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Term pack

Class 9 Maths

Name: ________________________

Skills in this pack

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

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Class 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.Write 3/8 as a decimal.

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

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

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

    Mark the line with an X.

    -5-4-3-2-1012345

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

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

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

    • 1.5
    • √2
    • 2
    • √3

  6. 6.Sort each number by whether it is rational or irrational.

    Groups: Rational · Irrational

    • √25
    • π
    • √7
    • 0.25
    • 3/7
    • √2

  7. 7.Which of these is irrational?

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

  8. 8.The decimal expansion of a rational number is always...

    • a) terminating, always
    • b) terminating or recurring
    • c) recurring, always
    • d) never-ending and never repeating

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

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

    Groups: Rational · Irrational

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

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

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

  3. 3.What is √3 × √12?

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

  5. 5.What is √5 × √5?

  6. 6.What is (√6 + √5)(√6 - √5)?

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

    • √2
    • √5
    • 3
    • √11

  8. 8.Match each surd to its simplest form.

    • √12
    • √18
    • √75
    • √32
    • √45
    • 2√3
    • 3√5
    • 5√3
    • 3√2
    • 4√2

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

Class 9 Maths · 3 of 16

Polynomials

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

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

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

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

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

  4. 4.Sort each polynomial by its degree.

    Groups: Linear · Quadratic

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

  5. 5.Match each polynomial to its degree.

    • 7
    • 4x + 1
    • x² - 3
    • 2x³ + x
    • x⁵
    • 0
    • 1
    • 3
    • 5
    • 2

  6. 6.What is the degree of the polynomial 3x⁴ + 2x - 7?

  7. 7.The factor theorem says that if p(k) = 0 then (x - k) is a factor of p(x). Given that p(2) = 0, what is k?

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

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

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

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

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

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

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

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

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

  6. 6.The x-axis itself is a line. Its equation is y = what?

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

  8. 8.Sort each equation by whether it is linear in two variables.

    Groups: Linear in two variables · Not linear in two variables

    • x³ = y
    • x + y = 0
    • y = 5x - 1
    • 2x + 3y = 6
    • xy = 12
    • x² + y = 4

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

Class 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 (3, 5) lie?

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

  2. 2.Sort each point by which axis it sits on.

    Groups: On the x-axis · On the y-axis

    • (0, -7)
    • (0, 5)
    • (0, 2)
    • (5, 0)
    • (-3, 0)
    • (9, 0)

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

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

  4. 4.Match each point to where it lies.

    • (2, 3)
    • (-2, 3)
    • (-2, -3)
    • (2, -3)
    • (0, 0)
    • Second quadrant
    • The origin
    • Fourth quadrant
    • First quadrant
    • Third quadrant

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

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

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

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

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

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

Class 9 Maths · 6 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 two distinct points?

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

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

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

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

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

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

  5. 5.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?

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

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

  7. 7.Match each of Euclid's statements to what he called it.

    • A point is that which has no part
    • A straight line may be drawn from any point to any other point
    • The whole is greater than the part
    • The angles of a triangle add up to 180°
    • Postulate
    • Definition
    • Axiom
    • Theorem

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

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

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

Class 9 Maths · 7 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.Sort each set of angles by whether they must be equal or must add up to 180°.

    Groups: Must be equal · Must add up to 180°

    • A linear pair
    • Alternate interior angles on parallel lines
    • Co-interior angles on parallel lines
    • Corresponding angles on parallel lines
    • Vertically opposite angles
    • The three angles of a triangle

  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.Match each description to the name of the angle pair.

    • Same side of the transversal, one inside the parallel lines and one outside
    • Opposite sides of the transversal, both inside the parallel lines
    • Same side of the transversal, both inside the parallel lines
    • Facing each other where two straight lines cross
    • Alternate interior angles
    • Co-interior angles
    • Corresponding angles
    • Vertically opposite angles

  5. 5.Parallel lines are cut by a transversal, and one of a pair of alternate interior angles is 73°. What is the other, in degrees?

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

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

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

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

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

Class 9 Maths · 8 of 16

Congruent triangles

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

  1. 1.Triangle ABC is congruent to triangle PQR. The area of ABC is 24 square centimetres. What is the area of PQR, in square centimetres?

  2. 2.Is knowing that all three pairs of angles are equal enough to prove two triangles congruent?

    • a) No — that only makes them the same shape, not the same size
    • b) Only for right-angled triangles
    • c) Only for isosceles triangles
    • d) Yes, always

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

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

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

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

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

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

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

  7. 7.Match each set of matching parts to the congruence rule it gives you.

