Mixed Topic Worksheet Grades 8/9: Day 002
- 1
Question 1. x = 120, correct to the nearest 10
y = 35, correct to the nearest 5
Work out the upper bound of x ÷ y
Give your answer correct to 3 significant figures.Combining Upper & Lower Bounds▶ Show worked answerHide worked answer
- M1: 120 + 5 and 35 - 2.5
- M1: 125 ÷ 32.5
- A1: 3.85
Answer: 3.85[3 marks] - 2
Question 2. Use sin 30° = 1/2 to find both values of x between 0° and 360° for which sin x = -1/2.
Sketching sin, cos & tan▶ Show worked answerHide worked answer
- M1: solutions lie in the third and fourth quadrants
- A1: 180° + 30° = 210°
- A1: 360° - 30° = 330°
Answer: 210° and 330°[3 marks] - 3
Question 3. 45% of voters are men. 40% of the men and 55% of the women vote Yes. A voter who voted Yes is chosen at random. Work out the probability that this voter is a man. Give your answer to 3 significant figures.
Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: men voting Yes = 0.45 × 0.4; women voting Yes = 0.55 × 0.55
- M1: 0.18 ÷ (0.18 + 0.3025)
- A1: 0.373
Answer: 0.373[3 marks] - 4
Question 4. Prove that the difference between the squares of any two consecutive odd numbers is always a multiple of 8
Proof with Consecutive Numbers▶ Show worked answerHide worked answer
- M1: two consecutive odd numbers written as 2n - 1 and 2n + 1
- M1: (2n + 1)² - (2n - 1)² = (4n² + 4n + 1) - (4n² - 4n + 1)
- A1: = 8n
- C1: 8n = 8 × n, so it is always a multiple of 8
Answer: (2n + 1)² - (2n - 1)² = 8n, a multiple of 8[4 marks] - 5
Question 5. The circle C, centre O, has equation x² + y² = 65
P is the point (7, -4) on C.
Find an equation of the tangent to C at P.Finding Equations of Tangents to Circles▶ Show worked answerHide worked answer
- M1: Gradient of OP = -dfrac47
- M1: Gradient of the tangent = dfrac74 and y + 4 = dfrac74(x - 7)
- A1: 7x - 4y = 65
Answer: 7x - 4y = 65[3 marks]
Mixed Topic Worksheet Grades 8/9: Day 002
- 1
Question 1. x = 120, correct to the nearest 10
[3 marks]
y = 35, correct to the nearest 5
Work out the upper bound of x ÷ y
Give your answer correct to 3 significant figures.Combining Upper & Lower Bounds▶ Show worked answerHide worked answer
- M1: 120 + 5 and 35 - 2.5
- M1: 125 ÷ 32.5
- A1: 3.85
Answer: 3.85 - 2
Question 2. Use sin 30° = 1/2 to find both values of x between 0° and 360° for which sin x = -1/2.
[3 marks]Sketching sin, cos & tan▶ Show worked answerHide worked answer
- M1: solutions lie in the third and fourth quadrants
- A1: 180° + 30° = 210°
- A1: 360° - 30° = 330°
Answer: 210° and 330° - 3
Question 3. 45% of voters are men. 40% of the men and 55% of the women vote Yes. A voter who voted Yes is chosen at random. Work out the probability that this voter is a man. Give your answer to 3 significant figures.
[3 marks]Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: men voting Yes = 0.45 × 0.4; women voting Yes = 0.55 × 0.55
- M1: 0.18 ÷ (0.18 + 0.3025)
- A1: 0.373
Answer: 0.373 - 4
Question 4. Prove that the difference between the squares of any two consecutive odd numbers is always a multiple of 8
[4 marks]Proof with Consecutive Numbers▶ Show worked answerHide worked answer
- M1: two consecutive odd numbers written as 2n - 1 and 2n + 1
- M1: (2n + 1)² - (2n - 1)² = (4n² + 4n + 1) - (4n² - 4n + 1)
- A1: = 8n
- C1: 8n = 8 × n, so it is always a multiple of 8
Answer: (2n + 1)² - (2n - 1)² = 8n, a multiple of 8 - 5
Question 5. The circle C, centre O, has equation x² + y² = 65
[3 marks]
P is the point (7, -4) on C.
Find an equation of the tangent to C at P.Finding Equations of Tangents to Circles▶ Show worked answerHide worked answer
- M1: Gradient of OP = -dfrac47
- M1: Gradient of the tangent = dfrac74 and y + 4 = dfrac74(x - 7)
- A1: 7x - 4y = 65
Answer: 7x - 4y = 65
Mixed Topic Worksheet Grades 8/9: Day 002
- 1
Question 1. x = 120, correct to the nearest 10
y = 35, correct to the nearest 5
Work out the upper bound of x ÷ y
Give your answer correct to 3 significant figures.Combining Upper & Lower Bounds▶ Show worked answerHide worked answer
- M1: 120 + 5 and 35 - 2.5
- M1: 125 ÷ 32.5
- A1: 3.85
Answer: 3.85[3 marks] - 2
Question 2. Use sin 30° = 1/2 to find both values of x between 0° and 360° for which sin x = -1/2.
Sketching sin, cos & tan▶ Show worked answerHide worked answer
- M1: solutions lie in the third and fourth quadrants
- A1: 180° + 30° = 210°
- A1: 360° - 30° = 330°
Answer: 210° and 330°[3 marks] - 3
Question 3. 45% of voters are men. 40% of the men and 55% of the women vote Yes. A voter who voted Yes is chosen at random. Work out the probability that this voter is a man. Give your answer to 3 significant figures.
Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: men voting Yes = 0.45 × 0.4; women voting Yes = 0.55 × 0.55
- M1: 0.18 ÷ (0.18 + 0.3025)
- A1: 0.373
Answer: 0.373[3 marks] - 4
Question 4. Prove that the difference between the squares of any two consecutive odd numbers is always a multiple of 8
Proof with Consecutive Numbers▶ Show worked answerHide worked answer
- M1: two consecutive odd numbers written as 2n - 1 and 2n + 1
- M1: (2n + 1)² - (2n - 1)² = (4n² + 4n + 1) - (4n² - 4n + 1)
- A1: = 8n
- C1: 8n = 8 × n, so it is always a multiple of 8
Answer: (2n + 1)² - (2n - 1)² = 8n, a multiple of 8[4 marks] - 5
Question 5. The circle C, centre O, has equation x² + y² = 65
P is the point (7, -4) on C.
Find an equation of the tangent to C at P.Finding Equations of Tangents to Circles▶ Show worked answerHide worked answer
- M1: Gradient of OP = -dfrac47
- M1: Gradient of the tangent = dfrac74 and y + 4 = dfrac74(x - 7)
- A1: 7x - 4y = 65
Answer: 7x - 4y = 65[3 marks]