Mixed Topic Worksheet Grades 8/9: Day 047
- 1
Question 1. The graph of y = sin x and the line y = 0.5 are shown for 0^° ≤ x ≤ 360^°. sin 30^° = 0.5. Use the graph to solve sin x = 0.5 in this range.
Solving Trigonometric Equations Using a Graph▶ Show worked answerHide worked answer
- M1: 180 - 30 for the second point where the line meets the curve
- A1: x = 30 and x = 150
Answer: x = 30 and x = 150[2 marks] - 2
Question 2. Of 50 students, 30 are girls. 12 of the girls and 8 of the boys play chess. A chess player is chosen. Find the probability that the player is a girl.
Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: chess players = 12 + 8 = 20
- A1: dfrac1220 = dfrac35
Answer: dfrac35[2 marks] - 3
Question 3. Solve x² - 2x - 8 ≥ 0
Show your working, using a sketch if you wish.Sketching Quadratic Inequalities▶ Show worked answerHide worked answer
- M1: factorises to (x - 4)(x + 2)
- M1: finds the critical values x = 4 and x = -2
- A1: x ≤ -2 or x ≥ 4
Answer: x ≤ -2 or x ≥ 4[3 marks] - 4
Question 4. Solve x² + 4kx - 3 = 0 by completing the square
Give your answer in terms of kSolving Quadratics by Completing the Square (a = 1)▶ Show worked answerHide worked answer
- M1: (x + 2k)² - 4k² - 3 = 0
- M1: (x + 2k)² = 4k² + 3
- M1: x + 2k = ±√4k² + 3
- A1: Both solutions: -2k + √4k² + 3 and -2k - √4k² + 3
Answer: x = -2k ± √4k² + 3[4 marks] - 5
Question 5. Show that 2x² - 12x + 25 is never less than 7
You must use completing the square.Solving Quadratics by Completing the Square (a > 1)▶ Show worked answerHide worked answer
- M1: 2(x² - 6x) + 25
- M1: 2≤ft((x - 3)² - 9) + 25 = 2(x - 3)² + 7
- A1: (x - 3)² ≥ 0 so 2(x - 3)² + 7 ≥ 7
Answer: 2(x - 3)² + 7 ≥ 7 because a square is never negative[3 marks]
Mixed Topic Worksheet Grades 8/9: Day 047
- 1
Question 1. The graph of y = sin x and the line y = 0.5 are shown for 0^° ≤ x ≤ 360^°. sin 30^° = 0.5. Use the graph to solve sin x = 0.5 in this range.
[2 marks]Solving Trigonometric Equations Using a Graph▶ Show worked answerHide worked answer
- M1: 180 - 30 for the second point where the line meets the curve
- A1: x = 30 and x = 150
Answer: x = 30 and x = 150 - 2
Question 2. Of 50 students, 30 are girls. 12 of the girls and 8 of the boys play chess. A chess player is chosen. Find the probability that the player is a girl.
[2 marks]Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: chess players = 12 + 8 = 20
- A1: dfrac1220 = dfrac35
Answer: dfrac35 - 3
Question 3. Solve x² - 2x - 8 ≥ 0
[3 marks]
Show your working, using a sketch if you wish.Sketching Quadratic Inequalities▶ Show worked answerHide worked answer
- M1: factorises to (x - 4)(x + 2)
- M1: finds the critical values x = 4 and x = -2
- A1: x ≤ -2 or x ≥ 4
Answer: x ≤ -2 or x ≥ 4 - 4
Question 4. Solve x² + 4kx - 3 = 0 by completing the square
[4 marks]
Give your answer in terms of kSolving Quadratics by Completing the Square (a = 1)▶ Show worked answerHide worked answer
- M1: (x + 2k)² - 4k² - 3 = 0
- M1: (x + 2k)² = 4k² + 3
- M1: x + 2k = ±√4k² + 3
- A1: Both solutions: -2k + √4k² + 3 and -2k - √4k² + 3
Answer: x = -2k ± √4k² + 3 - 5
Question 5. Show that 2x² - 12x + 25 is never less than 7
[3 marks]
You must use completing the square.Solving Quadratics by Completing the Square (a > 1)▶ Show worked answerHide worked answer
- M1: 2(x² - 6x) + 25
- M1: 2≤ft((x - 3)² - 9) + 25 = 2(x - 3)² + 7
- A1: (x - 3)² ≥ 0 so 2(x - 3)² + 7 ≥ 7
Answer: 2(x - 3)² + 7 ≥ 7 because a square is never negative
Mixed Topic Worksheet Grades 8/9: Day 047
- 1
Question 1. The graph of y = sin x and the line y = 0.5 are shown for 0^° ≤ x ≤ 360^°. sin 30^° = 0.5. Use the graph to solve sin x = 0.5 in this range.
Solving Trigonometric Equations Using a Graph▶ Show worked answerHide worked answer
- M1: 180 - 30 for the second point where the line meets the curve
- A1: x = 30 and x = 150
Answer: x = 30 and x = 150[2 marks] - 2
Question 2. Of 50 students, 30 are girls. 12 of the girls and 8 of the boys play chess. A chess player is chosen. Find the probability that the player is a girl.
Conditional Probability with Two-Way Tables▶ Show worked answerHide worked answer
- M1: chess players = 12 + 8 = 20
- A1: dfrac1220 = dfrac35
Answer: dfrac35[2 marks] - 3
Question 3. Solve x² - 2x - 8 ≥ 0
Show your working, using a sketch if you wish.Sketching Quadratic Inequalities▶ Show worked answerHide worked answer
- M1: factorises to (x - 4)(x + 2)
- M1: finds the critical values x = 4 and x = -2
- A1: x ≤ -2 or x ≥ 4
Answer: x ≤ -2 or x ≥ 4[3 marks] - 4
Question 4. Solve x² + 4kx - 3 = 0 by completing the square
Give your answer in terms of kSolving Quadratics by Completing the Square (a = 1)▶ Show worked answerHide worked answer
- M1: (x + 2k)² - 4k² - 3 = 0
- M1: (x + 2k)² = 4k² + 3
- M1: x + 2k = ±√4k² + 3
- A1: Both solutions: -2k + √4k² + 3 and -2k - √4k² + 3
Answer: x = -2k ± √4k² + 3[4 marks] - 5
Question 5. Show that 2x² - 12x + 25 is never less than 7
You must use completing the square.Solving Quadratics by Completing the Square (a > 1)▶ Show worked answerHide worked answer
- M1: 2(x² - 6x) + 25
- M1: 2≤ft((x - 3)² - 9) + 25 = 2(x - 3)² + 7
- A1: (x - 3)² ≥ 0 so 2(x - 3)² + 7 ≥ 7
Answer: 2(x - 3)² + 7 ≥ 7 because a square is never negative[3 marks]