equations; there are an infinite number of solutions.
y
3x − 2y = 5 and −6x + 4y = −10
4
3
2
1
−4
−3
−2
−1
1
2
3
4
x
−1
−2
−3
−4
14. area = 12 m2 and perimeter = 14 m
15. area = 6 cm2 and perimeter = 12 cm
16. area = 88 mm2 and circumference = 33 mm
17. Here is the plot:
y
4
(1, 3)
3
(−3, 2)
2
1
−4
−3
−2
−1
1
−1
(−1, −2)
−2
−3
−4
18. (a) 5 (b) 7.7 (c) 2
2
3
(2, −1)
4
x
19. 4.9 cm, 33.4◦ , and 56.6◦
20. 8.5 km, 33.3◦ , and 56.7◦
21. 2.8 mm and 2.4 mm
22. 12.7 m and 5.9 m
23. 44.4◦ , 57.1◦ , and 78.5◦
24. 66.5 m
25. x = 4.3 m and y = 6.3 m