MPM 1D1 Summative Review #1 PDF

# MPM 1D1 - Summative Review # 1 ## Topics Covered - BEDMAS - Fractions - Percents - Proportions - Independent/ Dependent Variables - Correlations - Relationships - Discrete/Cont. Data ## 1. Evaluate each of the following. | Question | Answer | |---|---| | (a) 18 + (-6) | 12 | | (b) (-5)(-3) |...

# MPM 1D1 - Summative Review # 1 ## Topics Covered - BEDMAS - Fractions - Percents - Proportions - Independent/ Dependent Variables - Correlations - Relationships - Discrete/Cont. Data ## 1. Evaluate each of the following. | Question | Answer | |---|---| | (a) 18 + (-6) | 12 | | (b) (-5)(-3) | 15 | | (c) -7 + 19 - 12 | 0 | | (d) \(\frac{16}{-4}\) | -4 | | (e) \((\frac{2}{3})^3\) | \(\frac{8}{27}\) | | (f) \(\sqrt{\frac{16}{25}}\) | \(\frac{4}{5}\) | | (g) -22 - 11 | -33 | | (h) (-5)² | 25 | | (i) -4³ | -64 | | (j) (-1)²⁴ | 1 | * | (k) -(-2)² | -4 | * | (l) -25 + 17 | -8 | (* Remember that an exponent applied to a negative number is not evaluated until *after* the absolute value) ## 2. Complete the following chart. Make sure all fractions are in LOWEST TERMS. | Fraction | Decimal | Percent | |---|---|---| | \(\frac{38}{100} = \frac{19}{50}\) | 0.38 | 38% | | \(\frac{-65}{100} = \frac{-13}{20}\) | -0.65 | -65% | | \(\frac{4}{5} = \frac{8}{10}\) | 4.2 | 420% | ## 3. Evaluate the following using BEDMAS. Show all work. | Question | Answer | |---|---| | (a) 5 × 3 - [18 - (-6)] ÷ 3 | 7 | | (b) 5 - \(\frac{2}{3}\) × \(\frac{1}{4}\) ÷ \(\frac{3}{4}\) | \(\frac{58}{15}\) | ## 4. Determine the value of x in the following proportions. ROUND TO 2 DECIMAL PLACES IF NECESSARY. | Question | Answer | |---|---| | (a) \(\frac{3}{x} = \frac{17}{35}\) | 6.18 | | (b) 4:x = 14:56 | 16 | ## 5. Football equipment, which regularly sells for $588, is on sale for 25% off. Determine the total cost of the equipment, including 13% tax. * Discount: $588 x 0.25 = $147 * New Cost: $588 - $147 = $441 * Total Cost: $441 x 1.13 = $498.33 **∴ The total cost including tax of the equipment is $498.33** ## 6. 25% of a number is 66.25. What is the number? * \(\frac{25}{100} = \frac{66.25}{x}\) * 25x = 6625 * x = 265 **∴ The number is 265** ## 7. Complete the table to the right. | x | y | First Difference | |---|---|---| | 1 | 70 | - | | 3 | 62 | -8 | | 5 | 54 | -8 | | 7 | 44 | -10 | | 9 | 32 | -12 | **(a) What type of relationship does this data represent? Explain how you know.** * **The relationship is non-linear, because the first differences are not consistent.** **(b) Does the relationship have a positive or negative correlation? Explain.** * **This relationship has a negative correlation because the first differences are negative.** ## 8. The graph below shows Mike's motion. Describe his motion using sentences. BE CLEAR AND SPECIFIC ABOUT DIRECTIONS, TIMES, ETC. *Include a graph in your response* * **A-B:** Between 0 and 3 minutes, Mike walks quickly, then slows down. This is a curved path, indicating he is not walking at a constant rate. He is walking away from the sensor. * **B-C:** From 3 to 4 minutes, Mike walks at a constant speed of 400 m/min (because the line is straight) away from the sensor. * **C-D:** Mike stops for 1 minute, between 4 and 5 minutes. * **D-E:** From 5 to 8 minutes, Mike walks at a constant speed of 75 m/min towards the sensor. * **E-F:** From 8 to 28 minutes, Mike walks at a constant speed of 75 m/min back to the sensor. ## 9. A group of friends collect data to see whether hours watching TV had an effect on their exam scores. The following results were obtained. | Hours Watching TV | Exam Score | |---|---| | 8 | 72 | | 7 | 67 | | 4 | 81 | | 3 | 93 | | 9 | 54 | | 10 | 66 | **(a) Create a scatter plot. Include a line of best fit.