SL MATH

SIXTH FORM ENTRANCE TEST FOR LOWER SIXTH (YEAR 12) MATHEMATICS 2022

Overview

• Test Details

  • Level: Standard Level

  • Time Allowed: 45 minutes

  • Equipment Needed: Pen, pencil, eraser, calculator.

  • Calculator Usage: Candidates may use a calculator.

  • Answer Submission: Indicate answers on the answer sheet provided.

  • Scoring: Each question is worth 1 mark; a total of 15 questions.

  • Diagrams: Not to scale.

Useful Formulas

Binomial Expansion

• The binomial expansion formula is given by: $(a + b)^n = ext{summation from } r=0 ext{ to } n ext{ of } C(n, r) a^{n-r} b^r$ Where:

  • $C(n, r) = \frac{n!}{(n-r)!r!}$

Quadratic Equation

• The quadratic equation is represented as:
  $<br>ax^2 + bx + c = 0 ext{ (where } a \neq 0) <br>$

• The solutions are given by:
  $<br>x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}<br>$

Triangle Properties

• In triangle ABC, we have the following properties:

  • Area of triangle:
    $<br>A = \frac{1}{2}ab \sin C<br>$

  • Sine Rule:
    $<br>\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}<br>$

  • Cosine Rule:
    $<br>a^2 = b^2 + c^2 - 2bc \cos A<br>$

Sphere Formulas

• Volume of a sphere:
  $<br>V = \frac{4}{3}\pi r^3<br>$

• Surface Area of a sphere:
  $<br>SA = 4\pi r^2<br>$

Cylinder Formulas

• Volume of a cylinder:
  $<br>V = \pi r^2 h<br>$

• Curved Surface Area of cylinder:
  $<br>CSA = 2\pi rh<br>$

Cone Formulas

• Volume of a cone:
  $<br>V = \frac{1}{3}\pi r^2 h<br>$

• Curved Surface Area of cone:
  $<br>CSA = \pi rl<br>$

Questions

1. Solve the Equation
   2(x - 2) = 3x + 7

   • [A] x = 11

   • [B] x = 3

   • [C] x = -11

   • [D] x = -3

   • [E] x = 11 5

2. Sphere Problem
   A sphere has a surface area of 450 cm². Find the volume of the sphere to 3 significant figures.

   • [A] 104 cm³

   • [B] 98.2 cm³

   • [C] 284 cm³

   • [D] 562 cm³

   • [E] 898 cm³

3. Equation of a Straight Line
   Find the equation of the straight line which passes through the points (2, 7) and (-1, 5).

   • [A] y = 3x + 4

   • [B] y = 2x + 3

   • [C] y = rac{3}{2}x + 7

   • [D] y = - rac{2}{3}x - 2

   • [E] y = rac{2}{3}x + rac{17}{3}

4. Estimate Mean Weight
   Work out an estimate for the mean weight of a tea bag using halfway values of 2.85 grams, 2.95 grams, etc. Round your answer to 3 significant figures.

   • [A] 2.95 g

   • [B] 3.10 g

   • [C] 3.14 g

   • [D] 3.72 g

   • [E] 4.11 g

5. Area of Kite
   Work out the area of this kite to 3 significant figures.

   • [A] 24.0 cm²

   • [B] 11.3 cm²

   • [C] 56.8 cm²

   • [D] 22.6 cm²

   • [E] 12.0 cm²

6. Percentage Increase
   Helen’s savings increased from £1051 to £5182. Work out the percentage increase in Helen’s savings to the nearest whole number.

   • [A] 174%

   • [B] 518%

   • [C] 295%

   • [D] 393%

   • [E] 493%

7. Total Surface Area
   A solid shape made from a cone on top of a cylinder has the following dimensions: the cone has a radius of 10 cm and a height of 10 cm, while the cylinder also has a radius of 10 cm and a height of 10 cm. The centre of the base of the cone coincides with the centre of the top face of the cylinder. Find the total surface area of the solid to 2 decimal places.

   • [A] 1072.61 cm²

   • [B] 1989.68 cm²

   • [C] 1386.77 cm²

   • [D] 1675.52 cm²

   • [E] 2406.26 cm²

8. Average Speed
   A plane flew from Bogota to Quito over a distance of 725 km in a time of 1 hour and 24 minutes. Work out the average speed of the plane. Round your answer to 3 significant figures.

   • [A] 585 km/h

   • [B] 518 km/h

   • [C] 668 km/h

   • [D] 529 km/h

   • [E] 613 km/h

9. Simplify Expression
   Simplify fully (2x + y)² − (2x − y)².

   • [A] 8x³

   • [B] 4x² + 2y²

   • [C] 2y²

   • [D] 8x³ + 2y²

   • [E] 4x³

10. Triangle Angles
    A triangle has sides of length 8 cm, 10 cm, and 14 cm. Work out the size of the largest angle in the triangle, correct to 1 decimal place.

    • [A] 97.8°

    • [B] 114.5°

    • [C] 84.2°

    • [D] 135.5°

    • [E] 101.5°

11. Solve Quadratic Equation
    Solve the equation 3x² + 6x - 5 = 0. Give your solutions correct to 3 significant figures.

    • [A] x = 0.345, -1.35

    • [B] x = 1.13, -2.13

    • [C] x = 1.90, -7.90

    • [D] x = 2.67, -3.67

    • [E] x = 0.633, -2.63

12. Value of k
    Work out the value of k in the equation 82 × √2 = 42k.

    • [A] k = 1.25

    • [B] k = 1.375

    • [C] k = 1.5

    • [D] k = 1.625

    • [E] k = 1.75

13. Solve Inequality
    Solve the inequality x² - 6x ≥ -8.

    • [A] 2 ≤ x ≤ 4

    • [B] x ≤ -4, x ≥ -2

    • [C] x ≤ 2, x ≥ 4

    • [D] -4 ≤ x ≤ -2

    • [E] x ≥ 4

14. Coefficient of x^5
    Find the coefficient of the x⁵ term in the expansion of (3x² + 1)^4.

    • [A] 3

    • [B] 9

    • [C] 27

    • [D] 81

    • [E] 108

15. Curve and Line Intersection
    The sketch shows the curve with equation y = x² + 4 and the line with equation y = x + 10. The line cuts the curve at points A and B. Find the coordinates of M, where M is the midpoint of AB.

    • [A] (0.25, 10.25)

    • [B] (0.5, 10.5)

    • [C] (1, 11)

    • [D] (1.25, 11.25)

    • [E] (1.5, 11.5)

Conclusion

• This document serves as a comprehensive guide for the Mathematics Standard Level Test for Sixth Form entrance, summarizing key formulas and presenting a complete set of questions for practice and review.