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ATAR Maths

1.2 Exam Paper (Level 02)

More complex algebraic manipulations, higher-level trigonometry, and multi-step problem-solving questions. The questions will require deeper understanding and application of mathematical concepts.

1. Exam Paper 01

Year 10 Advanced Mathematics Exam

Time Allowed: 2 Hours
Total Marks: 100
Instructions:

  1. Write all your answers in the space provided.
  2. Show all working for full marks.
  3. Calculators are allowed.
  4. Check your answers carefully.

Section A: Multiple Choice (20 marks)

  1. Simplify the expression:

    a)
    b)
    c)
    d)
    (1 mark)

  2. Solve the equation for :

    a)
    b)
    c)
    d)
    (1 mark)

  3. The area of a sector of a circle with radius 10 cm and angle is:

    a) cm²
    b) cm²
    c) cm²
    d) cm²
    (1 mark)

  4. If , what is the value of at ?

    a) 10
    b) 4
    c) 12
    d) 8
    (1 mark)

  5. The probability of rolling a sum of 8 with two dice is:

    a)
    b)
    c)
    d)
    (1 mark)

Section B: Short Answer (40 marks)

  1. Simplify the expression and factor completely:
    (4 marks)

  2. Solve the simultaneous equations:
    (6 marks)

  3. A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Prove whether it is a right-angled triangle, and find the area.
    (5 marks)

  4. Calculate the exact value of using trigonometric identities.
    (4 marks)

  5. The sum of the roots of the quadratic equation is 7, and the product of the roots is 12. Find the values of , , and .
    (6 marks)

  6. In a geometric sequence, the 3rd term is 24, and the 6th term is 192. Find the first term and the common ratio.
    (5 marks)

2. Exam Paper 02

Year 10 Mathematics Examination Paper

Time Allowed: 2 hours

Instructions:

  1. Answer all questions.
  2. Write your answers in the space provided.
  3. Use a calculator where necessary.
  4. Show all working out.

Section A: Algebra and Functions

  1. Quadratic Equations
    Solve the quadratic equation using the quadratic formula. Show all steps.

  2. Polynomial Functions
    Consider the polynomial function .

    a) Find the roots of using synthetic division.
    b) Determine the local maxima and minima of the function.

  3. Logarithms
    Solve for in the equation . Provide a detailed solution.

Section B: Geometry and Trigonometry

  1. Circle Geometry
    In a circle with center , the angle at the center is and the radius is 10 cm.

    a) Find the length of the arc subtended by this angle.
    b) Find the area of the sector formed by this angle.

  2. Trigonometric Identities
    Prove the identity: using the unit circle definition. Then, use this identity to solve for in the equation .

  3. 3D Geometry
    Given a rectangular prism with dimensions , , and , find the length of the diagonal of the prism. Express your answer in terms of , , and .

Section C: Probability and Statistics

  1. Probability
    A bag contains 5 red, 7 blue, and 8 green marbles. If two marbles are drawn randomly without replacement, what is the probability that both marbles are of different colors?

  2. Statistics
    The following data represents the scores of students on a test: 55, 70, 65, 80, 90, 75, 85.

    a) Calculate the mean, median, and mode of the data set.
    b) Determine the standard deviation of the scores.

Section D: Advanced Applications

  1. Simultaneous Equations
    Solve the system of equations:

  2. Sequences and Series
    The first term of an arithmetic sequence is 4, and the common difference is 6.

    a) Find the 10th term of the sequence.
    b) Calculate the sum of the first 15 terms of the sequence.

3. Exam Paper 03

Year 10 Advanced Mathematics Exam

Duration: 2 hours
Instructions: Answer all questions. Show all working out for full marks.

Section A: Algebra and Functions

  1. Quadratic Functions

    • Solve the quadratic equation using the quadratic formula.
    • Given the quadratic function , find the vertex and the axis of symmetry.

  2. Exponential and Logarithmic Functions

    • Solve the equation for .
    • Given , find .

  3. Simultaneous Equations

    • Solve the following system of equations:

Section B: Geometry and Measurement

  1. Trigonometry

    • In triangle , , , and cm. Calculate the length of using the sine rule.

