Mixed Topic Worksheet Grades 8/9: Day 015
- 1
Question 1. The graph of y = cos x and the line y = 0.5 are shown for 0^° ≤ x ≤ 360^°. cos 60^° = 0.5. Use the graph to solve cos x = 0.5 in this range.
Solving Trigonometric Equations Using a Graph▶ Show worked answerHide worked answer
- M1: 360 - 60 for the second point where the line meets the curve
- A1: x = 60 and x = 300
Answer: x = 60 and x = 300[2 marks] - 2
Question 2. Solve dfrac3xx + 2 × dfracx² - 46 = 12
Multiplying & Dividing Algebraic Fractions▶ Show worked answerHide worked answer
- M1: dfrac3x(x - 2)(x + 2)6(x + 2) = dfracx(x - 2)2
- M1: x² - 2x - 24 = 0 so (x - 6)(x + 4) = 0
- A1: x = 6 or x = -4
Answer: x = 6 or x = -4[3 marks] - 3
Question 3. Of 200 people, 60% are over 30. 25% of those over 30 own a dog and 40% of the others own a dog. A dog owner is chosen. Find the probability that they are over 30.
Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: over 30 owners = 30, others = 32
- M1: dfrac3062
- A1: dfrac1531
Answer: dfrac1531[3 marks] - 4
Question 4. A, B and C are on a circle, centre O.
Angle ACB = 33^°
Work out angle OAB.
Give reasons for each stage.Combining Circle Theorems▶ Show worked answerHide worked answer
- M1: angle AOB = 2 × 33
- M1: (180 - 66) ÷ 2
- A1: angle OAB = 57^°
- C1: angle at the centre is twice the angle at the circumference; OAB is isosceles (radii)
Answer: 57^°[4 marks] - 5
Question 5. Here are values of y = x³
smallbeginarrayc|c|c|c|c|c x & 0 & 0.5 & 1 & 1.5 & 2 hline y & 0 & 0.125 & 1 & 3.375 & 8 endarray
Use 4 trapezia of equal width to estimate the area under the curve from x = 0 to x = 2. Is it an overestimate or an underestimate? Give a reason.Estimating Areas Under Curves▶ Show worked answerHide worked answer
- M2: dfrac12 × 0.5 × (0 + 8 + 2(0.125 + 1 + 3.375))
- A1: 4.25
- B1: overestimate: the curve bends upwards, so the trapezia lie above it
Answer: 4.25, an overestimate (the curve bends upwards)[4 marks]
Mixed Topic Worksheet Grades 8/9: Day 015
- 1
Question 1. The graph of y = cos x and the line y = 0.5 are shown for 0^° ≤ x ≤ 360^°. cos 60^° = 0.5. Use the graph to solve cos x = 0.5 in this range.
[2 marks]Solving Trigonometric Equations Using a Graph▶ Show worked answerHide worked answer
- M1: 360 - 60 for the second point where the line meets the curve
- A1: x = 60 and x = 300
Answer: x = 60 and x = 300 - 2
Question 2. Solve dfrac3xx + 2 × dfracx² - 46 = 12
[3 marks]Multiplying & Dividing Algebraic Fractions▶ Show worked answerHide worked answer
- M1: dfrac3x(x - 2)(x + 2)6(x + 2) = dfracx(x - 2)2
- M1: x² - 2x - 24 = 0 so (x - 6)(x + 4) = 0
- A1: x = 6 or x = -4
Answer: x = 6 or x = -4 - 3
Question 3. Of 200 people, 60% are over 30. 25% of those over 30 own a dog and 40% of the others own a dog. A dog owner is chosen. Find the probability that they are over 30.
[3 marks]Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: over 30 owners = 30, others = 32
- M1: dfrac3062
- A1: dfrac1531
Answer: dfrac1531 - 4
Question 4. A, B and C are on a circle, centre O.
[4 marks]
Angle ACB = 33^°
Work out angle OAB.
Give reasons for each stage.Combining Circle Theorems▶ Show worked answerHide worked answer
- M1: angle AOB = 2 × 33
- M1: (180 - 66) ÷ 2
- A1: angle OAB = 57^°
- C1: angle at the centre is twice the angle at the circumference; OAB is isosceles (radii)
Answer: 57^° - 5
Question 5. Here are values of y = x³
[4 marks]
smallbeginarrayc|c|c|c|c|c x & 0 & 0.5 & 1 & 1.5 & 2 hline y & 0 & 0.125 & 1 & 3.375 & 8 endarray
Use 4 trapezia of equal width to estimate the area under the curve from x = 0 to x = 2. Is it an overestimate or an underestimate? Give a reason.Estimating Areas Under Curves▶ Show worked answerHide worked answer
- M2: dfrac12 × 0.5 × (0 + 8 + 2(0.125 + 1 + 3.375))
- A1: 4.25
- B1: overestimate: the curve bends upwards, so the trapezia lie above it
Answer: 4.25, an overestimate (the curve bends upwards)
Mixed Topic Worksheet Grades 8/9: Day 015
- 1
Question 1. The graph of y = cos x and the line y = 0.5 are shown for 0^° ≤ x ≤ 360^°. cos 60^° = 0.5. Use the graph to solve cos x = 0.5 in this range.
Solving Trigonometric Equations Using a Graph▶ Show worked answerHide worked answer
- M1: 360 - 60 for the second point where the line meets the curve
- A1: x = 60 and x = 300
Answer: x = 60 and x = 300[2 marks] - 2
Question 2. Solve dfrac3xx + 2 × dfracx² - 46 = 12
Multiplying & Dividing Algebraic Fractions▶ Show worked answerHide worked answer
- M1: dfrac3x(x - 2)(x + 2)6(x + 2) = dfracx(x - 2)2
- M1: x² - 2x - 24 = 0 so (x - 6)(x + 4) = 0
- A1: x = 6 or x = -4
Answer: x = 6 or x = -4[3 marks] - 3
Question 3. Of 200 people, 60% are over 30. 25% of those over 30 own a dog and 40% of the others own a dog. A dog owner is chosen. Find the probability that they are over 30.
Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: over 30 owners = 30, others = 32
- M1: dfrac3062
- A1: dfrac1531
Answer: dfrac1531[3 marks] - 4
Question 4. A, B and C are on a circle, centre O.
Angle ACB = 33^°
Work out angle OAB.
Give reasons for each stage.Combining Circle Theorems▶ Show worked answerHide worked answer
- M1: angle AOB = 2 × 33
- M1: (180 - 66) ÷ 2
- A1: angle OAB = 57^°
- C1: angle at the centre is twice the angle at the circumference; OAB is isosceles (radii)
Answer: 57^°[4 marks] - 5
Question 5. Here are values of y = x³
smallbeginarrayc|c|c|c|c|c x & 0 & 0.5 & 1 & 1.5 & 2 hline y & 0 & 0.125 & 1 & 3.375 & 8 endarray
Use 4 trapezia of equal width to estimate the area under the curve from x = 0 to x = 2. Is it an overestimate or an underestimate? Give a reason.Estimating Areas Under Curves▶ Show worked answerHide worked answer
- M2: dfrac12 × 0.5 × (0 + 8 + 2(0.125 + 1 + 3.375))
- A1: 4.25
- B1: overestimate: the curve bends upwards, so the trapezia lie above it
Answer: 4.25, an overestimate (the curve bends upwards)[4 marks]