IEB · Grade 11 · Mathematics IEB-style Grade 11 Mathematics Paper 2: Practice Set 2
In the print dialog, choose Save as PDF. IEB-style Grade 11 Mathematics Paper 2: Practice Set 2 Question 1 The heights (cm) of 50 Grade 11 learners are: $140 < h \leq 150$ : 3 $150 < h \leq 160$ : 9 $160 < h \leq 170$ : 18 $170 < h \leq 180$ : 14 $180 < h \leq 190$ : 6 1.1 [2] Write down the cumulative frequencies. 1.2 [3] Draw an ogive. 1.3 [2] Estimate the median height. 1.4 [3] Estimate the lower and upper quartiles. 1.5 [2] Estimate the percentage of learners taller than 175 cm. 1.6 [3] A separate data set is: $3; 5; 6; 6; 7; 8; 9; 20$ . An outlier is a value less than $Q_{1} - 1, 5 \times IQR$ or greater than $Q_{3} + 1, 5 \times IQR$ . Show that 20 is an outlier. 1.7 [2] Calculate the standard deviation of this data set. 1.8 [2] The value 20 is removed. Calculate the new mean and say what happens to the standard deviation. Explain. Question 2 $A (1; 6)$ , $B (- 3; - 2)$ and $C (5; - 1)$ are given. 2.1 [2] Determine the midpoint $M$ of $A B$ . 2.2 [4] Determine the equation of the perpendicular bisector of $A B$ . 2.3 [3] Show that $C$ lies on the perpendicular bisector, and hence explain why $C A = C B$ . Confirm by calculation. 2.4 [2] Calculate the angle of inclination of $A B$ . 2.5 [4] Calculate $\hat{A B C}$ . Question 3 $A$ , $B$ , $C$ and $D$ lie on a circle. $A B$ is produced to $E$ . $\hat{C B E} = 80°$ and $\hat{A C D} = 30°$ . 3.1 [2] Calculate $\hat{A D C}$ , with a reason. 3.2 [2] Calculate $\hat{A B D}$ , with a reason. 3.3 [2] Calculate $\hat{D B C}$ . 3.4 [2] Calculate $\hat{D A C}$ , with a reason. 3.5 [5] In the diagram for 3.5, $O$ is the centre of a different circle, $T A S$ is a tangent at $A$ , $A B$ is a chord and $C$ lies on the circle in the alternate segment. Prove the theorem that states that the angle between a tangent and a chord is equal to the angle in the alternate segment, that is, $\hat{T A B} = \hat{A C B}$ . Question 4 4.1 [5] Without using a calculator, determine the value of $sin^{2} 135° + cos 240° \cdot tan 300°$ . 4.2 [5] Simplify fully: $\frac{sin ( 360° - x ) \cdot tan ( - x )}{cos ( 180° + x ) \cdot tan ( 180° - x )}$ 4.3 [3] If $sin 25° = t$ , express $tan 65°$ in terms of $t$ . Question 5 A tent is a right pyramid with a square base of side 3 m and a perpendicular height of 2 m. It has no floor. The volume of a pyramid is $V = \frac{1}{3} \times area of base \times H$ . 5.1 [2] Calculate the volume of the tent. 5.2 [3] Calculate the slant height of each triangular face. 5.3 [3] Calculate the area of canvas needed. 5.4 [2] A cone-shaped tent has a radius of 1,5 m and the same height. Calculate its volume. Use $V = \frac{1}{3} π r^{2} H$ . 5.5 [2] Every dimension of the pyramid tent is multiplied by 1,5. By what factor does the canvas area change? Question 6 $D (- 2; k)$ , $E (1; 4)$ and $F (7; 8)$ are collinear. 6.1 [4] Determine $k$ . 6.2 [3] Determine the equation of line $D F$ . 6.3 [4] $G$ lies on the $y$ -axis and $E G ⊥ D F$ . Determine the coordinates of $G$ . 6.4 [4] Calculate the length $E G$ and the area of $△ E GF$ . Question 7 $O$ is the centre of a circle. Chord $P Q = 16$ cm. $O M ⊥ P Q$ , and $O M$ extended meets the circle at $R$ , with $M R = 4$ cm. $R S$ is a diameter. 7.1 [5] Calculate the radius of the circle. 7.2 [1] Write down $\hat{R P S}$ with a reason. 7.3 [5] Calculate $R P$ and $P S$ in simplest surd form. Question 8 In quadrilateral $A B C D$ , $A B = 7$ m, $B C = 9$ m, $\hat{A B C} = 120°$ , $C D = 6$ m and $\hat{A C D} = 50°$ . 8.1 [3] Calculate $A C$ . 8.2 [3] Calculate $\hat{B A C}$ . 8.3 [3] Calculate $A D$ . 8.4 [5] Calculate the area of $A B C D$ . 8.5 [5] Calculate the length of the diagonal $B D$ . Question 9 9.1 [2] Write down the period of $h (x) = tan 3 x$ . 9.2 [4] Solve $tan (x - 20°) = - 1, 2$ for $0° \leq x \leq 360°$ . 9.3 [4] Sketch $f (x) = sin (x - 30°)$ and $g (x) = cos x$ on the same axes for $- 180° \leq x \leq 180°$ . 9.4 [4] Determine the $x$ -values where $f$ and $g$ intersect for $- 180° \leq x \leq 180°$ . 9.5 [5] Prove that $tan x + \frac{cos x}{1 + sin x} = \frac{1}{cos x}$ . Question 10 $△ A B C$ is drawn in a circle. $T A$ is a tangent at $A$ . $D$ lies on $A B$ and $E$ lies on $A C$ with $D E ∥ T A$ . 10.1 [6] Prove that $B D E C$ is a cyclic quadrilateral. 10.2 [8] If $\hat{T A B} = 65°$ and $\hat{B A C} = 50°$ , calculate $\hat{A C B}$ , $\hat{A B C}$ , $\hat{A D E}$ and $\hat{A E D}$ , and show that $△ A B C$ is isosceles.
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IEB-style Grade 11 Mathematics Paper 2: Practice Set 2

