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Term pack

Year 9 Maths

Name: ________________________

Skills in this pack

  • Number systems
  • Surds and real numbers
  • Polynomials
  • Linear equations in two variables
  • Coordinate geometry
  • Heron's formula
  • Surface area and volume
  • Euclid's geometry
  • Proving angle facts
  • Congruent triangles
  • Quadrilaterals
  • Areas on the same base
  • Circles
  • Why constructions work
  • Mean, median and mode
  • Probability from experiments

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Year 9 Maths · 1 of 16

Number systems

Tell a rational number from an irrational one, and place any real number on the number line.

Before you start

Every tick on this line is a rational number. Move along it and watch where √2 (about 1.41), √3 (about 1.73) and π (about 3.14) would have to squeeze in.

0.00.51.01.52.02.53.03.54.0

No tick will ever land exactly on √2, however finely you chop the line up. That is what irrational means — not that the number is missing, but that no fraction can name it.

  1. 1.√25 turns out to be rational. Which whole number is it equal to?

  2. 2.Write the recurring decimal 0.777… as a fraction in simplest form.

  3. 3.Write 3/8 as a decimal.

  4. 4.Every square root of a perfect square is rational. Tap where -√4 sits.

    Mark the line with an X.

    -5-4-3-2-1012345

  5. 5.√2 is 1.414 to three decimal places. Tap the tick it is nearest to.

    Mark the line with an X.

    1.01.52.0

  6. 6.Which of these is irrational?

    • a) 0.6
    • b) √36
    • c) 2/9
    • d) √11

  7. 7.Is every whole number also a rational number?

    • a) Only when it is positive
    • b) Only when it is even
    • c) No — whole numbers are a separate kind of number
    • d) Yes — a whole number n can be written as n/1

  8. 8.Match each number to the smallest family it belongs to.

    • 12
    • -7
    • 5/6
    • √11
    • Rational, but not an integer
    • Irrational
    • Whole number
    • Integer, but not a whole number

Number systems · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 2 of 16

Surds and real numbers

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

  1. 1.What is √5 × √5?

  2. 2.What is (√6 + √5)(√6 - √5)?

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

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

  4. 4.Which of these is equal to 16 to the power 1/2?

    • a) 8
    • b) 256
    • c) 32
    • d) 4

  5. 5.3√7 + 4√7 can be written as a√7. What is a?

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

    • √5
    • 3
    • √2
    • √11

  7. 7.What is √3 × √12?

  8. 8.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?

Surds and real numbers · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 3 of 16

Polynomials

Find the degree of a polynomial, evaluate it at a value and recognise its zeroes.

  1. 1.If p(x) = x² - 5x + 6, what is p(3)?

  2. 2.If p(x) = x² - 3x + 2, what is p(2)?

  3. 3.What is the degree of the polynomial 5x + 1?

  4. 4.If p(x) = 2x + 6, what value of x makes p(x) = 0?

  5. 5.Sort each polynomial by its degree.

    Groups: Linear · Quadratic

    • x
    • 5 - x
    • x² + 1
    • 9x² + 4
    • 3x + 2
    • 2x² - 7x

  6. 6.If p(x) = x³ - 2x² + x, what is p(1)?

  7. 7.A polynomial of degree 2 is called a...

    • a) constant polynomial
    • b) cubic polynomial
    • c) linear polynomial
    • d) quadratic polynomial

  8. 8.Which of these is a zero of p(x) = x² - 9?

    • a) 3
    • b) -9
    • c) 9
    • d) 0

Polynomials · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 4 of 16

Linear equations in two variables

Find solutions of an equation like 2x + y = 7 and read its graph as a straight line.

  1. 1.The graph of every linear equation in two variables is...

    • a) a curve
    • b) a straight line
    • c) a single point
    • d) a circle

  2. 2.Written in the form ax + by + c = 0, the equation 3x = 5 has a = 3 and c = -5. What is b?

  3. 3.In the equation 2x + y = 7, what is y when x = 3?

  4. 4.Each pair of coordinates solves exactly one of these equations. Match them up.

    • (1, 4)
    • (2, 6)
    • (0, 7)
    • (3, 3)
    • x = y
    • x + y = 5
    • y = 3x
    • y = 7

  5. 5.Is (2, 3) a solution of x + y = 5?

