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Term pack

Year 10 Maths

Name: ________________________

Skills in this pack

  • Compound interest
  • Surds and real numbers
  • Pair of linear equations
  • Heights and distances
  • Combined solids

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Year 10 Maths · 1 of 5

Compound interest

Work out compound interest and the amount it grows to, and use the same idea for growth and depreciation.

  1. 1.$2000 is invested at 10 per cent per year, compounded yearly, for 3 years. What amount does it grow to? Give just the number.

  2. 2.Find the compound interest on $5000 at 10 per cent per year for 2 years. Give just the number.

  3. 3.$4000 is lent at 10 per cent per year, compounded half-yearly, for 1 year. What amount is owed at the end? Give just the number.

  4. 4.Why does compound interest come to more than simple interest after the first year, at the same rate?

    • a) Because each year's interest is worked out on a larger amount
    • b) Because the sum invested is doubled at the start
    • c) Because the time is counted twice over
    • d) Because the rate goes up a little every year

  5. 5.Sort each description by which kind of interest it belongs to.

    Groups: Simple interest · Compound interest

    • The interest is the same amount every year
    • Doubling the time doubles the interest exactly
    • The amount grows faster and faster as the years go by
    • A population rising by the same percentage every year
    • Each year's interest is added on before the next year is worked out
    • Interest is always worked out on the original sum only

  6. 6.A town of 10000 people grows by 10 per cent every year. How many people live there after 2 years?

  7. 7.The simple interest on $4000 at 10 per cent for 2 years is $800. What is the compound interest on the same sum at the same rate and time? Give just the number.

  8. 8.Each card is a rate at which $1000 is invested for 2 years, compounded yearly. Put them in order by the amount at the end, smallest first.

    • 15 per cent per year
    • 20 per cent per year
    • 5 per cent per year
    • 10 per cent per year
    • 8 per cent per year

Compound interest · Year 10 Maths · www.arenapublications.com/learn

Year 10 Maths · 2 of 5

Surds and real numbers

Simplify, add and multiply surds, use fractional powers, and clear a surd out of a denominator.

  1. 1.What is 27 to the power 1/3?

  2. 2.√50 can be written as a√2. What is a?

  3. 3.What is √3 × √12?

  4. 4.Work each expression out, then sort it by whether the result is rational or irrational.

    Groups: Rational · Irrational

    • √3 + √3
    • √6 × √6
    • √5 × √5
    • √2 × √3
    • 2 + √7
    • √2 × √8

  5. 5.What is 8 to the power 2/3?

  6. 6.Put these numbers in order, smallest first.

    • √2
    • 3
    • √11
    • √5

  7. 7.To clear the surd from the denominator of 1/√5, multiply top and bottom by √5. The result is √5 over b. What is b?

  8. 8.For positive real numbers a and b, √a × √b is always equal to...

    • a) √a + √b
    • b) √(ab)
    • c) a√b
    • d) √(a + b)

Surds and real numbers · Year 10 Maths · www.arenapublications.com/learn

Year 10 Maths · 3 of 5

Pair of linear equations

Solve a pair of linear equations in two variables by substitution and elimination, and say how many solutions a pair has.

Before you start

One scale, and every box on it hides the same weight. Move things about and find out what a single scale can and cannot tell you.

113

A scale can pin down one unknown, never two at once — which is exactly why a pair of equations is needed. Elimination is the step that turns the pair into a single scale like this one.

  1. 1.The sum of two numbers is 25 and their difference is 7. What is the larger number?

  2. 2.How many solutions does the pair x + 2y = 4 and 2x + 4y = 12 have?

    • a) Exactly one
    • b) None
    • c) Exactly two
    • d) Infinitely many

  3. 3.Two adult tickets and one child ticket cost 900 $. One adult ticket and one child ticket cost 500 $. What does one adult ticket cost? Give just the number.

  4. 4.Solve the pair 3x + 2y = 12 and x = 2. What is y?

  5. 5.For which of these pairs do the two lines cross at exactly one point?

    • a) x + y = 5 and x + y = 8
    • b) 2x + 3y = 6 and 4x + 6y = 5
    • c) x + 2y = 5 and 2x + y = 4
    • d) x + y = 5 and 2x + 2y = 10

  6. 6.Solve the pair x + y = 10 and x - y = 4. What is y?

  7. 7.Sort each pair of equations by whether its two lines ever meet.

    Groups: They meet · They never meet

    • 2x + y = 7 and x + 3y = 6
    • 2x - y = 1 and x + 2y = 8
    • x + y = 4 and 2x + 2y = 9
    • 3x + 2y = 6 and 6x + 4y = 5
    • x + y = 4 and x - y = 2
    • x + 5y = 2 and 2x + 10y = 9

  8. 8.The sum of two numbers is 25 and their difference is 7. What is the smaller number?

Pair of linear equations · Year 10 Maths · www.arenapublications.com/learn

Year 10 Maths · 4 of 5

Heights and distances

Use angles of elevation and depression with the standard angles to find a height or a distance you cannot measure directly.

  1. 1.A ladder 10 m long leans against a wall and makes an angle of 60° with the ground. Take √3 = 1.73. How far up the wall does it reach, in metres?

  2. 2.From a window you look down at a car, and from the car someone looks up at the window. How do the angle of depression and the angle of elevation compare?

    • a) They add up to 90°
    • b) The angle of elevation is larger
    • c) They are equal
    • d) The angle of depression is larger

  3. 3.From the top of a cliff, the angle of depression of a boat is 30°. As the boat sails straight towards the base of the cliff, what happens to the angle of depression?

