IEB · Grade 11 · Mathematics IEB-style Grade 11 Mathematics Paper 2: Practice Set 1
In the print dialog, choose Save as PDF. IEB-style Grade 11 Mathematics Paper 2: Practice Set 1 Question 1 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 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 $A (- 2; - 1)$ , $B (4; 2)$ 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 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 $O$ 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 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 $P (- 1; 3)$ , $Q (3; 5)$ , $R (5; 1)$ 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 $A$ , $B$ , $C$ 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 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 $P A$ 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 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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IEB-style Grade 11 Mathematics Paper 2: Practice Set 1

Question 1

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

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

$A (- 2; - 1)$ , $B (4; 2)$ 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

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

$O$ 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

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

$P (- 1; 3)$ , $Q (3; 5)$ , $R (5; 1)$ 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

$A$ , $B$ , $C$ 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

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

$P A$ 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

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 $x$ .