    • a) No — it solves x - y = 5 instead
    • b) Yes — 2 + 3 comes to 5
    • c) Only if x and y are swapped over
    • d) No — no pair of numbers can solve it

  6. 6.How many solutions does a linear equation in two variables have?

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

  7. 7.Two numbers add up to 12. Calling them x and y, the equation is x + y = c. What is c?

  8. 8.The line 2x + y = 6 crosses the y-axis. Give the y-coordinate of that crossing point.

Linear equations in two variables · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 5 of 16

Coordinate geometry

Read and plot points on the coordinate plane and name the quadrant they lie in.

Before you start

The two axes cut the page into four quarters. Find out what each one is called — all four, in any order, nothing to get right.

1234
  1. 1. First quadrant
  2. 2. Second quadrant
  3. 3. Third quadrant
  4. 4. Fourth quadrant

They are numbered anticlockwise from the top right. Both coordinates are positive in the first and both are negative in the third.

  1. 1.In which quadrant does the point (5, -1) lie?

    • a) Third
    • b) Fourth
    • c) First
    • d) Second

  2. 2.A rectangle has corners at (0, 0), (6, 0), (6, 4) and (0, 4). What is its area, in square units?

  3. 3.Where does the point (0, 4) lie?

    • a) On the y-axis
    • b) In the first quadrant
    • c) At the origin
    • d) On the x-axis

  4. 4.How far is the point (6, 8) from the y-axis? Give just the number.

  5. 5.In which quadrant does the point (-4, 2) lie?

    • a) Second
    • b) Fourth
    • c) First
    • d) Third

  6. 6.What is the x-coordinate of the point (7, -3)?

  7. 7.In which quadrant does the point (3, 5) lie?

    • a) First
    • b) Second
    • c) Third
    • d) Fourth

  8. 8.In which quadrant does the point (-2, -6) lie?

    • a) Fourth
    • b) First
    • c) Second
    • d) Third

Coordinate geometry · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 6 of 16

Heron's formula

Find the area of a triangle from its three sides, without knowing its height.

  1. 1.A triangle with sides of 6, 8 and 10 cm has an area of 24 square cm. Taking the side of 10 cm as the base, what is the height, in centimetres?

  2. 2.A triangle has sides of 5, 12 and 13 cm. What is its area, in square centimetres?

  3. 3.A triangle has sides of 9, 12 and 15 cm. What is its area, in square centimetres?

  4. 4.A triangle has sides of 13, 14 and 15 cm. What is its area, in square centimetres?

  5. 5.A triangle has sides of 7, 24 and 25 cm. What is its area, in square centimetres?

  6. 6.A triangle has sides of 3, 4 and 5 cm. What is its semi-perimeter, in centimetres?

  7. 7.Sort each set of three lengths by whether a triangle can be built from them at all.

    Groups: A triangle is possible · No triangle is possible

    • 4, 4 and 20
    • 3, 4 and 5
    • 1, 2 and 3
    • 5, 12 and 13
    • 7, 24 and 25
    • 2, 3 and 9

  8. 8.A triangle has sides of 3, 4 and 5 cm. What is its area, in square centimetres?

Heron's formula · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 7 of 16

Surface area and volume

Find surface areas and volumes of cones, spheres and hemispheres.

  1. 1.A cone has a radius of 7 cm and a slant height of 25 cm. Take π = 22/7. What is its curved surface area, in square centimetres?

  2. 2.Which of these gives the volume of a cone?

    • a) (4/3) × π × r × r × r
    • b) π × r × r × h
    • c) π × r × l
    • d) (1/3) × π × r × r × h

  3. 3.Match each solid to the formula for its volume.

    • Cuboid
    • Cylinder
    • Cone
    • Sphere
    • (4/3) × π × r × r × r
    • π × r × r × h
    • (1/3) × π × r × r × h
    • length × width × height

  4. 4.A cone has a radius of 7 cm and a height of 12 cm. Take π = 22/7. What is its volume, in cubic centimetres?

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

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

  7. 7.A sphere has a radius of 21 cm. Take π = 22/7. What is its volume, in cubic centimetres?

  8. 8.A sphere has a radius of 14 cm. Take π = 22/7. What is its surface area, in square centimetres?

Surface area and volume · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 8 of 16

Euclid's geometry

Tell an axiom from a postulate, and use Euclid's first assumptions to justify a step.

