IEB practice

IEB-style Grade 11 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.
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 written to IEB standards. The IEB does not set Grade 11 papers; your school's paper may differ in length and layout.

Question 1

19 marks

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

15 marks

$A (1; 6)$ , $B (- 3; - 2)$ 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

13 marks

$A$ , $B$ , $C$ 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

13 marks

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

12 marks

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

15 marks

$D (- 2; k)$ , $E (1; 4)$ 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

11 marks

$O$ 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

19 marks

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

19 marks

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

14 marks

$△ A B C$ 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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