Year 11 Maths

1. 1.Euclid's division lemma says that for whole numbers a and b there are whole numbers q and r with a = bq + r, where r is smaller than b. Divide 455 by 42. What is the remainder?

2. 2.Which of these numbers is irrational?

   • a) √2
   • b) 7/2
   • c) 0.25
   • d) √4

3. 3.A number is written as 2³ × 3² × 5. What is the number?

4. 4.Find the HCF of 6, 72 and 120.

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

   Groups: Rational · Irrational

   • 0.75
   • √3
   • √5
   • √7
   • 22/7
   • √9

6. 6.Find the LCM of 12 and 18.

7. 7.Find the HCF of 616 and 32 using Euclid's algorithm.

8. 8.For any two numbers, HCF × LCM equals the product of the numbers. The HCF of two numbers is 9 and their LCM is 360. One of the numbers is 45. What is the other?

Real numbers · Year 11 Maths · www.arenapublications.com/learn

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

2. 2.Sort each polynomial by its degree.

   Groups: Linear · Quadratic

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

3. 3.Match each polynomial to its degree.

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

   • 5
   • 3
   • 1
   • 0
   • 2

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

5. 5.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?

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

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

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

Polynomials · Year 11 Maths · www.arenapublications.com/learn

1. 1.Find the sum of the first 10 terms of 2, 4, 6, 8, ...

2. 2.In the AP 3, 7, 11, 15, ... what is the common difference?

3. 3.Find the sum of the first 15 terms of 1, 3, 5, 7, ...

4. 4.In the AP 20, 17, 14, ... what is the common difference?

5. 5.In the AP 5, 8, 11, ... what is the 20th term?

6. 6.Match each progression to its common difference.

   • 2, 5, 8, 11
   • 1, 3, 5, 7
   • 10, 6, 2, -2
   • 7, 8, 9, 10
   • 0, 10, 20, 30

   • -4
   • 3
   • 2
   • 1
   • 10

7. 7.Sort each list by whether it is an arithmetic progression.

   Groups: Is an AP · Is not an AP

   • 10, 20, 30, 40
   • 9, 6, 3, 0
   • 2, 3, 5, 7
   • 1, 3, 5, 7
   • 1, 4, 9, 16
   • 1, 2, 4, 8

8. 8.In the AP 20, 17, 14, ... what is the 8th term?

Arithmetic progressions · Year 11 Maths · www.arenapublications.com/learn

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

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

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

4. 4.Solve the pair 5x - 2y = 16 and x + 2y = 8. What is x?

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

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

6. 6.Solve the pair 2x + y = 11 and x - y = 1. What is x?

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

   Groups: They meet · They never meet

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

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

Pair of linear equations · Year 11 Maths · www.arenapublications.com/learn

1. 1.What is sin 30°? Give your answer as a fraction.

2. 2.In a right-angled triangle the side opposite an angle is 3 cm and the hypotenuse is 5 cm. What is the sine of that angle? Give your answer as a fraction.

3. 3.What is sin 90°?

4. 4.Match each trigonometric ratio to its value.

   • sin 30°
   • cos 30°
   • tan 45°
   • sin 0°
   • tan 60°

   • 0
   • √3
   • √3/2
   • 1/2
   • 1

5. 5.In a right-angled triangle the side next to an angle is 4 cm and the hypotenuse is 5 cm. What is the cosine of that angle? Give your answer as a fraction.

6. 6.What is tan 45°?

7. 7.What is cos 60°? Give your answer as a fraction.

8. 8.For any angle θ, what does sin²θ + cos²θ come to?

Introduction to trigonometry · Year 11 Maths · www.arenapublications.com/learn

1. 1.Match each angle to its tangent.

   • 30°
   • 45°
   • 60°

   • √3
   • 1
   • 1/√3

2. 2.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?

3. 3.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?

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

   • 75°
   • 30°
   • 45°
   • 60°

5. 5.A tree is 20 m tall. From a point on level ground the angle of elevation of its top is 30°. Take √3 = 1.73. How far is that point from the base of the tree, in metres?

6. 6.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?

7. 7.The Sun is 45° above the horizon. How long a shadow does an upright pole 12 m tall cast on level ground, in metres?

8. 8.A ladder 10 m long leans against a wall and makes an angle of 60° with the ground. How far is the bottom of the ladder from the wall, in metres?

Heights and distances · Year 11 Maths · www.arenapublications.com/learn

1. 1.Find the distance between the points (0, 0) and (6, 8). Give the answer in units.

2. 2.Find the area of the triangle whose vertices are (2, 1), (8, 1) and (2, 9). Give the answer in square units.

3. 3.Match each pair of points to the distance between them, in units.

   • (0, 0) and (5, 12)
   • (0, 0) and (9, 12)
   • (0, 0) and (8, 6)
   • (1, 1) and (4, 5)

   • 15
   • 13
   • 10
   • 5

4. 4.Find the midpoint of the segment joining (2, 3) and (8, 11). What is its x-coordinate?

5. 5.Find the midpoint of the segment joining (2, 3) and (8, 11). What is its y-coordinate?

6. 6.The point P divides the segment joining (1, 2) and (7, 11) in the ratio 2 : 1, measured from (1, 2). What is the y-coordinate of P?

7. 7.What is the area of the triangle whose vertices are (1, 2), (2, 4) and (3, 6)? Give the answer in square units.

   • a) 0
   • b) 3
   • c) 1
   • d) 2

8. 8.This line runs from 0 to 10. Tap the point that divides it in the ratio 2 : 3, measured from 0.

   Mark the line with an X.