Question 1

The heights (cm) of 50 Grade 11 learners are:

$140 < h \leq 150$ : 3
$150 < h \leq 160$ : 9
$160 < h \leq 170$ : 18
$170 < h \leq 180$ : 14
$180 < h \leq 190$ : 6

1.1 [2]

Write down the cumulative frequencies.

1.2 [3]

Draw an ogive.

1.3 [2]

Estimate the median height.

1.4 [3]

Estimate the lower and upper quartiles.

1.5 [2]

Estimate the percentage of learners taller than 175 cm.

1.6 [3]

A separate data set is: $3; 5; 6; 6; 7; 8; 9; 20$ . An outlier is a value less than $Q_{1} - 1, 5 \times IQR$ or greater than $Q_{3} + 1, 5 \times IQR$ . Show that 20 is an outlier.

1.7 [2]

Calculate the standard deviation of this data set.

1.8 [2]

The value 20 is removed. Calculate the new mean and say what happens to the standard deviation. Explain.

Question 2

$A (1; 6)$ , $B (- 3; - 2)$ and $C (5; - 1)$ are given.

2.1 [2]

Determine the midpoint $M$ of $A B$ .

2.2 [4]

Determine the equation of the perpendicular bisector of $A B$ .

2.3 [3]

Show that $C$ lies on the perpendicular bisector, and hence explain why $C A = C B$ . Confirm by calculation.

2.4 [2]

Calculate the angle of inclination of $A B$ .

2.5 [4]

Calculate $\hat{A B C}$ .

Question 3

$A$ , $B$ , $C$ and $D$ lie on a circle. $A B$ is produced to $E$ . $\hat{C B E} = 80°$ and $\hat{A C D} = 30°$ .

