Mixed Topic Worksheet Grades 8/9: Day 078
- 1
Question 1. A cuboid has a 6 cm by 8 cm base and height 5 cm. Work out the angle between a space diagonal and the base. Give your answer to 1 decimal place.
3D Trigonometry▶ Show worked answerHide worked answer
- M1: base diagonal 10, tan θ = 5/10
- A1: 26.6°
Answer: 26.6°[2 marks] - 2
Question 2. Find the minimum value of x² - 3x + 4.
Solving Quadratics by Completing the Square (a = 1)▶ Show worked answerHide worked answer
- M1: ≤ft(x - frac32)² - frac94 + 4
- A1: minimum value frac74
Answer: frac74 (or 1.75)[2 marks] - 3
Question 3. A force of 450 newtons, correct to 2 significant figures, acts on a circular area of radius 4.2 cm, correct to 2 significant figures.
Work out the lower bound of the pressure in newtons per square centimetre.
pressure = dfracforcearea
Give your answer to 3 significant figures.Combining Upper & Lower Bounds▶ Show worked answerHide worked answer
- M1: 450 - 5 and 4.2 + 0.05
- M1: π × 4.25²
- M1: 445 ÷ (π × 4.25²)
- A1: 7.84
Answer: 7.84[4 marks] - 4
Question 4. The sum of the squares of two consecutive positive integers is 313.
Use algebra to work out the two integers.Proof with Consecutive Numbers▶ Show worked answerHide worked answer
- M1: n² + (n + 1)² = 2n² + 2n + 1
- M1: 2n² + 2n + 1 = 313 so n² + n - 156 = 0
- M1: (n + 13)(n - 12) = 0
- A1: n = 12 so the integers are 12 and 13
Answer: 12 and 13[4 marks] - 5
Question 5. A circle has centre O and equation x² + y² = 10
The point (1, 3) lies on the circle.
The tangent to the circle at (1, 3) meets the x-axis at R.
Work out the coordinates of R.Finding Equations of Tangents to Circles▶ Show worked answerHide worked answer
- M1: Gradient of the radius = 3, so gradient of the tangent = -dfrac13
- M1: y - 3 = -dfrac13(x - 1) and put y = 0
- A2: x = 10, so R(10, 0)
Answer: R(10, 0)[4 marks]
Mixed Topic Worksheet Grades 8/9: Day 078
- 1
Question 1. A cuboid has a 6 cm by 8 cm base and height 5 cm. Work out the angle between a space diagonal and the base. Give your answer to 1 decimal place.
[2 marks]3D Trigonometry▶ Show worked answerHide worked answer
- M1: base diagonal 10, tan θ = 5/10
- A1: 26.6°
Answer: 26.6° - 2
Question 2. Find the minimum value of x² - 3x + 4.
[2 marks]Solving Quadratics by Completing the Square (a = 1)▶ Show worked answerHide worked answer
- M1: ≤ft(x - frac32)² - frac94 + 4
- A1: minimum value frac74
Answer: frac74 (or 1.75) - 3
Question 3. A force of 450 newtons, correct to 2 significant figures, acts on a circular area of radius 4.2 cm, correct to 2 significant figures.
[4 marks]
Work out the lower bound of the pressure in newtons per square centimetre.
pressure = dfracforcearea
Give your answer to 3 significant figures.Combining Upper & Lower Bounds▶ Show worked answerHide worked answer
- M1: 450 - 5 and 4.2 + 0.05
- M1: π × 4.25²
- M1: 445 ÷ (π × 4.25²)
- A1: 7.84
Answer: 7.84 - 4
Question 4. The sum of the squares of two consecutive positive integers is 313.
[4 marks]
Use algebra to work out the two integers.Proof with Consecutive Numbers▶ Show worked answerHide worked answer
- M1: n² + (n + 1)² = 2n² + 2n + 1
- M1: 2n² + 2n + 1 = 313 so n² + n - 156 = 0
- M1: (n + 13)(n - 12) = 0
- A1: n = 12 so the integers are 12 and 13
Answer: 12 and 13 - 5
Question 5. A circle has centre O and equation x² + y² = 10
[4 marks]
The point (1, 3) lies on the circle.
The tangent to the circle at (1, 3) meets the x-axis at R.
Work out the coordinates of R.Finding Equations of Tangents to Circles▶ Show worked answerHide worked answer
- M1: Gradient of the radius = 3, so gradient of the tangent = -dfrac13
- M1: y - 3 = -dfrac13(x - 1) and put y = 0
- A2: x = 10, so R(10, 0)
Answer: R(10, 0)
Mixed Topic Worksheet Grades 8/9: Day 078
- 1
Question 1. A cuboid has a 6 cm by 8 cm base and height 5 cm. Work out the angle between a space diagonal and the base. Give your answer to 1 decimal place.
3D Trigonometry▶ Show worked answerHide worked answer
- M1: base diagonal 10, tan θ = 5/10
- A1: 26.6°
Answer: 26.6°[2 marks] - 2
Question 2. Find the minimum value of x² - 3x + 4.
Solving Quadratics by Completing the Square (a = 1)▶ Show worked answerHide worked answer
- M1: ≤ft(x - frac32)² - frac94 + 4
- A1: minimum value frac74
Answer: frac74 (or 1.75)[2 marks] - 3
Question 3. A force of 450 newtons, correct to 2 significant figures, acts on a circular area of radius 4.2 cm, correct to 2 significant figures.
Work out the lower bound of the pressure in newtons per square centimetre.
pressure = dfracforcearea
Give your answer to 3 significant figures.Combining Upper & Lower Bounds▶ Show worked answerHide worked answer
- M1: 450 - 5 and 4.2 + 0.05
- M1: π × 4.25²
- M1: 445 ÷ (π × 4.25²)
- A1: 7.84
Answer: 7.84[4 marks] - 4
Question 4. The sum of the squares of two consecutive positive integers is 313.
Use algebra to work out the two integers.Proof with Consecutive Numbers▶ Show worked answerHide worked answer
- M1: n² + (n + 1)² = 2n² + 2n + 1
- M1: 2n² + 2n + 1 = 313 so n² + n - 156 = 0
- M1: (n + 13)(n - 12) = 0
- A1: n = 12 so the integers are 12 and 13
Answer: 12 and 13[4 marks] - 5
Question 5. A circle has centre O and equation x² + y² = 10
The point (1, 3) lies on the circle.
The tangent to the circle at (1, 3) meets the x-axis at R.
Work out the coordinates of R.Finding Equations of Tangents to Circles▶ Show worked answerHide worked answer
- M1: Gradient of the radius = 3, so gradient of the tangent = -dfrac13
- M1: y - 3 = -dfrac13(x - 1) and put y = 0
- A2: x = 10, so R(10, 0)
Answer: R(10, 0)[4 marks]