** *Include a scatter plot in your response* **(b) Fill in the table below regarding this information.** | Question | Answer | |---|---| | What type of correlation seems to exist? | Negative | | Is the relationship linear or non-linear? | Linear | | Is the data discrete or continuous? | Continuous | **(c) Describe the relationship that seems to exist with these variables.** * **A weak negative correlation seems to exist between the number of hours watching TV and exam scores.** **(d) Is the origin (0, 0) a reasonable element of the data? Explain.** * **No. The y-intercept is above 100, so a line of best fit will not go through the origin.** ## EQAO Sample Questions **1. Theo plans to purchase a new long distance telephone plan called the Silver Plan. Under this plan, the telephone company determines the monthly cost using the following charges:** * **The base fee is $30/month, which includes up to 150 minutes of long distance.** * **The cost for all minutes over 150 each month is $0.15/minute.** **With the Silver Plan, how much will it cost Theo to talk long-distance for 230 minutes over one month?** * **Equation: y = mx + b** where: * y = total cost * m = cost per minute * x = number of minutes **over** 150 * b = base cost * **y = 0.15x + 30** * **y = 0.15 (80) + 30** (because 230 - 150 = 80) * **y = 12 + 30 = 42** **∴ It will cost Theo $42.00 to talk long-distance for 230 minutes in one month.** **2. Meg has been asked to determine the value of the numerical expression below. Which of the following is the value of Meg's expression? ** \(\frac{2400}{2^3 - 2^2}\) * **Simplify:** * \(\frac{2400}{8 - 4}\) * \(\frac{2400}{4}\) * **Calculate:** 600 **∴ The value of Meg's expression is 600.** **3. The ratio of girls to boys in Mr. Dilworth's music class is 18:12. What percent of the students are boys?** * **The ratio 18:12 can also be expressed as \(\frac{18}{12}\) which simplifies to \(\frac{3}{2}\)** * **We know that \(\frac{2}{5}\) of the students are boys.** * **\(\frac{2}{5} = \frac{40}{100} = 40%\)** **∴ 40% of the students in Mr. Dilworth's music class are boys.** **4. Which of the following expressions is equal to 49?** A. (4 + 3)² B. (3 + 4²) C. 3² + 4² D. 3³ + 4² * **Simplify each expression:** * (4 + 3)² = 7² = 49 * (3 + 4²) = (3 + 16) = 19 * 3² + 4² = 9 + 16 = 25 * 3³ + 4² = 27 + 16 = 43 **∴ (4 + 3)² = 49** **5. The Lucas family is going on a trip to Manitoba and back. It is 1624 km from home to Manitoba. Determine the total cost of the gas for the Lucas family's trip to Manitoba and back, if the cost of gas is 83¢ per litre and the car uses 9.7 L per 100 km. SHOW YOUR WORK.