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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