IEB · Grade 12 · Mathematics IEB-style Grade 12 Mathematics Paper 2: Practice Set 2
In the print dialog, choose Save as PDF. IEB-style Grade 12 Mathematics Paper 2: Practice Set 2 Question 1 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 $A B$ 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 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 $P (- 4; 1)$ , $Q (2; 5)$ , $R (6; - 1)$ 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 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 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 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 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 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 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) (c) Calculate $\frac{area △ A P S}{area △ A B R}$ . (2) Question 11 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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IEB-style Grade 12 Mathematics Paper 2: Practice Set 2

Question 1

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

$A B$ 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

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

$P (- 4; 1)$ , $Q (2; 5)$ , $R (6; - 1)$ 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

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

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

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

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

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

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

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.