Before you start

Four drawings, and the only difference between them is where they stop. Find out what each one is called.

1234
  1. 1. A point — no length at all
  2. 2. A line segment — an end at each side
  3. 3. A ray — one end, and no end the other way
  4. 4. A straight line — no ends at all

A point has no length, a segment has two ends, a ray has one and a line has none. Euclid's opening definitions are this list and very little else — everything afterwards is built out of these four.

  1. 1.How many straight lines can be drawn through a single point?

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

  2. 2.Sort each statement by whether Euclid assumed it or proved it.

    Groups: Assumed without proof · Proved from the others

    • The base angles of an isosceles triangle are equal
    • The whole is greater than the part
    • All right angles are equal to one another
    • A straight line may be drawn from any point to any other point
    • Two distinct lines cannot have more than one point in common
    • The angles of a triangle add up to 180°

  3. 3.If equals are subtracted from equals, the remainders are equal. Taking 8 from both sides of y + 8 = 15 gives what value of y?

  4. 4.Euclid set all of this out in thirteen books, collected under one title. What is that title? One word.

  5. 5."Two distinct straight lines cannot have more than one point in common." In Euclid's scheme this is...

    • a) one of the axioms
    • b) one of the five postulates
    • c) a definition
    • d) a theorem, proved from the postulates

  6. 6.Euclid's axiom says the whole is greater than the part. A segment 12 units long is cut into two parts and one of them is 5 units. How long is the other, in units?

  7. 7.In Euclid's system, what is a postulate?

    • a) A word being given a meaning
    • b) A statement proved from earlier ones
    • c) An assumption made about geometry in particular
    • d) A measurement taken off a drawing

  8. 8.If a = b and b = c, which of Euclid's axioms lets you write a = c?

    • a) Things which are equal to the same thing are equal to one another
    • b) If equals are added to equals, the wholes are equal
    • c) The whole is greater than the part
    • d) Things which coincide with one another are equal

Euclid's geometry · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 9 of 16

Proving angle facts

Use the linear-pair axiom to prove vertically opposite angles equal, and apply the angle-sum and exterior-angle theorems.

Before you start

One side of this triangle has been carried straight on past the corner B. Find out what each of the four angles here is called.

1234
  1. 1. The interior angle at A
  2. 2. The interior angle at B
  3. 3. The interior angle at C
  4. 4. The exterior angle at B

The exterior angle at B equals the two interior angles it does not touch — the ones at A and C — added together. That is the exterior angle theorem, and it follows straight from the angle sum.

  1. 1.Two straight lines cross, making four angles: one at the top, one at the bottom, one on the left and one on the right. Tap the angle vertically opposite the top one.

    Write the number of the part.

    1234

  2. 2.The three angles of a triangle are in the ratio 1 : 2 : 3. How big is the largest, in degrees?

  3. 3.Parallel lines are cut by a transversal. Two co-interior angles measure 2x degrees and x degrees. What is x?

  4. 4.The exterior angle theorem says an exterior angle of a triangle is equal to...

    • a) the largest interior angle
    • b) the two interior angles opposite it, added together
    • c) the interior angle next to it
    • d) half of 180°

  5. 5.Two angles form a linear pair. One of them is 118°. What is the other, in degrees?

  6. 6.Two straight lines cross. One of the four angles is 47°. What is the angle vertically opposite it, in degrees?

  7. 7.An exterior angle of a triangle is 110°, and one of the two interior angles opposite it is 45°. What is the other one, in degrees?

  8. 8.The proof that vertically opposite angles are equal reaches the line a + b = b + c. What is taken from both sides next?

    • a) a
    • b) b
    • c) 180
    • d) c

Proving angle facts · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 10 of 16

Congruent triangles

Decide when two triangles are congruent using SSS, SAS, ASA and RHS.

  1. 1.In an isosceles triangle, the line from the apex to the midpoint of the base splits it into two triangles that are...

    • a) congruent, by SSS
    • b) congruent, by AAA
    • c) the same shape but different sizes
    • d) not related to each other

  2. 2.Two right-angled triangles have equal hypotenuses and one pair of equal shorter sides. Which rule proves they are congruent?

    • a) ASA
    • b) SSS
    • c) RHS
    • d) SAS

  3. 3.Triangle ABC is congruent to triangle PQR. Angle A is 55°. How big is angle P, in degrees?