3.1 [2]

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

3.2 [2]

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

3.3 [2]

Calculate $\hat{D B C}$ .

3.4 [2]

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

3.5 [5]

In the diagram for 3.5, $O$ is the centre of a different circle, $T A S$ is a tangent at $A$ , $A B$ is a chord and $C$ lies on the circle in the alternate segment. Prove the theorem that states that the angle between a tangent and a chord is equal to the angle in the alternate segment, that is, $\hat{T A B} = \hat{A C B}$ .

Question 4

4.1 [5]

Without using a calculator, determine the value of $sin^{2} 135° + cos 240° \cdot tan 300°$ .

4.2 [5]

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

4.3 [3]

If $sin 25° = t$ , express $tan 65°$ in terms of $t$ .

Question 5

A tent is a right pyramid with a square base of side 3 m and a perpendicular height of 2 m. It has no floor. The volume of a pyramid is $V = \frac{1}{3} \times area of base \times H$ .

5.1 [2]

Calculate the volume of the tent.

5.2 [3]

Calculate the slant height of each triangular face.

5.3 [3]

Calculate the area of canvas needed.

5.4 [2]

A cone-shaped tent has a radius of 1,5 m and the same height. Calculate its volume. Use $V = \frac{1}{3} π r^{2} H$ .

5.5 [2]

Every dimension of the pyramid tent is multiplied by 1,5. By what factor does the canvas area change?

Question 6

$D (- 2; k)$ , $E (1; 4)$ and $F (7; 8)$ are collinear.

6.1 [4]

Determine $k$ .

6.2 [3]

Determine the equation of line $D F$ .

6.3 [4]

$G$ lies on the $y$ -axis and $E G ⊥ D F$ . Determine the coordinates of $G$ .

6.4 [4]

Calculate the length $E G$ and the area of $△ E GF$ .

Question 7

$O$ is the centre of a circle. Chord $P Q = 16$ cm. $O M ⊥ P Q$ , and $O M$ extended meets the circle at $R$ , with $M R = 4$ cm. $R S$ is a diameter.

7.1 [5]

Calculate the radius of the circle.

7.2 [1]

Write down $\hat{R P S}$ with a reason.

7.3 [5]

Calculate $R P$ and $P S$ in simplest surd form.

Question 8

In quadrilateral $A B C D$ , $A B = 7$ m, $B C = 9$ m, $\hat{A B C} = 120°$ , $C D = 6$ m and $\hat{A C D} = 50°$ .

8.1 [3]

Calculate $A C$ .

8.2 [3]

Calculate $\hat{B A C}$ .

8.3 [3]

Calculate $A D$ .

8.4 [5]

Calculate the area of $A B C D$ .

8.5 [5]

Calculate the length of the diagonal $B D$ .

Question 9

9.1 [2]

Write down the period of $h (x) = tan 3 x$ .

9.2 [4]

Solve $tan (x - 20°) = - 1, 2$ for $0° \leq x \leq 360°$ .

9.3 [4]

Sketch $f (x) = sin (x - 30°)$ and $g (x) = cos x$ on the same axes for $- 180° \leq x \leq 180°$ .

9.4 [4]

Determine the $x$ -values where $f$ and $g$ intersect for $- 180° \leq x \leq 180°$ .

9.5 [5]

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

Question 10

$△ A B C$ is drawn in a circle. $T A$ is a tangent at $A$ . $D$ lies on $A B$ and $E$ lies on $A C$ with $D E ∥ T A$ .

10.1 [6]

Prove that $B D E C$ is a cyclic quadrilateral.

10.2 [8]

If $\hat{T A B} = 65°$ and $\hat{B A C} = 50°$ , calculate $\hat{A C B}$ , $\hat{A B C}$ , $\hat{A D E}$ and $\hat{A E D}$ , and show that $△ A B C$ is isosceles.