Mathematics Review Exercises. (answers at end)
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1 Brock University Physics 1P21/1P91 Mathematics Review Exercises (answers at end) Work each exercise without using a calculator. 1. Express each number in scientific notation. (a) (b) 563, 000 (c) Multiply the numbers. (a) ( ) ( ) (b) ( ) ( ) (c) ( ) ( ) ( ) ( ) 3. Divide the numbers. (a) ( ) ( ) (b) ( ) ( ) (c) ( ) ( ) ( ) (10 2 ) 4. Determine the result. (a) (b) (c) Decompose each number as a product of prime factors. (a) 48 (b) 180 (c) Fill in the blanks to determine equivalent fractions.
2 (a) 5 3 = 12 = 63 = 108 (b) 2 7 = 35 = 8 = 30 (c) 6 9 = 3 = 60 = = 2 = 143 = Express each fraction in lowest terms. 48 (a) 180 (b) (c) Multiply the fractions. (a) (b) 3 4 t 2v (c) a b c d 9. Divide the fractions. (a) (b) 3 4 t 2v (c) a b c d ( a ) b ( c d) 10. Add or subtract the fractions. (a) (b) (c) a b + c d a b + c d e f
3 11. Determine the unknown quantity. (a) If a book costs $17 and you are offered a 20% discount, what is the dollar value of the discount? (b) If a 30% discount amounts to a $24 price reduction, what is the original price? (c) If a book has an original price of $20, and the sale price is $17, how much of a discount is offered (in percent)? 12. Simplify each expression by expanding. (a) (x + 2)(x 3) (b) (x vt)(x + vt) (c) 2(x 1 + x 2 )(m + 3) 13. Solve each equation. (a) 2x + 5 = 7 (b) 3 2t = 4t (c) 2(2t + 3) = 3(t 4) Solve each equation. (a) 2t 3 = 4 9 (b) 7 4t = 2 5 (c) 4 3t = 5t Solve each quadratic equation. (a) (t 2)(t + 3) = 0 (b) t 2 5t + 6 = 0 (c) 2t 2 + t 3 = t 2 + 5t 6 2t 2 5t + 3 = 0 (e) 3t 2 7t + 5 = 0 (f) 4t 2 28t + 49 = Solve each equation to obtain a formula for t. (a) x = vt (b) at = b t (c) a t t b = 0 x f = x i + v i t at2
4 17. Consider the equation AB = C + D E. (a) Solve the equation for A. (b) Solve the equation for B. (c) Solve the equation for C. Solve the equation for D. (e) Solve the equation for E. 18. Determine the area and perimeter of a rectangle that has dimensions 4 m and 3 m. 19. Determine the area and perimeter of a triangle that has sides of lengths 3 cm, 4 cm, and 5 cm. 20. Determine the area and circumference of a circle that has diameter 5 mm. 21. Plot each point on a rectangular coördinate system: (1, 3), (2, 1), ( 3, 2), and ( 1, 2). 22. Evaluate each quantity. (a) (b) (c) 72 ( 5 3 ) ( ) 23. Evaluate each quantity. (a) 4 3/2 (b) 2 7/2 (c) ( 4 5/4) ( 8 3/2) 6 5/2 18 3/2 24. 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. 25. 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. 26. 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. 27. Bisect an equilateral triangle, and then use one of the resulting right triangles to calculate exact values for the sine, cosine, and tangent of each of the angles 30 and 60. Then use a right isoscles triangle to calculate exact values for the sine, cosine, and tangent of 45.
5 28. Consider the points (1, 4) and (3, 1) plotted on the usual Cartesian coördinate plane. (a) Determine the slope of the line joining the two points. (b) Determine the distance between the two points. (c) Determine an equation for the line joining the two points. 29. Consider two variables x and y that are related by the formula y = 5x. (a) If x increases by 4, what is the corresponding increase in y? (b) If x increase by 30%, what is the corresponding percentage increase in y? 30. Solve the system of equations. (a) (b) (c) (e) (f) 2x + 3y = 1 5x + 2y = 8 6x 4y = 6 3x + 8y = 2 4x 5y = 7 8x + 10y = 6 2x + 3y + 4z = 8 5x + 2y 2z = 5 6x y + 3z = 17 3x + 2y + 3z = 0 4x + 8y z = 3 5x + 2y 2z = 2 x + 2y 3z = 0 2x + 4y + 3z = 1 5x 6y 9z = Grace runs a business that produces jewelry bracelets. Her fixed costs per month (rent, utilities, etc.) are $1000, and her material costs average $10 per bracelet. Let x represent the number of bracelets she produces in a month, and let y represent Grace s costs in a month.
6 (a) Write a formula for y in terms of x. (b) Determine Grace s costs in a month in which she produces 200 bracelets. Table of Greek letters There are many concepts in physics, and not enough letters in the English alphabet for all of them, even though the same letter is often used for many different concepts. The Greek alphabet is therefore heavily used in physics, and it s worthwhile becoming familiar with it. Alpha A α Nu N ν Beta B β Xi Ξ ξ Gamma Γ γ Omicron O o Delta δ Pi Π π Epsilon E ε Rho P ρ Zeta Z ζ Sigma Σ σ Eta H η Tau T τ Theta Θ θ Upsilon Y υ Iota I ι Phi Φ φ Kappa K κ Chi X χ Lambda Λ λ Psi Ψ ψ Mu M µ Omega Ω ω Answers 1. (a) (b) (c) (a) (b) (c) (a) (b) 200 (c) (a) (b) = (c) = (a) 48 = (b) 180 = (c) 66 = = (a) 5 3 = = = = 2 11 = = (a) (a) (a) (a) (b) 5 8 (b) 3 t 8v (b) 3v 2 t (b) 1 12 (c) (c) ac bd (c) ad bc (c) 4 9 (b) 2 7 = = 8 28 = ad bc ad + bc bd adf + bcf bde bdf (c) 6 9 = 2 3 = = 26 39
7 11. (a) $3.40 (b) $80 (c) 15% 12. (a) x 2 x 6 (b) x 2 v 2 t 2 (c) 2mx 1 + 2mx 2 + 6x 1 + 6x (a) 6 (b) 1 2 (c) (a) 2 3 (b) 35 8 (c) (a) 3 and 2 (b) 2 and 3 (c) 1 and 3 1 and 3 2 (e) no solutions (f) (a) t = x v 17. (a) A = CE + D BE D E = AB C b (b) t = ± a (c) t = ± ab t = v i ± v 2 i + 2a(x f x i ) a (b) B = CE + D AE 18. area = 12 m 2 and perimeter = 14 m (c) C = ABE D E D = E(AB C) (e) 19. area = 6 cm 2 and perimeter = 12 cm 20. area = 6.25π mm 2 and circumference = 5π mm y ( 3, 2) (1, 3) (2, 1) 2 ( 1, 2) 3 x (a) 5 (b) 2 (c) (a) 8 (b) 8 2 (c) 2 7 =
8 cm, tan 1 ( km, sin 1 ( sin 40 mm and 3.7 cos 40 mm ) ( ), and tan ), and cos 1 ( ) 27. θ sin θ cos θ tan θ (a) 2.5 (b) 29 (c) y = 2.5x (a) 20 (b) 30% 30. (a) x = 2, y = 1 (b) x = 2/3, y = 1/2 (c) no solution x = 1, y = 2, z = 3 (e) x = 1/6, y = 1/4, z = 1/3 (f) no solution 31. (a) y = 10x (b) $3000
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