  4. 4.Two triangles have two pairs of angles equal, and the side between those angles is equal too. Which rule proves they are congruent?

    • a) SSS
    • b) ASA
    • c) SAS
    • d) RHS

  5. 5.Triangle ABC is congruent to triangle PQR. AB is 7 cm long. How long is PQ, in centimetres?

  6. 6.Sort each set of matching parts by whether it is enough to prove congruence.

    Groups: Proves congruence · Does not prove congruence

    • SAS
    • ASA
    • RHS
    • SSA
    • AAA
    • SSS

  7. 7.Two triangles have two pairs of sides equal, and the angle between those sides is equal too. Which rule proves they are congruent?

    • a) RHS
    • b) SSS
    • c) SAS
    • d) ASA

  8. 8.Two triangles have all three pairs of sides equal. Which rule proves they are congruent?

    • a) SSS
    • b) SAS
    • c) ASA
    • d) RHS

Congruent triangles · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 11 of 16

Quadrilaterals

Use the properties of a parallelogram and the mid-point theorem.

Before you start

A parallelogram with one diagonal drawn across it. Find the two pieces the diagonal makes, and the diagonal itself.

123
  1. 1. One of the two triangles
  2. 2. The other triangle — the same size and shape, turned round
  3. 3. The diagonal that makes them

Turn either triangle half a turn about the middle of the diagonal and it lands exactly on the other one. Opposite sides equal and opposite angles equal both fall straight out of that single fact.

  1. 1.One angle of a parallelogram is 65°. How big is the angle next to it, in degrees?

  2. 2.One side of a parallelogram is 8 cm long. How long is the side opposite it, in centimetres?

  3. 3.In triangle ABC, D is the mid-point of AB and E is the mid-point of AC. BC is 12 cm long. How long is DE, in centimetres?

  4. 4.A quadrilateral has diagonals that are equal and cut each other in half at right angles. What is its most specific name?

    • a) rectangle
    • b) square
    • c) rhombus
    • d) parallelogram

  5. 5.In triangle PQR, the segment joining the mid-points of PQ and PR is 9 cm long. How long is QR, in centimetres?

  6. 6.The mid-point theorem says the segment joining the mid-points of two sides of a triangle is...

    • a) perpendicular to the third side
    • b) equal in length to the third side
    • c) parallel to the third side and half its length
    • d) twice the length of the third side

  7. 7.Sort each property by whether every parallelogram has it.

    Groups: Every parallelogram · Not every parallelogram

    • Opposite sides are equal
    • The diagonals are equal in length
    • The diagonals cut each other in half
    • Every angle is a right angle
    • All four sides are equal
    • Opposite angles are equal

  8. 8.Which of these is true of every rhombus but not of every parallelogram?

    • a) Opposite angles are equal
    • b) All four sides are equal
    • c) Opposite sides are parallel
    • d) The diagonals cut each other in half

Quadrilaterals · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 12 of 16

Areas on the same base

Use the fact that figures on the same base and between the same parallels have equal areas.

  1. 1.Two triangles stand on the same base and between the same parallels. One has an area of 25 square centimetres. What is the area of the other, in square centimetres?

  2. 2.Two parallelograms on the same base and between the same parallels always have...

    • a) equal perimeters
    • b) equal diagonals
    • c) equal areas
    • d) the same shape

  3. 3.All four triangles have bases of the same length on the lower line, and three of them have their tips on the upper line. Tap the triangle with the smallest area.

    Write the number of the part.

    1234

  4. 4.A diagonal splits a parallelogram of area 90 square centimetres into two triangles. What is the area of each, in square centimetres?

  5. 5.Two figures stand on the same base and between the same parallels. Sort each quantity by whether it must be the same for both.

    Groups: Must be the same · Need not be the same

    • The length of their base
    • Their shapes
    • Their areas, if both are parallelograms
    • Their perimeters
    • Their heights
    • The lengths of their slanting sides

  6. 6.A parallelogram and a rectangle stand on the same base and between the same pair of parallel lines. The rectangle's area is 48 square centimetres. What is the parallelogram's area, in square centimetres?

  7. 7.Each triangle's base and height are given in centimetres. Match it to its area in square centimetres.