    • a) It increases
    • b) It stays the same
    • c) It becomes an angle of elevation
    • d) It decreases

  4. 4.A pole stands upright on level ground. From a point 10 m from its base, the angle of elevation of the top of the pole is 45°. How tall is the pole, in metres?

  5. 5.The angle of elevation of the top of a tower from a point on level ground is 60°, and the point is 45 m from the base of the tower. Take √3 = 1.73. How tall is the tower, in metres?

  6. 6.An upright pole casts a shadow. Put these angles of elevation of the Sun in order, shortest shadow first.

    • 45°
    • 75°
    • 30°
    • 60°

  7. 7.From the top of a cliff 40 m high, the angle of depression of a boat at sea is 45°. How far is the boat from the base of the cliff, in metres?

  8. 8.Someone at the top of a cliff is looking down at a boat. Tap the angle of depression.

    Write the number of the part.

    123

Heights and distances · Year 10 Maths · www.arenapublications.com/learn

Year 10 Maths · 5 of 5

Combined solids

Find the surface area and volume of a solid made by joining a cylinder, a cone or a hemisphere.

  1. 1.A hemisphere has a radius of 7 cm. Take π = 22/7. What is its curved surface area, in square centimetres?

  2. 2.Sort each formula by what it measures.

    Groups: A surface area · A volume

    • πrl
    • 4/3 πr³
    • 1/3 πr²h
    • 2πrh
    • 3πr²
    • πr²h

  3. 3.A toy is a cone standing on a hemisphere, both of radius 7 cm. The cone is 24 cm tall. What is the cone's slant height, in centimetres?

  4. 4.A cone, a hemisphere and a cylinder all have a radius of 7 cm, and the cone and the cylinder are both 7 cm tall. Which holds the most?

    • a) The hemisphere
    • b) The cone
    • c) The cylinder
    • d) They all hold the same

  5. 5.Match each solid to the formula for its curved surface area.

    • A sphere
    • A hemisphere
    • A cone
    • A cylinder
    • 2πrh
    • πrl
    • 2πr²
    • 4πr²

  6. 6.A sphere of radius 7 cm is cut exactly in half. Take π = 22/7. What is the total surface area of one hemisphere, counting its flat face, in square centimetres?

  7. 7.A cylinder of radius 7 cm and height 10 cm has a cone of radius 7 cm and slant height 25 cm standing on top of it. Its outside is the cone's curved surface, the cylinder's curved surface and the circular base. Take π = 22/7. What is the total surface area, in square centimetres?

  8. 8.A solid is a cylinder of radius 7 cm and height 10 cm with a cone of radius 7 cm and height 12 cm standing on top of it. Take π = 22/7. What is the volume of the cylinder part, in cubic centimetres?

Combined solids · Year 10 Maths · www.arenapublications.com/learn

Answer keys — Year 10 Maths

In the same order as the worksheets.

Compound interest

  1. 1. 2662
  2. 2. 1050
  3. 3. 4410
  4. 4. a) Because each year's interest is worked out on a larger amount
  5. 5. The interest is the same amount every year → Simple interest; Each year's interest is added on before the next year is worked out → Compound interest; Interest is always worked out on the original sum only → Simple interest; The amount grows faster and faster as the years go by → Compound interest; Doubling the time doubles the interest exactly → Simple interest; A population rising by the same percentage every year → Compound interest
  6. 6. 12100
  7. 7. 840
  8. 8. 1. 5 per cent per year 2. 8 per cent per year 3. 10 per cent per year 4. 15 per cent per year 5. 20 per cent per year

Surds and real numbers

  1. 1. 3
  2. 2. 5
  3. 3. 6
  4. 4. √2 × √8 → Rational; √3 + √3 → Irrational; √5 × √5 → Rational; 2 + √7 → Irrational; √6 × √6 → Rational; √2 × √3 → Irrational
  5. 5. 4
  6. 6. 1. √2 2. √5 3. 3 4. √11
  7. 7. 5
  8. 8. b) √(ab)

Pair of linear equations

  1. 1. 16
  2. 2. b) None
  3. 3. 400
  4. 4. 3
  5. 5. c) x + 2y = 5 and 2x + y = 4
  6. 6. 3
  7. 7. x + y = 4 and x - y = 2 → They meet; x + y = 4 and 2x + 2y = 9 → They never meet; 2x + y = 7 and x + 3y = 6 → They meet; 3x + 2y = 6 and 6x + 4y = 5 → They never meet; 2x - y = 1 and x + 2y = 8 → They meet; x + 5y = 2 and 2x + 10y = 9 → They never meet
  8. 8. 9

Heights and distances

  1. 1.
    • 8.65
    • 8.66
  2. 2. c) They are equal
  3. 3. a) It increases
  4. 4. 10
  5. 5.
    • 77.85
    • 77.94
  6. 6. 1. 75° 2. 60° 3. 45° 4. 30°
  7. 7. 40
  8. 8. 1 — The angle of depression at the cliff top

Combined solids

  1. 1. 308
  2. 2. 2πrh → A surface area; πr²h → A volume; 4/3 πr³ → A volume; πrl → A surface area; 3πr² → A surface area; 1/3 πr²h → A volume
  3. 3. 25
  4. 4. c) The cylinder
  5. 5. A sphere → 4πr²; A hemisphere → 2πr²; A cone → πrl; A cylinder → 2πrh
  6. 6. 462
  7. 7. 1144
  8. 8. 1540

Year 10 Maths · www.arenapublications.com/learn