** * **Calculate total distance:** * Each way is 1624 km * Total: 1624 x 2 = 3248 km * **Calculate amount of gas needed:** * The car uses 9.7 L every 100 km * Total litres needed: 9.7 x 32.48 = 315.056 L * **Calculate cost of gas:** * Cost per litre: $0.83 * Total cost: 315.056 x $0.83 = $261.50 **∴ The total cost of gas for the Lucas family's trip to Manitoba and back is $261.50** ## 6. The graph below represents 4 segments of Selena's morning walk. *Include a graph* **A-B** Selena walks away from home at a constant speed of 60 m/min from 0 to 5 minutes. **B-C** Selena stops between 5 and 8 minutes for 3 minutes. **C-D** Selena walks away from home at a constant speed of 75 m/min from 8 to 16 minutes. **D-E** Selena walks back home at a constant speed of 75 m/min for 12 minutes (from 16 to 28 minutes). ## EQAO Sample Questions **1. Expressions for the base area and volume of a prism are given. Which expression represents the height of the prism?** * **Volume = 64a 3 b 6** * **Base Area = 16ab 3** * **Height = Volume / Base Area** * **Height = 64a 3 b 6 / 16ab 3 = 4a 2 b 3** **∴ The expression representing the height of the prism is 4a 2 b 3** **2. A rectangular field has a perimeter of (10a - 6) m and a width of 2a m. Which expression represents the length of this field?** * **Perimeter = 2l + 2w** * **(10a - 6) = 2l + 2(2a)** * **10a - 6 = 2l + 4a** * **10a - 4a -6 = 2l** * **6a - 6 = 2l** * **l = (6a - 6)/2** * **l = 3a - 3** **∴ The expression representing the length of the field is 3a - 3** **3. Tyler, Raven and Deb are discussing the number of CDs they each own. They find that the following statements are true:** * **Tyler owns five more than twice the number of CDs Raven owns.** * **Deb owns three times as many CDs as Tyler.** **Using x to represent the number of CDs Raven owns, write an expression for the total number of CDs the three friends own. SHOW YOUR WORK AND SIMPLIFY YOUR ANSWER.** * **Raven: x** * **Tyler: 2x + 5** * **Deb: 3(2x + 5)** * **Total CDs = x + (2x + 5) + 3(2x + 5)** * **Total CDs = x + 2x + 5 + 6x + 15** * **Total CDs = 9x + 20** **∴ The expression representing the total number of CDs owned by Tyler, Raven and Deb is 9x + 20** # MPM 1D1 - Summative Review # 2 ## Topics Covered - Exponent Rules - Polynomials - Like Terms - Distributive Property - Solving Equations - Translating Words to Math - Word Problems ## 1. Write each of the following as a single power. | Question | Answer | |---|---| | (a) 4⁴ × 4³ | 4⁷ | | (b) (-2)⁵ × (-2)² | (-2)⁷ | | (c) 6²⁶ ÷ 6²¹ | 6⁵ | | (d) \(\frac{5^{5}}{5^{3}}\) | 5² | | (e) (5³)-⁶ | 5⁻¹⁸ | | (f) [(-8)⁴]⁵ | (-8)20 | ## 2. Simplify the following expressions. | Question | Answer | |---|---| | (a) (7x⁴)(-5x²) | -35x⁶ | | (b) \(\frac{28a^{9}}{4a^{5}}\) | 7a⁴ | | (c) (-5ab²c⁵)(4ab³c⁴) | -20a²b⁵c⁹ | | (d) \(\frac{32x^{6}y^{4}z^{7}}{-4x^{2}y^{2}z^{4}}\) | -8x⁴y²z³ | ## 3. Simplify this expression. SHOW ALL STEPS! \( (2g²h³) × (-3g²h²)² ÷ (3gh)(6g²h²)\) * \((2g²h³) × (9g⁴h⁴)\) ÷ (3gh)(6g²h²) * \((18g⁶h⁷)\) ÷ (3gh)(6g²h²) * \((18g⁶h⁷ )\) ÷ (18g³h³) * \(g³h⁴\) **∴ The simplified expression is g³h⁴** ## 4. Read each statement below carefully. Determine whether each is TRUE or FALSE by writing a T or F in the blank provided. | Statement | T/F | |---|---| | (a) The coefficient of 11x²y³z is x²y³z. | F | | (b) a² - 3a + 3 is a polynomial. | T | | (c) The coefficient of -8mn⁵ is mn. | F | | (d) 11xy² and -13x²y are like terms. | F | | (e) (2d²e)(-5d³e) = -7d⁵e⁷ | T | | (f) 3t⁵ + 2t² + 4tu - v has 4 terms and 4 variables. | F | ## 5. Simplify each of the following to a single power. THEN EVALUATE. | Question | Answer | |---|---| | (a) 7⁸ x 7⁴ | 7¹² = 117 649 | | (b) (7⁶)⁵ × (7³)⁻⁵ ÷ (7⁻²)⁻⁶ | 7³ = 343 | ## 6. Simplify the following expressions. | Question | Answer | |---|---| | (a) 8m² + 5n² - 9n² - 6m² | 2m² - 4n² | | (b) 11x² - 2x - 7 + 2x - 17 + 2x² | 13x² - 24 | ## 7. Simplify the following. | Question | Answer | |---|---| | (a) (2x - 3) - (3x + 1) | -x - 4 | | (b) 2x(4x - 8) - 3x(3x + 2) | -x² - 22x | ## 8. Write an algebraic expression or equation for each of the following statements. | Statement | Expression/Equation | |---|---| | (a) Double a number decreased by three. | 2x - 3 | | (b) The value of n dimes (in cents). | 10n | | (c) A number tripled gives 24. | 3n = 24 | | (d) 10 more than quadruple a number. | 4x + 10 | | (e) Martin's age in six years will be thirteen. | x + 6 = 13 | ## 9. Solve the following equations. | Question | Answer | |---|---| | (a) n + 18 = 25 | n = 7 | | (b) m - 13 = 18 | m = 31 | | (c) 7x = 63 | x = 9 | | (d) \(\frac{-13}{5} = r\) | r = -65 | | (e) \(\frac{x}{2} = 20\) | x = 40 | | (f) -30 = 6n | n = -5 | ## 10. Solve the following equations. SHOW ALL WORK. | Question | Answer | |---|---| | (a) -4(3x + 4) = -28 | x = 1 | | (b) 5(2t - 1) + 9 = 2(t - 2) | t = -1 | ## 11. Solve the following equations. SHOW ALL WORK. | Question | Answer | |---|---| | (a) \(\frac{3a}{2} + \frac{7}{5} = -3\) | a = \(\frac{-30}{17}\)| | (b) \(\frac{x-2}{3} + \frac{x + 2}{4} = \frac{x + 1}{12}\) | x = 2 | ## 12. Solve and check the following equation. * **Equation:** 5(2t - 1) + 9 = 2(t - 2) * **Solve:** * 10t - 5 + 9 = 2t - 4 * 10t + 4 = 2t - 4 * 8t = -8 * t = -1 * **Check:** * **Left side:** 5(2(-1)-1) + 9 = 5(-2 - 1) + 9 = 5(-3) + 9 = -15 + 9 = -6 * **Right side:** 2(-1 -2) = 2(-3) = - 6 **Since the left side and the right side are the same, the answer t = -1 is correct.** ##13. Three times a number decreased by eleven is thirty-seven. Determine the number. You must solve this by writing an equation. * **Let x represent the number.** * **Equation:** 3x - 11 = 37 * **Solve:** * 3x - 11 = 37 * 3x = 37 + 11 * 3x = 48 * x = 16 **∴ The number is 16** ## 14. Find two consecutive odd numbers so that the sum of two times the smaller number and five times the larger number is 87. SOLVE FULLY BY WRITING AN EQUATION * **Let x represent the smaller number.** * **Let x + 2 represent the larger number.** * **Equation:** 2x + 5(x + 2) = 87 * **Solve:** * 2x + 5(x + 2) = 87 * 2x + 5x + 10 = 87 * 7x + 10 = 87 * 7x = 87 - 10 * 7x = 77 * x = 11 * **Find the larger number:** 11 + 2 = 13 **∴ The smaller number is 11 and the larger number is 13.** ## 15. A soccer team goes to Starr Burger after a game. The number of drinks ordered is 2 less than the number of burgers. If burgers cost $3.25 and drinks cost $1.50, how many of each were ordered if the coach paid a total of $49.25? SOLVE FULLY. * **Let x represent the number of burgers.** * **Let x - 2 represent the number of drinks.** * **Equation:** $3.25x + $1.50(x - 2) = $49.25 * **Solve:** * $3.25x + $1.50x - $3.00 = $49.25 * $4.75x = $49.25 + $3 * $4.75x = $52.25 * x = 11 * **Find the number of drinks:** 11 - 2 = 9 **∴ The coach ordered 11 burgers and 9 drinks.** ## EQAO Sample Questions **1. Expressions for the base area and volume of a prism are given. Which expression represents the height of the prism?** * **Volume = 64a³b⁶** * **Base Area = 16ab³** * **Height = Volume / Base Area** * **Height = 64a³b⁶ / 16ab³ = 4a²b³** **∴ The expression representing the height of the prism is 4a²b³** **2. A rectangular field has a perimeter of (10a - 6) m and a width of 2a m. Which expression represents the length of this field?