    • base 10, height 6
    • base 8, height 5
    • base 12, height 9
    • base 7, height 4
    • base 14, height 6
    • 54
    • 20
    • 42
    • 14
    • 30

  8. 8.A triangle has a base of 12 cm and a height of 7 cm. What is its area, in square centimetres?

Areas on the same base · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 13 of 16

Circles

Use the chord, arc and cyclic-quadrilateral facts to find angles and lengths in a circle.

  1. 1.Match each word to what it names in a circle.

    • Major arc
    • Minor arc
    • Segment
    • Sector
    • Cyclic quadrilateral
    • The longer of the two arcs a chord cuts off
    • A four-sided shape with every corner on the circle
    • The shorter of the two arcs a chord cuts off
    • The region trapped between a chord and its arc
    • The region between two radii and the arc joining them

  2. 2.The chord AB has an angle drawn on it at the centre of the circle, and another at the point C up on the circle. Tap the angle at the centre.

    Write the number of the part.

    12

  3. 3.In cyclic quadrilateral ABCD, angle A is 70° and angle B is 100°. How big is angle C, in degrees?

  4. 4.The longest chord that can be drawn in a circle has a name of its own. What is it? One word.

  5. 5.Two chords of a circle are equal in length. What else must be equal?

    • a) Their distances from the centre
    • b) Their distances from each other
    • c) The two arcs on either side of each chord
    • d) Their angles at every point of the circle

  6. 6.An angle drawn in a semicircle is always how many degrees?

  7. 7.One angle of a cyclic quadrilateral is 95°. How big is the angle opposite it, in degrees?

  8. 8.Sort each statement about a circle by whether it is always true.

    Groups: Always true · Not always true

    • All the chords of a circle are the same length
    • The perpendicular from the centre to a chord cuts the chord in half
    • The angle in a semicircle is a right angle
    • The angle at the centre equals the angle at the circumference on the same arc
    • Opposite angles of a cyclic quadrilateral add up to 180°
    • Every chord passes through the centre

Circles · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 14 of 16

Why constructions work

Explain what each compass step guarantees, and decide when a triangle can be constructed at all.

  1. 1.Why must the compass opening stay the same when you swing arcs from each end of a segment?

    • a) So each crossing point is equally far from both ends
    • b) So the finished line comes out the same length as the segment
    • c) So each arc reaches both ends of the segment
    • d) So the finished line comes out horizontal

  2. 2.Two sides of a triangle are 6 and 9 cm. What is the smallest whole number the third side could be, in centimetres?

  3. 3.To bisect an angle you mark equal distances along both arms, then swing equal arcs from those two marks. The two triangles this makes are congruent by which rule?

    • a) AAA
    • b) ASA
    • c) SSA
    • d) SSS

  4. 4.Why does the line through those two crossing points pass through the mid-point of the segment?

    • a) Because both arcs share a centre
    • b) Because it is the longest line that can be drawn
    • c) Because it is drawn vertically
    • d) Every point on it is the same distance from both ends

  5. 5.An angle of 60° is bisected, and one of the halves is then bisected again. How big is the smallest angle now, in degrees?

  6. 6.Two sides of a triangle differ by 9 cm and the base is 7 cm. The difference of two sides must be less than the third side for the triangle to exist. Can it be built? Answer yes or no.

  7. 7.A triangle can be built only if any two of its sides add up to more than the third. Two sides are 6 and 9 cm. What is the largest whole number the third side could be, in centimetres?

  8. 8.A triangle is to be built on a base of 8 cm, with the other two sides adding up to 12 cm. What would its perimeter be, in centimetres?

Why constructions work · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 15 of 16

Mean, median and mode

Find the mean, median, mode and range of a small set of numbers.

  1. 1.Find the median of 2, 4, 6 and 8.

  2. 2.Five children scored 6, 7, 8, 9 and 10. What is the mean score?

  3. 3.Find the mode of 5, 8, 8, 2, 9, 8 and 1.

  4. 4.Which average is the middle value once the data has been put in order?

    • a) The median
    • b) The mean
    • c) The mode
    • d) The range

  5. 5.Find the mean of 10, 20, 30 and 40.

  6. 6.Find the mean of 4, 8, 6, 10 and 2.

  7. 7.Find the range of 12, 4, 9 and 20.

  8. 8.Find the median of 12, 5, 9, 3 and 7.

