Mixed Topic Worksheet Grades 8/9: Day 005
- 1
Question 1. sin 30^° = dfrac12. Without a calculator, find the two values of x, between 0^° and 360^°, for which sin x = -dfrac12
Calculating Values of Trig Functions Using Graphs▶ Show worked answerHide worked answer
- M1: 180^° + 30^° or 360^° - 30^°
- A1: 210^° and 330^°
Answer: x = 210^° and x = 330^°[2 marks] - 2
Question 2. Show that the equation x³ - 5x + 1 = 0 has a solution between x = 0.2 and x = 0.3
Change of Signs▶ Show worked answerHide worked answer
- M1: f(x) = x³ - 5x + 1, work out f(0.2) and f(0.3)
- C1: f(0.2) = 0.008 > 0 and f(0.3) = -0.473 < 0, sign change, so a solution lies between 0.2 and 0.3
Answer: Shown[2 marks] - 3
Question 3. The histogram is part drawn. It shows the times, t minutes, that 104 students took to cycle to school. The bar for 35 < t ≤ k is not drawn and has frequency density 2.6. Work out k.
Interpreting Histograms▶ Show worked answerHide worked answer
- M1: 3.0 × 5 + 1.8 × 15 + 2.4 × 15 = 78
- M1: 104 - 78 = 26
- M1: 26 ÷ 2.6 = 10
- A1: k = 45
Answer: 45[4 marks] - 4
Question 4. Kofi cycles to school with probability 0.3, otherwise he walks. He is late with probability 0.1 when he cycles and 0.25 when he walks. Given that he is late, work out the probability that he walked.
Conditional Probability with Tree Diagrams▶ Show worked answerHide worked answer
- M1: P(walks and late) = 0.7 × 0.25
- M1: P(late) = 0.3 × 0.1 + 0.175 = 0.205
- M1: 0.175 ÷ 0.205 A1: 35/41
Answer: 35/41[4 marks] - 5
Question 5. Write dfrac3 + √53 - √5 in the form dfraca + b√5c where a, b and c are integers.
Rationalising Denominators of the Form a ± √b▶ Show worked answerHide worked answer
- M1: multiply top and bottom by 3 + √5
- M1: numerator 9 + 6√5 + 5, denominator 9 - 5 = 4
- A1: dfrac7 + 3√52
Answer: dfrac7 + 3√52[3 marks]
Mixed Topic Worksheet Grades 8/9: Day 005
- 1
Question 1. sin 30^° = dfrac12. Without a calculator, find the two values of x, between 0^° and 360^°, for which sin x = -dfrac12
[2 marks]Calculating Values of Trig Functions Using Graphs▶ Show worked answerHide worked answer
- M1: 180^° + 30^° or 360^° - 30^°
- A1: 210^° and 330^°
Answer: x = 210^° and x = 330^° - 2
Question 2. Show that the equation x³ - 5x + 1 = 0 has a solution between x = 0.2 and x = 0.3
[2 marks]Change of Signs▶ Show worked answerHide worked answer
- M1: f(x) = x³ - 5x + 1, work out f(0.2) and f(0.3)
- C1: f(0.2) = 0.008 > 0 and f(0.3) = -0.473 < 0, sign change, so a solution lies between 0.2 and 0.3
Answer: Shown - 3
Question 3. The histogram is part drawn. It shows the times, t minutes, that 104 students took to cycle to school. The bar for 35 < t ≤ k is not drawn and has frequency density 2.6. Work out k.
[4 marks]Interpreting Histograms▶ Show worked answerHide worked answer
- M1: 3.0 × 5 + 1.8 × 15 + 2.4 × 15 = 78
- M1: 104 - 78 = 26
- M1: 26 ÷ 2.6 = 10
- A1: k = 45
Answer: 45 - 4
Question 4. Kofi cycles to school with probability 0.3, otherwise he walks. He is late with probability 0.1 when he cycles and 0.25 when he walks. Given that he is late, work out the probability that he walked.
[4 marks]Conditional Probability with Tree Diagrams▶ Show worked answerHide worked answer
- M1: P(walks and late) = 0.7 × 0.25
- M1: P(late) = 0.3 × 0.1 + 0.175 = 0.205
- M1: 0.175 ÷ 0.205 A1: 35/41
Answer: 35/41 - 5
Question 5. Write dfrac3 + √53 - √5 in the form dfraca + b√5c where a, b and c are integers.
[3 marks]Rationalising Denominators of the Form a ± √b▶ Show worked answerHide worked answer
- M1: multiply top and bottom by 3 + √5
- M1: numerator 9 + 6√5 + 5, denominator 9 - 5 = 4
- A1: dfrac7 + 3√52
Answer: dfrac7 + 3√52
Mixed Topic Worksheet Grades 8/9: Day 005
- 1
Question 1. sin 30^° = dfrac12. Without a calculator, find the two values of x, between 0^° and 360^°, for which sin x = -dfrac12
Calculating Values of Trig Functions Using Graphs▶ Show worked answerHide worked answer
- M1: 180^° + 30^° or 360^° - 30^°
- A1: 210^° and 330^°
Answer: x = 210^° and x = 330^°[2 marks] - 2
Question 2. Show that the equation x³ - 5x + 1 = 0 has a solution between x = 0.2 and x = 0.3
Change of Signs▶ Show worked answerHide worked answer
- M1: f(x) = x³ - 5x + 1, work out f(0.2) and f(0.3)
- C1: f(0.2) = 0.008 > 0 and f(0.3) = -0.473 < 0, sign change, so a solution lies between 0.2 and 0.3
Answer: Shown[2 marks] - 3
Question 3. The histogram is part drawn. It shows the times, t minutes, that 104 students took to cycle to school. The bar for 35 < t ≤ k is not drawn and has frequency density 2.6. Work out k.
Interpreting Histograms▶ Show worked answerHide worked answer
- M1: 3.0 × 5 + 1.8 × 15 + 2.4 × 15 = 78
- M1: 104 - 78 = 26
- M1: 26 ÷ 2.6 = 10
- A1: k = 45
Answer: 45[4 marks] - 4
Question 4. Kofi cycles to school with probability 0.3, otherwise he walks. He is late with probability 0.1 when he cycles and 0.25 when he walks. Given that he is late, work out the probability that he walked.
Conditional Probability with Tree Diagrams▶ Show worked answerHide worked answer
- M1: P(walks and late) = 0.7 × 0.25
- M1: P(late) = 0.3 × 0.1 + 0.175 = 0.205
- M1: 0.175 ÷ 0.205 A1: 35/41
Answer: 35/41[4 marks] - 5
Question 5. Write dfrac3 + √53 - √5 in the form dfraca + b√5c where a, b and c are integers.
Rationalising Denominators of the Form a ± √b▶ Show worked answerHide worked answer
- M1: multiply top and bottom by 3 + √5
- M1: numerator 9 + 6√5 + 5, denominator 9 - 5 = 4
- A1: dfrac7 + 3√52
Answer: dfrac7 + 3√52[3 marks]