IEB practice

IEB-style Grade 11 Mathematics Paper 2: Practice Set 1

Original practice paper · 150 marks · 180 minutes

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Paper info

150 marks180 minutes

Show all working. Diagrams are not to scale.

Instructions

This paper has 150 marks and you have 3 hours.
Answer all the questions.
Number your answers exactly as the questions are numbered.
Show all calculations, diagrams and graphs you used to find your answers.
You may use an approved scientific calculator (non-programmable and non-graphical) unless a question says otherwise.
Round answers to two decimal places unless a question says otherwise.
Diagrams are not necessarily drawn to scale.
This is an original Global Tutors practice paper written to IEB standards. The IEB does not set Grade 11 papers; your school's paper may differ in length and layout.

Question 1

14 marks

The test marks (%) of 10 learners are: $42; 48; 55; 58; 60; 63; 65; 70; 74; 85$

1.1[2]

Calculate the mean.

1.2[2]

Calculate the standard deviation.

1.3[3]

The teacher adds 5 marks to every learner's mark. Without recalculating from the data, write down the new mean and standard deviation. Explain your answer.

1.4[3]

Write down the five-number summary.

1.5[2]

Draw a box-and-whisker diagram.

1.6[2]

Describe the skewness of the data. Give a reason.

Question 2

6 marks

The masses (kg) of 30 parcels are:

$0 < m \leq 2$ : 5
$2 < m \leq 4$ : 9
$4 < m \leq 6$ : 10
$6 < m \leq 8$ : 4
$8 < m \leq 10$ : 2

2.1[3]

Estimate the mean mass.

2.2[1]

Write down the modal class.

2.3[2]

In which class does the median lie? Explain.

Question 3

15 marks

$A (- 2; - 1)$ , $B (4; 2)$ and $C (0; - 5)$ are the vertices of $△ A B C$ .

3.1[1]

Calculate the gradient of $A B$ .

3.2[3]

Prove that $\hat{B A C} = 90°$ .

3.3[4]

Calculate the area of $△ A B C$ .

3.4[2]

Determine the coordinates of $M$ , the midpoint of $B C$ .

3.5[3]

Determine the equation of the median $A M$ .

3.6[2]

Calculate the angle of inclination of $A B$ .

Question 4

18 marks

4.1[5]

Without using a calculator, determine the value of $\frac{cos 300° \cdot tan 135°}{sin ( - 30° )}$ .

4.2[4]

Simplify: $\frac{cos ( 90° - x ) \cdot cos ( 180° + x )}{sin ( - x )}$

4.3[5]

$5 sin θ + 4 = 0$ and $90° < θ < 270°$ . Without a calculator, determine:

(a) $cos θ$ (2)

(b) $tan θ + cos (180° - θ)$ (3)

4.4[4]

Prove that $\frac{1 - sin ^{2} x}{1 - sin x} = 1 + sin x$ , and state the values of $x$ for which it is undefined.

Question 5

12 marks

$O$ is the centre of a circle with radius 13 cm. Chord $A B = 24$ cm and chord $C D = 10$ cm are parallel and lie on the same side of $O$ . $M$ and $N$ are the midpoints of $A B$ and $C D$ .

5.1[5]

Prove the theorem that states that the line drawn from the centre of a circle to the midpoint of a chord is perpendicular to the chord.

5.2[2]

Calculate $O M$ .

5.3[5]

Calculate the distance between the two chords.

Question 6

14 marks

6.1[3]

In $△ A B C$ , $a = 12$ cm, $\hat{B} = 40°$ and $\hat{C} = 75°$ . Calculate $b$ .

6.2[4]

A triangle has sides 6 cm, 8 cm and 11 cm. Calculate its largest angle.

6.3[2]

Calculate the area of the triangle in 6.2.

6.4[5]

In $△ A B C$ , $b = 9$ cm, $c = 12$ cm and $\hat{B} = 42°$ . Calculate the two possible sizes of $\hat{C}$ and explain why both are possible.

Question 7

15 marks

$P (- 1; 3)$ , $Q (3; 5)$ , $R (5; 1)$ and $S (1; - 1)$ are the vertices of quadrilateral $P QR S$ .

7.1[3]

Prove that $P QR S$ is a parallelogram.

7.2[2]

Prove that $P QR S$ is a rhombus.

7.3[2]

Prove that $P QR S$ is a square.

7.4[3]

$T (k; 7)$ lies on the line through $P$ and $Q$ . Determine $k$ .

7.5[2]

Calculate the angle of inclination of $QR$ .

7.6[3]

Calculate the acute angle between the diagonal $P R$ and the side $P Q$ . Explain why this answer is expected.

Question 8

14 marks

$A$ , $B$ , $C$ and $D$ lie on a circle with centre $O$ . $\hat{A D C} = 112°$ . $C T$ is a tangent at $C$ , with $T$ on the same side of $A C$ as $D$ .

8.1[2]

Calculate $\hat{A B C}$ , with a reason.

8.2[2]

Calculate the non-reflex angle $\hat{A O C}$ , with a reason.

8.3[2]

Calculate $\hat{O A C}$ .

8.4[3]

Calculate $\hat{A C T}$ and $\hat{O C T}$ , with reasons.

8.5[5]

It is also given that $A D = D C$ . Calculate $\hat{D A C}$ and prove that $C D$ bisects $\hat{A C T}$ .

Question 9

12 marks

A grain silo is a cylinder with a radius of 3 m and a height of 10 m, with a hemisphere on top. The volume of a sphere is $V = \frac{4}{3} π r^{3}$ and its surface area is $S = 4 π r^{2}$ .

9.1[4]

Calculate the volume of the silo.

9.2[4]

The outside of the silo, but not the base, is painted. Calculate the area to be painted.

9.3[2]

How many litres does the silo hold?

9.4[2]

A second silo has every dimension doubled. By what factor is its volume larger?

Question 10

12 marks

$P A$ and $P B$ are tangents to a circle with centre $O$ at $A$ and $B$ . $C$ lies on the major arc $A B$ . $\hat{A P B} = 54°$ .

10.1[3]

Prove that $P A O B$ is a cyclic quadrilateral.

10.2[6]

Calculate $\hat{A O B}$ , $\hat{A C B}$ and $\hat{P A B}$ , with reasons.

10.3[3]

Prove that $O P ⊥ A B$ .

Question 11

18 marks

11.1[6]

Determine the general solution of $2 cos^{2} x + cos x - 1 = 0$ .

11.2[4]

Sketch $f (x) = 2 sin x$ and $g (x) = cos 2 x + 1$ on the same axes for $0° \leq x \leq 360°$ .

11.3[4]

Use algebra to find the $x$ -values where $f$ and $g$ intersect for $0° \leq x \leq 360°$ . Give answers to two decimal places.

11.4[4]

The sides of a triangle are $x$ , $x + 1$ and $x + 2$ cm long, and its largest angle is $120°$ . Determine $x$ .

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