** * **Perimeter = 2l + 2w** * **(10a - 6) = 2l + 2(2a)** * **10a - 6 = 2l + 4a** * **10a - 4a -6 = 2l** * **6a - 6 = 2l** * **l = (6a - 6)/2** * **l = 3a - 3** **∴ The expression representing the length of the field is 3a - 3** **3. Tyler, Raven and Deb are discussing the number of CDs they each own. They find that the following statements are true:** * **Tyler owns five more than twice the number of CDs Raven owns.** * **Deb owns three times as many CDs as Tyler.** **Using x to represent the number of CDs Raven owns, write an expression for the total number of CDs the three friends own. SHOW YOUR WORK AND SIMPLIFY YOUR ANSWER.** * **Raven: x** * **Tyler: 2x + 5** * **Deb: 3(2x + 5)** * **Total CDs = x + (2x + 5) + 3(2x + 5)** * **Total CDs = x + 2x + 5 + 6x + 15** * **Total CDs = 9x + 20** **∴ The expression representing the total number of CDs owned by Tyler, Raven and Deb is 9x + 20** # MPM 1D1 - Summative Review # 3 ## Topics Covered - Slope - 1st Differences - Linear vs. Non-Linear - Equations of Lines (Slope-Int., Standard) - Parallel/Perp. Lines - Direct/Partial Variations - Intercepts ## 1. Calculate the slope of the line segments through the points shown. DO NOT CONVERT ANSWERS TO DECIMALS. **(a)** It appears the points are (-6, 8) and (2, -2) **(b)** Q(1, -5), R(6, 2) * **(a) Slope = \(\frac{-2 - 8}{2 - (-6)}\) = \(\frac{-10}{8}\) = \(\frac{-5}{4}\)** * **(b) Slope = \(\frac{4 - 5 }{8 - 1)}\) = \(\frac{-1}{7}\)** ## 2. Each table below represents a linear relation. Determine the slope of each. **(a)** | x | y | |---|---| | -2 | 16 | | -1 | 12 | | 0 | 8 | | 1 | 4 | | 2 | 0 | **(b)** | x | y | |---|---| | -2 | -5 | | 1 | -3 | | 4 | -1 | | 7 | 1 | | 10 | 3 | **(a) slope = \(\frac{4 - 0}{1 - 2}\) = \(\frac{4}{-1}\) = -4** **(b) slope = \(\frac{-5 - 3}{7 - 10}\) = \(\frac{-8}{-3}\) = \(\frac {8}{3}\)** ## 3. Identify the following relations as linear (L) or non-linear (N). | Equation | Linear/Non-linear | |---|---| | (a) y = 2 - 5x | L | | (b) 6x + y = 7 | L | | (c) y = 2x² + 4 | N | | (d) 2x³ - y = 4 | N | | (e) x = -3 | L | | (f) y = -2√x | N | | (f) y = -2x² | N | ## 4. Write the equation that would represent each table of values below. **(a)** | x | y | |---|---| | 0 | 3 | | 1 | 4 | | 2 | 5 | | 3 | 6 | | 4 | 7 | **(b)** | x | y | |---|---| | -4 | 0 | | -2 | 1 | | 0 | 2 | | 2 | 3 | | 4 | 4 | **(c)** | x | y | |---|---| | 8 | -3 | | 6 | -1 | | 4 | 1 | | 2 | 3 | | 0 | 5 | (**a**) * **Slope (m):** \(\frac{4 - 3}{1 - 0}\) = \(\frac{1}{1}\) = 1 * **y-intercept (b):** (0, 3) * **Equation: y = 1x + 3 or y = x + 3** (**b**) * **Slope (m):** \(\frac{1-0}{-2 - (-4)}\) = \(\frac {1}{2}\) * **y-intercept (b):** (0, 2) * **Equation: y = \(\frac{1}{2}\)x + 2** (**c**) * **Slope (m):** \(\frac{-3 - 1}{8-6}\) = \(\frac{-4}{2}\) = -2 * **y-intercept (b):** (0, 5) * **Equation: y = -2x + 5** ## 5. Graph these equations USING THE METHOD INDICATED. LABEL AND SCALE YOUR AXES! **(a) y = \(\frac{2}{3}\)x - 3 (using slope and y-intercept)** **(b) 4x + y - 4 = 0 (calculate x and y intercepts)** ## 6. Write the equation of the line that passes through the points (-3, -5) and (1, 7). * **Slope (m):** \(\frac{7 - (-5)}{1 - (-3)}\) = \(\frac{12}{4}\) = 3 * **y-intercept (b):** Let's substitute the point (1, 7) into the equation y = 3x + b to solve for b. * 7 = 3(1) + b * 7 = 3 + b * b = 7 - 3 * b = 4 * **Equation:** y = 3x + 4 **or** * **Standard form:** 3x - y + 4 = 0 ## 7. Consider each equation below. **(a) Identify the form that the equation is in.