    • Three pairs of sides
    • Two sides and the angle between them
    • Two angles and the side between them
    • Hypotenuse and one side, in right-angled triangles
    • RHS
    • SAS
    • ASA
    • SSS

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

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

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

Class 9 Maths · 9 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.The four angles of any quadrilateral add up to how many degrees?

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

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

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

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

  6. 6.The diagonals of a parallelogram cut each other in half. One diagonal is 14 cm long. How long is each of its two halves, in centimetres?

  7. 7.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?

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

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

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

Class 9 Maths · 10 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.A triangle has a base of 12 cm and a height of 7 cm. What is its area, in square centimetres?

  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 median splits a triangle into two smaller triangles of 18 square centimetres each. What is the area of the whole triangle, in square centimetres?

  5. 5.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
    • 30
    • 54
    • 20
    • 42
    • 14

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

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

  7. 7.A parallelogram has an area of 96 square centimetres on a base of 16 cm. What is its height, in centimetres?

  8. 8.A triangle and a parallelogram stand on the same base and between the same parallels. The parallelogram's area is 60 square centimetres. What is the triangle's area, in square centimetres?

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

Class 9 Maths · 11 of 16

Circles

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

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

  2. 2.An angle at the circumference standing on an arc is 35°. What is the angle at the centre standing on the same arc, in degrees?

  3. 3.A chord 8 cm long is drawn in a circle of radius 5 cm. Drop a perpendicular from the centre to the chord: the radius to the chord's end is the hypotenuse of the right-angled triangle it makes. How far is the chord from the centre, in centimetres?

  4. 4.The line from the centre of a circle to the mid-point of a chord that is not a diameter is always...

    • a) longer than the radius
    • b) perpendicular to the chord
    • c) the same length as the chord
    • d) parallel to the chord

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

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

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

  7. 7.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

  8. 8.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
    • The region between two radii and the arc joining them
    • The region trapped between a chord and its arc
    • The shorter of the two arcs a chord cuts off
    • A four-sided shape with every corner on the circle

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

Class 9 Maths · 12 of 16

Why constructions work

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

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

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

  3. 3.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?

  4. 4.Sort each set of measurements by whether it pins down exactly one triangle.

    Groups: Fixes exactly one triangle · Does not fix one triangle

    • Three sides: 5, 6 and 7
    • Two sides and the angle between them
    • Three angles: 60°, 60° and 60°
    • Two angles and the side between them
    • One side on its own
    • Three angles: 40°, 60° and 80°

  5. 5.Put these steps in order to build the perpendicular bisector of a segment AB.

    • Open the compasses to more than half of AB.
    • Keeping that same opening, put the point on B and draw two more arcs, crossing the first two.
    • With the point on A, draw an arc above AB and another below it.
    • Join the two crossing points with a straight line.

  6. 6.Swing an arc from a point, then from where it cuts the arm swing another arc with the compasses opened exactly as before, and join up. All three lengths come out equal, so the triangle is equilateral. What angle have you just built, in degrees?

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

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

  8. 8.Match each set of compass moves to what it produces.

    • Equal arcs above and below, swung from each end of a segment
    • Equal arcs swung from two marks on the arms of an angle
    • An arc of the same radius stepped once round from a point
    • A perpendicular raised on a line, then bisected
    • An angle of 60°
    • The perpendicular bisector of that segment
    • The bisector of that angle
    • An angle of 45°

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

Class 9 Maths · 13 of 16

Heron's formula

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

  1. 1.Heron's formula is the one to reach for when you know...

    • a) two sides and the angle between them
    • b) the perimeter and nothing else
    • c) all three sides but no height
    • d) the base and the height

  2. 2.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?

  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 5, 12 and 13 cm. What is its area, in square centimetres?

  5. 5.Each card gives a triangle's three sides. Put them in order of area, smallest first.

    • 6, 8 and 10
    • 9, 12 and 15
    • 3, 4 and 5
    • 13, 14 and 15

  6. 6.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

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

  7. 7.For a triangle with sides a, b and c, the semi-perimeter s is...

    • a) (a + b + c) ÷ 2
    • b) (a × b × c) ÷ 2
    • c) (a + b + c) ÷ 3
    • d) a + b + c

  8. 8.A triangle has sides of 13, 14 and 15 cm. What is its semi-perimeter, in centimetres?

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

Class 9 Maths · 14 of 16

Surface area and volume

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

  1. 1.Which of these gives the volume of a sphere?

