IEB · Grade 12 · Mathematics IEB-style Grade 12 Mathematics Paper 2: Practice Set 1
In the print dialog, choose Save as PDF. IEB-style Grade 12 Mathematics Paper 2: Practice Set 1 Question 1 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 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 $A$ , $B$ , $C$ 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 $A (- 3; 4)$ , $B (5; 8)$ 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 $P A$ 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 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 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 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 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 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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IEB-style Grade 12 Mathematics Paper 2: Practice Set 1

Question 1

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

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

$A$ , $B$ , $C$ 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

$A (- 3; 4)$ , $B (5; 8)$ 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

$P A$ 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

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

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

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

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

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 $[- 180°; 180°]$ ?