  2. Coordinate Geometry

    • Find the distance between the points and .
    • Determine the equation of the line passing through the points and .

  3. Volume and Surface Area

    • Calculate the volume of a cone with a radius of 4 cm and a height of 9 cm.
    • Find the surface area of a sphere with a radius of 7 cm.

Section C: Statistics and Probability

  1. Probability

    • A bag contains 5 red balls, 3 blue balls, and 2 green balls. If a ball is picked at random, what is the probability that it is either red or green?

  2. Statistics

    • The following data set represents the scores of 10 students on a test:
      • Calculate the mean, median, and mode of the scores.

Section D: Number and Algebra

  1. Complex Numbers

    • Simplify the expression and express the result in standard form.

  2. Sequences and Series

    • Find the 10th term of the arithmetic sequence where and the common difference .
    • Determine the sum of the first 8 terms of the geometric sequence where and the common ratio .

End of Exam

4. Exam Paper 04

Year 10 Advanced Mathematics Exam

Duration: 2 hours
Instructions: Answer all questions. Show all working out.

Section A: Algebra and Functions

(20 marks)

  1. Quadratic Equations
    Solve the following quadratic equations by factoring:

    a.
    b.

  2. Functions and Graphs
    Consider the function .

    a. Find the vertex of the parabola.
    b. Determine the x-intercepts and y-intercept.
    c. Sketch the graph of the function.

  3. Simultaneous Equations
    Solve the following simultaneous equations using the elimination method:

    a.
    b.

Section B: Geometry and Trigonometry

(20 marks)

  1. Geometry
    In a triangle ABC, angle A = 30°, angle B = 45°.
    a. Calculate angle C.
    b. If side AB = 7 cm and side AC = 10 cm, use the Law of Cosines to find the length of side BC.

  2. Trigonometry

    a. Solve for in the equation where .
    b. A ladder leans against a wall forming a 75° angle with the ground. If the ladder is 5 meters long, how high up the wall does it reach?

Section C: Probability and Statistics

(20 marks)

  1. Probability

    A bag contains 4 red, 6 blue, and 5 green marbles.
    a. What is the probability of drawing a red marble?
    b. If two marbles are drawn at random without replacement, what is the probability that both are blue?

  2. Statistics

    The following data set represents the scores of 10 students on a test: 45, 55, 60, 70, 75, 80, 85, 90, 95, 100.
    a. Calculate the mean score.
    b. Find the median score.
    c. Determine the mode of the data set.

Section D: Number and Algebra

(20 marks)

  1. Linear Algebra

    a. Express the following as a single fraction and simplify:
    b. Solve the inequality .

  2. Polynomials

    a. Perform polynomial long division on .
    b. Expand and simplify .

End of Exam

5. Exam Paper 05

Year 10 Advanced Mathematics Exam

Duration: 2 hours

Instructions:

  • Answer all questions.
  • Show all working out.
  • Use a scientific calculator where necessary.

Part A: Algebra and Functions (25 marks)

  1. Solve the following equations:

    a)
    b)
    c)

  2. Factorise the following expressions:

    a)
    b)

  3. Given the function , find:

    a)
    b) The vertex of the parabola

Part B: Geometry and Measurement (25 marks)

  1. In a right-angled triangle, one angle is and the hypotenuse is 10 cm. Find:

    a) The length of the side opposite the angle.
    b) The length of the side adjacent to the angle.

  2. The volume of a cylinder is cubic centimeters. If the height of the cylinder is 10 cm, find: a) The radius of the cylinder.
    b) The surface area of the cylinder.

Part C: Statistics and Probability (25 marks)

  1. The following data set represents the scores of 10 students in a math test: 56, 67, 78, 45, 89, 76, 90, 55, 70, 80. Find:

    a) The mean score.
    b) The median score.
    c) The range of the scores.

  2. A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random, find the probability that:

    a) The ball is red.
    b) The ball is not blue.

Part D: Calculus and Rates of Change (25 marks)

  1. Differentiate the following functions with respect to :

    a)
    b)

  2. Find the integral of the following functions with respect to :

    a)
    b)

  3. A car travels at a velocity described by m/s. Find the total distance traveled by the car from seconds to seconds.

End of Exam