    • a) (4/3) × π × r × r × r
    • b) 4 × π × r × r
    • c) 2 × π × r × r
    • d) (1/3) × π × r × r × h

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

  3. 3.A cone has a radius of 7 cm and a height of 24 cm. What is its slant height, in centimetres?

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

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

  5. 5.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?

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

  7. 7.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?

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

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

Class 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 mode of 2, 3, 3, 5, 7 and 3.

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

  3. 3.The mean of four numbers is 12. What do the four numbers add up to?

  4. 4.Find the median of 3, 7, 9, 11 and 15.

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

  6. 6.Match each word to what it tells you about a set of numbers.

    • Mean
    • Median
    • Mode
    • Range
    • The value that turns up most often
    • The gap between the largest and the smallest
    • The middle value, once in order
    • The total shared out equally

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

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

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

Class 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 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.

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

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

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

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

  4. 4.Over 200 school days a bus was late on 30 of them. What is the empirical probability that it is late? Give your answer as a decimal.

  5. 5.A die was rolled 120 times. Match each result to its empirical probability, written as a fraction.

    • A six came up 20 times
    • An even number came up 66 times
    • A one came up 15 times
    • A number above 4 came up 42 times
    • 66/120
    • 42/120
    • 15/120
    • 20/120

  6. 6.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

  7. 7.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.

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

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

Answer keys — Class 9 Maths

In the same order as the worksheets.

Number systems

  1. 1.
    • 0.375
    • .375
  2. 2. a) Yes — a whole number n can be written as n/1
  3. 3. -2
  4. 4. 12 → Whole number; -7 → Integer, but not a whole number; 5/6 → Rational, but not an integer; √11 → Irrational
  5. 5. 1. √2 2. 1.5 3. √3 4. 2
  6. 6. 3/7 → Rational; √2 → Irrational; 0.25 → Rational; π → Irrational; √25 → Rational; √7 → Irrational
  7. 7. b) √11
  8. 8. b) terminating or recurring

Surds and real numbers

  1. 1. √2 × √8 → Rational; √3 + √3 → Irrational; √5 × √5 → Rational; 2 + √7 → Irrational; √6 × √6 → Rational; √2 × √3 → Irrational
  2. 2. c) 4
  3. 3. 6
  4. 4. 4
  5. 5. 5
  6. 6. 1
  7. 7. 1. √2 2. √5 3. 3 4. √11
  8. 8. √12 → 2√3; √18 → 3√2; √75 → 5√3; √32 → 4√2; √45 → 3√5

Polynomials

  1. 1. b) quadratic polynomial
  2. 2. 1
  3. 3. 0
  4. 4. 3x + 2 → Linear; 5 - x → Linear; x → Linear; x² + 1 → Quadratic; 2x² - 7x → Quadratic; 9x² + 4 → Quadratic
  5. 5. 7 → 0; 4x + 1 → 1; x² - 3 → 2; 2x³ + x → 3; x⁵ → 5
  6. 6. 4
  7. 7. 2
  8. 8. 0

Linear equations in two variables

  1. 1. 6
  2. 2. b) Infinitely many
  3. 3. 1
  4. 4. 0
  5. 5. b) a straight line
  6. 6. 0
  7. 7. 12
  8. 8. 2x + 3y = 6 → Linear in two variables; x² + y = 4 → Not linear in two variables; y = 5x - 1 → Linear in two variables; xy = 12 → Not linear in two variables; x + y = 0 → Linear in two variables; x³ = y → Not linear in two variables

Coordinate geometry

  1. 1. c) First
  2. 2. (5, 0) → On the x-axis; (0, 5) → On the y-axis; (-3, 0) → On the x-axis; (0, -7) → On the y-axis; (9, 0) → On the x-axis; (0, 2) → On the y-axis
  3. 3. c) Fourth
  4. 4. (2, 3) → First quadrant; (-2, 3) → Second quadrant; (-2, -3) → Third quadrant; (2, -3) → Fourth quadrant; (0, 0) → The origin
  5. 5. 24
  6. 6. c) Second
  7. 7. 6
  8. 8. 7

Euclid's geometry

  1. 1. a) Exactly one
  2. 2. a) An assumption made about geometry in particular
  3. 3. b) Things which are equal to the same thing are equal to one another
  4. 4. 7
  5. 5. 7
  6. 6. c) Infinitely many
  7. 7. A point is that which has no part → Definition; A straight line may be drawn from any point to any other point → Postulate; The whole is greater than the part → Axiom; The angles of a triangle add up to 180° → Theorem
  8. 8. c) a theorem, proved from the postulates