Mean, median and mode · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 16 of 16

Probability from experiments

Work out the empirical probability of an event from the results a trial actually gave.

Before you start

Every probability there has ever been lives somewhere on this line. Move along it and read off the values.

0.00.51.0

Zero means it never happened and one means it happened every single time. Nothing sits outside the two ends — a probability of 1.4 or of -0.2 is not unlikely, it is impossible.

  1. 1.A die is rolled 60 times and shows a six on 12 of them. What is the empirical probability of a six? Give your answer as a fraction.

  2. 2.A coin is tossed 100 times and lands heads 54 times. What is the empirical probability of a head? Give your answer as a fraction.

  3. 3.An event has an empirical probability of 0.7. Tap where that sits on the probability line.

    Mark the line with an X.

    0.00.51.0

  4. 4.A batch of 50 bulbs was tested and 3 of them were faulty. What is the empirical probability that a bulb is faulty? Give your answer as a decimal.

  5. 5.In a trial, seeds germinated with an empirical probability of 0.8. Out of 500 more seeds, how many would you expect to germinate?

  6. 6.Sort each number by whether it could possibly be a probability.

    Groups: Could be a probability · Could not be a probability

    • 0.25
    • 7/5
    • -0.2
    • 1.4
    • 1
    • 0

  7. 7.A coin gives 7 heads in 10 tosses, and 503 heads in 1000 tosses. Which empirical probability is closer to the theoretical one half?

    • a) They are exactly as close as each other
    • b) Neither is anywhere near
    • c) The one from 10 tosses
    • d) The one from 1000 tosses

  8. 8.A coin is tossed 100 times and lands heads 54 times. What is the empirical probability of a tail? Give your answer as a fraction.

Probability from experiments · Year 9 Maths · www.arenapublications.com/learn

Answer keys — Year 9 Maths

In the same order as the worksheets.

Number systems

  1. 1. 5
  2. 2. 7/9
  3. 3.
    • 0.375
    • .375
  4. 4. -2
  5. 5. 1.4
  6. 6. d) √11
  7. 7. d) Yes — a whole number n can be written as n/1
  8. 8. 12 → Whole number; -7 → Integer, but not a whole number; 5/6 → Rational, but not an integer; √11 → Irrational

Surds and real numbers

  1. 1. 5
  2. 2. 1
  3. 3. d) √(ab)
  4. 4. d) 4
  5. 5. 7
  6. 6. 1. √2 2. √5 3. 3 4. √11
  7. 7. 6
  8. 8. 5

Polynomials

  1. 1. 0
  2. 2. 0
  3. 3. 1
  4. 4. -3
  5. 5. 3x + 2 → Linear; 5 - x → Linear; x → Linear; x² + 1 → Quadratic; 2x² - 7x → Quadratic; 9x² + 4 → Quadratic
  6. 6. 0
  7. 7. d) quadratic polynomial
  8. 8. a) 3

Linear equations in two variables

  1. 1. b) a straight line
  2. 2. 0
  3. 3. 1
  4. 4. (1, 4) → x + y = 5; (2, 6) → y = 3x; (0, 7) → y = 7; (3, 3) → x = y
  5. 5. b) Yes — 2 + 3 comes to 5
  6. 6. b) Infinitely many
  7. 7. 12
  8. 8. 6

Coordinate geometry

  1. 1. b) Fourth
  2. 2. 24
  3. 3. a) On the y-axis
  4. 4. 6
  5. 5. a) Second
  6. 6. 7
  7. 7. a) First
  8. 8. d) Third

Heron's formula

  1. 1. 4.8
  2. 2. 30
  3. 3. 54
  4. 4. 84
  5. 5. 84
  6. 6. 6
  7. 7. 3, 4 and 5 → A triangle is possible; 1, 2 and 3 → No triangle is possible; 5, 12 and 13 → A triangle is possible; 2, 3 and 9 → No triangle is possible; 7, 24 and 25 → A triangle is possible; 4, 4 and 20 → No triangle is possible
  8. 8. 6

Surface area and volume

  1. 1. 550
  2. 2. d) (1/3) × π × r × r × h
  3. 3. Cuboid → length × width × height; Cylinder → π × r × r × h; Cone → (1/3) × π × r × r × h; Sphere → (4/3) × π × r × r × r
  4. 4. 616
  5. 5. 462
  6. 6. 308
  7. 7. 38808
  8. 8. 2464

