IEB practice

IEB-style Grade 12 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.
Answers only will not necessarily earn full marks.
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. It is not an IEB paper.

Question 1

12 marks

The hours that 12 learners spent on social media on one day are:

$1, 5; 2; 2, 5; 3; 3; 3, 5; 4; 4; 4, 5; 5; 6; 9$

1.1[2]

Calculate the mean and the median.

1.2[2]

Calculate the standard deviation.

1.3[3]

Write down the five-number summary.

1.4[3]

An outlier is a value less than $Q_{1} - 1, 5 \times IQR$ or greater than $Q_{3} + 1, 5 \times IQR$ . Show that 9 is an outlier.

1.5[2]

Two more learners are added to the group. They spent the same number of hours on social media, and the mean of the 14 learners is still 4 hours. How many hours did each of them spend, and what happens to the standard deviation? Explain.

Question 2

8 marks

A tuck shop recorded the midday temperature ( $x$ , in °C) and the number of cold drinks sold ( $y$ ) on seven days:

$x$ : 18 ; 21 ; 24 ; 26 ; 28 ; 31 ; 33
$y$ : 120 ; 145 ; 170 ; 182 ; 205 ; 230 ; 248

2.1[3]

Determine the equation of the least squares regression line.

2.2[2]

Write down the correlation coefficient and describe the correlation.

2.3[2]

Predict the number of cold drinks sold on a day when it is 30 °C.

2.4[1]

Explain why the line should not be used to predict sales at 45 °C.

Question 3

14 marks

$A$ , $B$ , $C$ and $D$ lie on a circle with centre $O$ . $\hat{B A D} = 64°$ and $A B = A D$ . $O B$ , $O D$ and $B D$ are drawn.

3.1[2]

Calculate $\hat{B O D}$ , with a reason.

3.2[2]

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

3.3[2]

Calculate $\hat{O B D}$ .

3.4[3]

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

3.5[5]

Prove the theorem that states that the opposite angles of a cyclic quadrilateral are supplementary.

Question 4

17 marks

$A (- 3; 4)$ , $B (5; 8)$ and $C (8; 2)$ are the vertices of $△ A B C$ .

4.1[2]

Calculate the length of $A B$ in simplest surd form.

4.2[2]

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

4.3[3]

Prove that $\hat{A B C} = 90°$ .

4.4[4]

Determine the equation of the circle through $A$ , $B$ and $C$ .

4.5[3]

Determine the coordinates of $D$ such that $A B C D$ is a rectangle.

4.6[3]

Calculate the area of rectangle $A B C D$ .

Question 5

14 marks

$P A$ is a tangent to the circle at $A$ . The line $P B C$ cuts the circle at $B$ and $C$ . Let $\hat{P A B} = x$ .

5.1[2]

Name, with a reason, an angle inside the circle equal to $x$ .

5.2[3]

Prove that $△ P A B ∣∣∣ △ P C A$ .

5.3[2]

Hence prove that $P A^{2} = P B \cdot P C$ .

5.4[3]

If $P B = 4$ cm and $B C = 5$ cm, calculate $P A$ and the ratio $A B : A C$ .

5.5[4]

If $\hat{A P C} = 30°$ and $x = 40°$ , calculate $\hat{P A C}$ and $\hat{B A C}$ .

Question 6

13 marks

A vertical tower $T F$ stands at $F$ . $A$ , $B$ and $F$ lie in the same horizontal plane. $A B = 60$ m, $\hat{F A B} = 50°$ and $\hat{A B F} = 70°$ . The angle of elevation of $T$ from $B$ is 32°.

6.1[1]

Calculate $\hat{A F B}$ .

6.2[3]

Calculate $B F$ .

6.3[3]

Calculate the height of the tower.

6.4[3]

Calculate the angle of elevation of $T$ from the point on $A B$ that is closest to $F$ .

6.5[3]

Calculate the distance from $A$ to the top of the tower, $T A$ .

Question 7

18 marks

7.1[4]

Without using a calculator, determine the value of $\frac{sin 210° \cdot tan 315°}{cos 120°}$ .

7.2[4]

Prove that $\frac{1 - cos 2 x}{sin 2 x} = tan x$ .

7.3[2]

For which values of $x$ is the identity in 7.2 undefined?

7.4[4]

Simplify to a single trigonometric ratio: $sin (180° + x) \cdot cos (90° - x) + cos^{2} (- x)$

7.5[4]

If $cos 37° = p$ , express $sin 74°$ in terms of $p$ .

Question 8

13 marks

In $△ A B C$ , $D$ lies on $A B$ and $E$ lies on $A C$ with $D E ∥ B C$ . $F$ lies on $B C$ with $E F ∥ A B$ . $A D = 6$ cm, $D B = 4$ cm and $A E = 7, 5$ cm.

8.1[3]

Calculate $E C$ .

8.2[3]

If $B C = 20$ cm, calculate $B F$ .

8.3[4]

Prove that $△ A D E ∣∣∣ △ A B C$ .

8.4[3]

Calculate $\frac{area △ A D E}{area △ A B C}$ .

Question 9

23 marks

The circle $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ has centre $N$ .

9.1[4]

Determine the coordinates of $N$ and the radius.

9.2[1]

Show that $P (5; 1)$ lies on the circle.

9.3[4]

Determine the equation of the tangent to the circle at $P$ .

9.4[4]

Calculate the length of a tangent from $Q (10; 3)$ to the circle. Leave your answer in simplest surd form.

9.5[3]

For which values of $k$ is the line $y = k$ a tangent to the circle?

9.6[4]

Determine the values of $c$ for which the line $y = x + c$ cuts the circle in two distinct points.

9.7[3]

A second circle, also with radius 5, touches the given circle externally at $P (5; 1)$ . Determine its equation.

Question 10

18 marks

Given $f (x) = cos 2 x$ and $g (x) = sin x$ .

10.1[2]

Write down the period and amplitude of $f$ .

10.2[4]

Sketch $f$ and $g$ on the same set of axes for $- 180° \leq x \leq 180°$ .

10.3[6]

Determine the general solution of $cos 2 x = sin x$ .

10.4[3]

Use your sketch to find the values of $x$ in $[- 180°; 180°]$ for which $f (x) \geq g (x)$ .

10.5[3]

For which value(s) of $k$ does the equation $cos 2 x = k$ have exactly two solutions in $[- 180°; 180°]$ ?

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