Proving angle facts

  1. 1. Vertically opposite angles → Must be equal; A linear pair → Must add up to 180°; Alternate interior angles on parallel lines → Must be equal; Co-interior angles on parallel lines → Must add up to 180°; Corresponding angles on parallel lines → Must be equal; The three angles of a triangle → Must add up to 180°
  2. 2. 90
  3. 3. 60
  4. 4. Same side of the transversal, one inside the parallel lines and one outside → Corresponding angles; Opposite sides of the transversal, both inside the parallel lines → Alternate interior angles; Same side of the transversal, both inside the parallel lines → Co-interior angles; Facing each other where two straight lines cross → Vertically opposite angles
  5. 5. 73
  6. 6. 62
  7. 7. a) b
  8. 8. 47

Congruent triangles

  1. 1. 24
  2. 2. a) No — that only makes them the same shape, not the same size
  3. 3. b) RHS
  4. 4. a) SAS
  5. 5. a) ASA
  6. 6. 7
  7. 7. Three pairs of sides → SSS; Two sides and the angle between them → SAS; Two angles and the side between them → ASA; Hypotenuse and one side, in right-angled triangles → RHS
  8. 8. d) congruent, by SSS

Quadrilaterals

  1. 1. 115
  2. 2. 360
  3. 3. b) All four sides are equal
  4. 4. b) square
  5. 5. 8
  6. 6. 7
  7. 7. 6
  8. 8. c) parallel to the third side and half its length

Areas on the same base

  1. 1. 25
  2. 2. 42
  3. 3. 3 — The third triangle
  4. 4. 36
  5. 5. 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
  6. 6. a) equal areas
  7. 7. 6
  8. 8. 30

Circles

  1. 1. 110
  2. 2. 70
  3. 3. 3
  4. 4. b) perpendicular to the chord
  5. 5. 85
  6. 6. b) Their distances from the centre
  7. 7. 1 — The angle at the centre
  8. 8. 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

Why constructions work

  1. 1. 4
  2. 2. 15
  3. 3. 14
  4. 4. Three sides: 5, 6 and 7 → Fixes exactly one triangle; Two sides and the angle between them → Fixes exactly one triangle; Two angles and the side between them → Fixes exactly one triangle; Three angles: 60°, 60° and 60° → Does not fix one triangle; Three angles: 40°, 60° and 80° → Does not fix one triangle; One side on its own → Does not fix one triangle
  5. 5. 1. Open the compasses to more than half of AB. 2. With the point on A, draw an arc above AB and another below it. 3. Keeping that same opening, put the point on B and draw two more arcs, crossing the first two. 4. Join the two crossing points with a straight line.
  6. 6. 60
  7. 7. d) So each crossing point is equally far from both ends
  8. 8. Equal arcs above and below, swung from each end of a segment → The perpendicular bisector of that segment; Equal arcs swung from two marks on the arms of an angle → The bisector of that angle; An arc of the same radius stepped once round from a point → An angle of 60°; A perpendicular raised on a line, then bisected → An angle of 45°

Heron's formula

  1. 1. c) all three sides but no height
  2. 2. 4.8
  3. 3. 54
  4. 4. 30
  5. 5. 1. 3, 4 and 5 2. 6, 8 and 10 3. 9, 12 and 15 4. 13, 14 and 15
  6. 6. 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
  7. 7. a) (a + b + c) ÷ 2
  8. 8. 21

Surface area and volume

  1. 1. a) (4/3) × π × r × r × r
  2. 2. 2464
  3. 3. 25
  4. 4. Cuboid → length × width × height; Cylinder → π × r × r × h; Cone → (1/3) × π × r × r × h; Sphere → (4/3) × π × r × r × r
  5. 5. 550
  6. 6. 616
  7. 7. 616
  8. 8. 462

Mean, median and mode

  1. 1. 3
  2. 2. 8
  3. 3. 48
  4. 4. 9
  5. 5. 6
  6. 6. Mean → The total shared out equally; Median → The middle value, once in order; Mode → The value that turns up most often; Range → The gap between the largest and the smallest
  7. 7. 5
  8. 8. 8

Probability from experiments

  1. 1. 0.06
  2. 2. 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
  3. 3. 400
  4. 4. 0.15
  5. 5. A six came up 20 times → 20/120; An even number came up 66 times → 66/120; A one came up 15 times → 15/120; A number above 4 came up 42 times → 42/120
  6. 6. 0.7
  7. 7. 46/100
  8. 8. b) The one from 1000 tosses

Class 9 Maths · www.arenapublications.com/learn