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IEB-style Grade 12 Mathematics Paper 2: Practice Set 2

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

13 marks

The times (in minutes) of 60 runners in a 5 km fun run are shown:

$15 < t \leq 20$ : 4
$20 < t \leq 25$ : 11
$25 < t \leq 30$ : 20
$30 < t \leq 35$ : 15
$35 < t \leq 40$ : 7
$40 < t \leq 45$ : 3

1.1[2]

Write down the cumulative frequencies.

1.2[4]

Draw an ogive for the data.

1.3[2]

Use the ogive to estimate the median time.

1.4[3]

Estimate the mean time.

1.5[2]

What percentage of runners took longer than 35 minutes?

Question 2

14 marks

$A B$ is a diameter of a circle with centre $O$ . $C$ lies on the circle with $\hat{C A B} = 25°$ . The tangent at $C$ meets $A B$ extended at $T$ .

2.1[5]

Prove the theorem that states that the angle between a tangent and a chord equals the angle in the alternate segment.

2.2[2]

Write down $\hat{A C B}$ and $\hat{A B C}$ , with reasons.

2.3[2]

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

2.4[3]

Calculate $\hat{T}$ .

2.5[2]

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

Question 3

7 marks

The ages ( $x$ , in years) and prices ( $y$ , in thousands of rand) of seven used cars of the same model are:

$x$ : 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 8
$y$ : 285 ; 250 ; 228 ; 205 ; 180 ; 162 ; 120

3.1[3]

Determine the equation of the least squares regression line.

3.2[2]

Write down the correlation coefficient and describe the correlation.

3.3[2]

Interpret the value of $b$ in this context.

Question 4

22 marks

$P (- 4; 1)$ , $Q (2; 5)$ , $R (6; - 1)$ and $S (0; - 5)$ are the vertices of quadrilateral $P QR S$ .

4.1[3]

Prove that $P QR S$ is a parallelogram.

4.2[3]

Prove that $P QR S$ is a rhombus.

4.3[3]

Prove that $P QR S$ is a square.

4.4[2]

Calculate the area of $P QR S$ .

4.5[2]

Calculate the angle of inclination of $P Q$ .

4.6[3]

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

4.7[3]

Determine the equation of the circle that passes through $P$ , $Q$ , $R$ and $S$ .

4.8[3]

Determine the equation of the tangent to this circle at $Q$ .

Question 5

15 marks

In $△ A B C$ , $\hat{A B C} = 90°$ and $B D ⊥ A C$ . $A D = 9$ cm and $D C = 16$ cm.

5.1[4]

Prove that $△ A B D ∣∣∣ △ A C B$ .

5.2[3]

Prove that $B D^{2} = A D \cdot D C$ .

5.3[4]

Calculate $B D$ and $A B$ .

5.4[4]

Calculate the length of the perpendicular from $D$ to $A B$ .

Question 6

11 marks

A circle has the equation $(x + 1)^{2} + (y - 2)^{2} = 25$ and centre $C$ . $A (3; 5)$ is a point on the plane.

6.1[1]

Show that $A$ lies on the circle.

6.2[4]

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

6.3[3]

The tangent cuts the $y$ -axis at $B$ . Determine the coordinates of $B$ and the length $A B$ .

6.4[3]

Calculate the area of $△ C A B$ .

Question 7

14 marks

7.1[3]

Simplify $cos 3 x cos x + sin 3 x sin x$ to a single trigonometric ratio, and hence write down its value when $x = 30°$ .

7.2[6]

Solve $3 sin x + 2 = cos 2 x$ for $0° \leq x \leq 360°$ .

7.3[5]

Prove that $\frac{sin 2 x}{1 - cos 2 x} = \frac{1}{tan x}$ , and state one value of $x$ in $(0°; 360°)$ for which it is undefined.

Question 8

18 marks

A triangular field $P QR$ has $P Q = 85$ m, $P R = 120$ m and $\hat{QP R} = 48°$ .

8.1[3]

Calculate $QR$ .

8.2[3]

Calculate $\hat{P R Q}$ , and hence $\hat{P QR}$ . Explain why you find $\hat{P R Q}$ first.

8.3[3]

Calculate the area of the field.

8.4[3]

A straight path runs from $P$ to $QR$ , perpendicular to $QR$ . Calculate its length.

8.5[6]

In a different triangle $A B C$ , $a = 7$ cm, $c = 10$ cm and $\hat{A} = 35°$ .

(a) Calculate the two possible sizes of $\hat{C}$ . (4)

(b) Explain why both triangles are possible. (2)

Question 9

16 marks

9.1[6]

Simplify fully: $\frac{cos ( 360° - x ) \cdot sin ( x - 180° )}{tan ( - x ) \cdot sin ^{2} ( 90° + x )}$

9.2[4]

Without using a calculator, determine the exact value of $cos 15°$ .

9.3[6]

$sin A = - \frac{5}{13}$ and $180° < A < 270°$ . Without a calculator, determine:

(a) $cos 2 A$ (3)

(b) $sin (A + 45°)$ (3)

Question 10

13 marks

10.1[5]

$A B C D$ is a cyclic quadrilateral with $A B ∥ D C$ . Diagonal $A C$ is drawn. Prove that $A D = B C$ .

10.2[8]

In $△ A B C$ , $P$ and $Q$ are the midpoints of $A B$ and $A C$ . $R$ lies on $B C$ with $B R : R C = 1 : 2$ , and $A R$ cuts $P Q$ at $S$ . $B C = 18$ cm.

(a) Calculate $P Q$ . (2)

(b) Calculate $P S$ and $S Q$ . (4)

Question 11

7 marks

The circle $(x + 1)^{2} + (y - 2)^{2} = 25$ has centre $C (- 1; 2)$ .

11.1[7]

For which values of $k$ does the line $y = x + k$ cut the circle at two points? Leave your answer in simplest surd form.

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