** **(b) Convert the equation to the other form.** **(i) 4x - 6y + 15 = 0** * **(a) Standard Form** * **(b) Slope-Intercept Form** * -6y = -4x - 15 * y = \(\frac{2}{3}\)x + \(\frac{5}{2}\) **(ii) y = \(\frac {3}{4}\)x + \(\frac {1}{2}\)** * **(a) Slope-Intercept Form** * **(b) Standard Form** * 4y = 4(\(\frac{3}{4}\)x) + 4(\(\frac{1}{2}\)) * 4y = 3x + 2 * 3x - 4y + 2 = 0 ## 8. Point A lies on the line shown. Determine the equation of ANOTHER line that is perpendicular to the one shown, going through Point A. * **Slope of given line (m₁):** \(\frac{2}{3}\) * **Slope of perpendicular line (m₂):** The negative reciprocal of \(\frac{2}{3}\) is \(\frac{-3}{2}\) * **Point A:** (-2,-2) * **Equation of perpendicular line:** - y = \(\frac{-3}{2}\)x + b - -2 = \(\frac{-3}{2}\)(-2) + b - - 2 = 3 + b - b = -5 **∴ The equation of the perpendicular line is y = \(\frac{-3}{2}\)x - 5** ## 9. A used car purchased for $9,500 decreases in value by approximately $1,200 per year (*Let V represent the value of the car and let n represent the number of years that pass after purchase. **(a) Write an equation for the relation between V and n.** * **V = -1200n + 9500** **(b) What is the slope?** * **-1200** **(c) What does the slope represent?** * **The decrease in the *value of the car* per year** **(d) Does this relationship represent a direct or partial variation? Explain** * **This represents a *partial variation*. The initial value of the car is $9500, but the value decreases at a constant rate per year.** ## EQAO Sample Questions **1. Which of the following equations is not represented by a straight line on a graph?** A. x = 3y - 4 B. y = - 2x C. x = 4 D. y = 2x² - 2 **∴ y = 2x² - 2 is the only equation that is not represented by a straight line on a graph because it contains a squared term.** **2. Alex's Rose Shop makes up bouquets and charges for the vase, plus a cost per rose.** * **The shop charges $32.85 for a bouquet of 12 roses.** * **The shop charges $50.85 for a bouquet of 20 roses.** **What does Alex's Rose Shop charge for a vase?** * **Let x represent the cost per rose.** * **Let b represent the cost of the vase.** * **Equation 1 (12 roses):** 32.85 = 12x + b * **Equation 2 (20 roses):** 50.85 = 20x + b * **Solve for x:** * 50.85 - (32.85) = 20x - 12x * 18 = 8x * x = 2.25 * **solve for b :** * 32.85 = 12(2.25) + b * 32.85 = 27 + b * b = 5.85 **∴ Alex's Rose Shop charges $5.85 for a vase.** **3. The charges on a monthly water bill are $0.86 per m³ of water used plus a service charge of $4.49. Which equation represents the relationship between C and w?** * **C = 0.86w + 4.49** (**Slope-intercept form**) **where:** * **C = total cost** * **w = amount of water (in m³) used** # MPM 1D1 - Summative Review # 4 ## Topics Covered - Pythagorean Theorem - Perimeter - Area - Surface Area - Volume - Angle Theorems - Triangle Properties - Quadrilateral Properties ## 1. Calculate the area of the shaded region shown. * **Area of parallelogram** = bh = (10.6)(12) = 127.2 * **Area of triangle** = \(\frac{bh}{2}\) = \(\frac{5.3(12)}{2}\) = 31.8 * **Area of shaded region:** 127.2 - 31