Distance and section formulas · Year 11 Maths · www.arenapublications.com/learn

1. 1.Match each circle and outside point to the length of the tangent from that point, in centimetres.

   • radius 3 cm, point 5 cm from the centre
   • radius 8 cm, point 10 cm from the centre
   • radius 12 cm, point 13 cm from the centre
   • radius 6 cm, point 10 cm from the centre

   • 5
   • 4
   • 6
   • 8

2. 2.PA and PB are the two tangents drawn from a point P, and PA is 12 cm long. What is the total length of PA and PB, in centimetres?

3. 3.Three straight lines are drawn on this circle. Tap the tangent.

   Write the number of the part.

4. 4.A tangent touches a circle with centre O at the point P. What is the angle between the tangent and the radius OP?

   • a) 45°
   • b) 180°
   • c) 60°
   • d) 90°

5. 5.Two tangents are drawn from a point P to a circle, touching it at A and B. How does PB compare with PA?

   • a) PB is half of PA
   • b) It depends on the radius
   • c) PB is the same length as PA
   • d) PB is twice PA

6. 6.A tangent PA is 24 cm long and the hypotenuse OP of triangle OAP is 25 cm. How long is the radius OA, in centimetres?

7. 7.Sort each point by how many tangents can be drawn from it to a given circle.

   Groups: No tangents · Exactly one tangent · Exactly two tangents

   • A point inside the circle
   • A point on the circle itself
   • The centre of the circle
   • A point outside the circle

8. 8.A tangent from a point P touches a circle with centre O at A. The radius OA is 5 cm and the hypotenuse OP is 13 cm. How long is the tangent PA, in centimetres?

Circles and tangents · Year 11 Maths · www.arenapublications.com/learn

1. 1.A sector of a circle of radius 7 cm has an angle of 90° at the centre. Take π = 22/7. How long is its arc, in centimetres?

2. 2.A sector of a circle of radius 21 cm has an angle of 60° at the centre. Take π = 22/7. How long is its arc, in centimetres?

3. 3.Doubling a sector's radius while keeping its angle at the centre the same multiplies its area by what?

   • a) 2
   • b) 4
   • c) 3
   • d) 8

4. 4.A sector of a circle of radius 6 cm has an angle of 60° at the centre. Take π = 3.14. What is its area, in square centimetres?

5. 5.A quadrant is a sector with an angle of 90° at the centre. Take π = 22/7. What is the area of a quadrant of a circle of radius 14 cm, in square centimetres?

6. 6.A sector of a circle of radius 21 cm has an angle of 60° at the centre. Take π = 22/7. What is its area, in square centimetres?

7. 7.A circle has been cut into three sectors. Tap the sector with the largest area.

   Write the number of the part.

8. 8.Match each sector to its area, in square centimetres. Take π = 22/7.

   • radius 7 cm, angle 45°
   • radius 7 cm, angle 135°
   • radius 14 cm, angle 180°
   • radius 21 cm, angle 45°

   • 19.25
   • 173.25
   • 308
   • 57.75

Areas related to circles · Year 11 Maths · www.arenapublications.com/learn

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

   Groups: A surface area · A volume

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

2. 2.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?

3. 3.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?

4. 4.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 total volume of the solid, in cubic centimetres?

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

7. 7.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 cone part, in cubic centimetres?

8. 8.A toy is a cone of slant height 25 cm standing on a hemisphere, both of radius 7 cm. Take π = 22/7. What is the total surface area of the toy, in square centimetres?

Combined solids · Year 11 Maths · www.arenapublications.com/learn

1. 1.Match each class to its class mark.

   • 0–10
   • 20–40
   • 25–45
   • 50–70

   • 30
   • 5
   • 35
   • 60

2. 2.A grouped table has the classes 0–10, 10–20, 20–30 and 30–40 with frequencies 2, 4, 3 and 1. Which class has the highest frequency? Give its lower limit.

3. 3.What is the class mark — the midpoint — of the class 10–20?

4. 4.What is the class mark of the class 40–70?

5. 5.A grouped table has the classes 0–10, 10–20, 20–30 and 30–40 with frequencies 2, 4, 3 and 1. What is the total frequency?

6. 6.In a grouped table the modal class is 30–40 with a frequency of 20. The class before it has a frequency of 10 and the class after it has a frequency of 5, and every class is 10 wide. The mode is l + ((f₁ - f₀) ÷ (2f₁ - f₀ - f₂)) × h. What is the mode?

7. 7.This histogram shows a grouped frequency table. Tap the modal class.

   Write the number of the part.

8. 8.A grouped table has the classes 0–10, 10–20, 20–30 and 30–40 with frequencies 2, 4, 3 and 1. What is the mean, using class marks?

Statistics of grouped data · Year 11 Maths · www.arenapublications.com/learn

1. 1.A fair die is rolled. What is the probability of getting an even number? Give your answer as a fraction.

2. 2.A bag holds 3 red balls and 5 blue balls. One is taken out without looking. What is the probability it is blue? Give your answer as a fraction.

3. 3.The probability that it rains tomorrow is 0.3. What is the probability that it does not rain?

4. 4.A card is drawn from a full pack of 52. What is the probability it is a king? Give your answer as a fraction.

5. 5.A fair coin is tossed. What is the probability of getting a head? Give your answer as a fraction.

6. 6.Which of these could never be a probability?

   • a) 1.5
   • b) 0.25
   • c) 0
   • d) 1

7. 7.What is the probability of an event that is certain to happen?

   • a) 0
   • b) 0.5
   • c) It depends on the event
   • d) 1

8. 8.A fair die is rolled. What is the probability of getting a number greater than 4? Give your answer as a fraction.

Probability · Year 11 Maths · www.arenapublications.com/learn