Euclid's geometry

  1. 1. a) Infinitely many
  2. 2. A straight line may be drawn from any point to any other point → Assumed without proof; All right angles are equal to one another → Assumed without proof; The whole is greater than the part → Assumed without proof; The angles of a triangle add up to 180° → Proved from the others; Two distinct lines cannot have more than one point in common → Proved from the others; The base angles of an isosceles triangle are equal → Proved from the others
  3. 3. 7
  4. 4.
    • Elements
    • The Elements
  5. 5. d) a theorem, proved from the postulates
  6. 6. 7
  7. 7. c) An assumption made about geometry in particular
  8. 8. a) Things which are equal to the same thing are equal to one another

Proving angle facts

  1. 1. 3 — The bottom angle
  2. 2. 90
  3. 3. 60
  4. 4. b) the two interior angles opposite it, added together
  5. 5. 62
  6. 6. 47
  7. 7. 65
  8. 8. b) b

Congruent triangles

  1. 1. a) congruent, by SSS
  2. 2. c) RHS
  3. 3. 55
  4. 4. b) ASA
  5. 5. 7
  6. 6. SSS → Proves congruence; SAS → Proves congruence; ASA → Proves congruence; RHS → Proves congruence; AAA → Does not prove congruence; SSA → Does not prove congruence
  7. 7. c) SAS
  8. 8. a) SSS

Quadrilaterals

  1. 1. 115
  2. 2. 8
  3. 3. 6
  4. 4. b) square
  5. 5. 18
  6. 6. c) parallel to the third side and half its length
  7. 7. Opposite sides are equal → Every parallelogram; Opposite angles are equal → Every parallelogram; The diagonals cut each other in half → Every parallelogram; All four sides are equal → Not every parallelogram; The diagonals are equal in length → Not every parallelogram; Every angle is a right angle → Not every parallelogram
  8. 8. b) All four sides are equal

Areas on the same base

  1. 1. 25
  2. 2. c) equal areas
  3. 3. 3 — The third triangle
  4. 4. 45
  5. 5. Their heights → Must be the same; The length of their base → Must be the same; Their areas, if both are parallelograms → Must be the same; Their perimeters → Need not be the same; The lengths of their slanting sides → Need not be the same; Their shapes → Need not be the same
  6. 6. 48
  7. 7. base 10, height 6 → 30; base 8, height 5 → 20; base 12, height 9 → 54; base 7, height 4 → 14; base 14, height 6 → 42
  8. 8. 42

Circles

  1. 1. Major arc → The longer of the two arcs a chord cuts off; Minor arc → The shorter of the two arcs a chord cuts off; Segment → The region trapped between a chord and its arc; Sector → The region between two radii and the arc joining them; Cyclic quadrilateral → A four-sided shape with every corner on the circle
  2. 2. 1 — The angle at the centre
  3. 3. 110
  4. 4. diameter
  5. 5. a) Their distances from the centre
  6. 6. 90
  7. 7. 85
  8. 8. The perpendicular from the centre to a chord cuts the chord in half → Always true; The angle in a semicircle is a right angle → Always true; Opposite angles of a cyclic quadrilateral add up to 180° → Always true; Every chord passes through the centre → Not always true; The angle at the centre equals the angle at the circumference on the same arc → Not always true; All the chords of a circle are the same length → Not always true

Why constructions work

  1. 1. a) So each crossing point is equally far from both ends
  2. 2. 4
  3. 3. d) SSS
  4. 4. d) Every point on it is the same distance from both ends
  5. 5. 15
  6. 6. no
  7. 7. 14
  8. 8. 20

Mean, median and mode

  1. 1. 5
  2. 2. 8
  3. 3. 8
  4. 4. a) The median
  5. 5. 25
  6. 6. 6
  7. 7. 16
  8. 8. 7

Probability from experiments

  1. 1. 12/60
  2. 2. 54/100
  3. 3. 0.7
  4. 4. 0.06
  5. 5. 400
  6. 6. 0.25 → Could be a probability; 1.4 → Could not be a probability; 0 → Could be a probability; -0.2 → Could not be a probability; 1 → Could be a probability; 7/5 → Could not be a probability
  7. 7. d) The one from 1000 tosses
  8. 8. 46/100

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