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GR 11 MATHEMATICS EXAMINATION 31st MAY 2021

TIME: 3 Hours 150 MARKS

EXAMINER: Mr. Courau

MODERATORS : Mrs. Bolton

NAME : ___________________________ TEACHER’S NAME:_________________

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY:

• This question paper consists of 14 pages (including the cover page) and 2 blank pages at

the end if necessary.

• All questions must be answered on the question paper.

• Read questions carefully.

• Answer all questions and show all working.

• All answers must be given correct to one decimal place if necessary, unless stated

otherwise.

• Diagrams are not drawn to scale.

USEFUL FORMULAE:

SECTION A

SECTION B

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 TOTAL

30 12 18 15 7 22 10 14 12 5 5 150

82 marks 68 marks

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

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SECTION A:

Question 1: [30 marks]

1.1) Solve for 𝑥:

a) 𝑥(𝑥 − 3) = 4 , if 𝑥 ∈ ℕ (3)

b) 4𝑥2 − 4𝑥 − 1 = 0 (leave your answers in simplest surd form) (3)

c) 2 − 𝑥 = √𝑥 − 2 (5)

d) 2𝑥2 ≥ 18 (4)

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1.2) The roots of a quadratic equation are given as 𝑥 =−2±√1−3𝑘

4 .

a) For which value(s) of 𝑘 will the roots be real? (2)

b) For which value(s) of 𝑘 will the roots be equal? (2)

1.3) Given that 2𝑥2 + (𝑝 − 3)𝑥 + 8 = 0 , determine the value(s) of 𝑝 for which the roots will be non-real. (5)

1.4) Solve for 𝑥 and 𝑦 if: (6)

81𝑥 = 3𝑦 and 𝑦 = 𝑥2 − 6𝑥 + 9

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Question 2: [12 marks]

2.1) Simplify the following, without the use of a calculator. Remember to show all working.

a) √9𝑥+2− √9𝑥−2

3𝑥−3 (4)

b) 𝑥2

1+𝑥 if 𝑥 = 1 + √3 (4)

2.2) Solve for 𝑥: 2𝑥4

3 = 32 (4)

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Question 3: [18 marks]

3.1) Given the number pattern 50 ; 47 ; 44 ; ……….

a) Determine the nth term of this pattern. (2)

b) Determine which term of the pattern is equal to −67. (2)

c) Calculate the number of positive terms in the pattern. (3)

3.2) A sequence of isosceles triangles is drawn. The 1st triangle has a base length of 2cm and a height of 2cm. The base of each successive triangle is increased by 2cm and the height is increased by 1cm. The diagram below shows the first three triangles.

1) Determine the 𝑛𝑡ℎ term of the pattern representing the areas of the triangles (𝐴 =1

2𝑏ℎ). (5)

2) Find the area of the 50th triangle. (2)

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3) Which triangle will have an area of 240𝑐𝑚2? (4)

Question 4: [15 marks]

The graphs of 𝑓(𝑥) = 𝑐𝑥 + 𝑑 and 𝑔(𝑥) =𝑎

𝑥+𝑝+ 𝑞 are sketched below. The graph 𝑓 passes through

the point 𝐵(2 ; 5) and the two graphs intersect at 𝐴(0 ; −3).

4.1) Determine the equation of the asymptote of 𝑓(𝑥). (2)

4.2) Hence, determine the equation of 𝑓(𝑥). (2)

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4.3) Determine the values of 𝑝 and 𝑞. (2)

4.4) Hence, determine the equation of 𝑔(𝑥). (3)

4.5) Determine the domain and range of 𝑓(𝑥). (2)

4.6) Determine the new coordinates of 𝐵(2 ; 5) if 𝑦 = 𝑓(𝑥) becomes:

a) 𝑦 = −𝑓(𝑥) (1)

b) 𝑦 = 𝑓(−𝑥) (1)

c) 𝑦 = 𝑓(𝑥 − 3) + 4 (2)

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Question 5: [7 marks]

The graphs of 𝑓(𝑥) = −𝑥2 + 7𝑥 + 8 and 𝑔(𝑥) = −3𝑥 + 24 are sketched below. A and B are the 𝑥-intercepts of 𝑓 and S, T and R are on the same vertical line.

5.1) Determine the length of AB (3)

5.2) If 𝑆𝑇 = 9 𝑢𝑛𝑖𝑡𝑠, determine the length of 𝑂𝑅. (4)

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SECTION B

Question 6: [22 marks]

The graph of 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 and the straight-line 𝑔(𝑥) are sketched below. A and B are the points of intersection of 𝑓 and 𝑔. A is also the turning point of 𝑓. The graph of 𝑓 intersects the x-axis at B and C(-1 ; 0) and the point 𝐷(6; 7) lies on 𝑓. The axis of symmetry of 𝑓 is 𝑥 = 2.

6.1) Determine the co-ordinates of 𝐵. (1)

6.2) Show that the equation of 𝑓 is given as 𝑓(𝑥) = 𝑥2 − 4𝑥 − 5 (4)

6.3) Determine domain and the range of 𝑓. (4)

6.4) Determine the equation of 𝑔. (4)

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6.5) Use the graph to determine the value(s) of 𝑥 for which,

a) 𝑔(𝑥) − 𝑓(𝑥) ≥ 0 (2)

b) 𝑓(𝑥)

𝑔(𝑥)> 0 (2)

6.6) Use the graph to determine the value(s) of 𝑘 for which 𝑓(𝑥 + 𝑘) = 0 will have two positive, real roots. (2)

6.7) Use the graph to determine the value(s) of 𝑡 for which the equation 𝑥2 − 4𝑥 = 𝑡 will have non-real roots. (3)

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Question 7: [10 marks]

In the diagram below, a circle is drawn passing through the points A, B, C, D and E. The tangents

at B and C meet at T. 𝐵�̂�𝐶 = 𝑥 and �̂� = 𝑦 .

Determine, with reasons, the following angles in terms of 𝑥 and/or 𝑦:

7.1) �̂�2 (2)

7.2) �̂� (2)

7.3) �̂� (6)

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Question 8: [14 marks]

8.1) In the diagram below, PR is a chord of the circle with centre O. Diameter ST is perpendicular to PR at M. 𝑃𝑅 = 8 𝑐𝑚 , 𝑀𝑇 = 2 𝑐𝑚 and 𝑂𝑀 = 𝑥 𝑐𝑚.

a) What is the length of OP in terms of 𝑥? (1) b) Determine the length of PM. Give a reason for your answer. (2) c) Hence, determine the radius of the circle. (4)

8.2) In the figure alongside, DC is the diameter of a circle with centre O. CE is a tangent to the circle at C. The diagonals of cyclic quadrilateral ABCD intersect at F. a) Prove that FAPB is a cyclic quadrilateral. (4)

b) If �̂�𝐸 = 𝑥 , determine three other angles equal to 𝑥, giving reasons where necessary. (3)

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Question 9: [12 marks]

In the diagram below, AB = BF. Chords AF and BC are extended to meet at E. The line FD is a

tangent to the circle at F.

Prove that:

9.1) �̂�2 = �̂�2 (3)

9.2) �̂�1 = �̂�1 + �̂� (4)

9.3) You are required to prove that line FB is a tangent to the circle passing through F, C and E. Complete the proof below by filling in the missing information: (5)

______ + �̂� + �̂�1 + �̂�2 = _______ ( _________________________________________________)

and ______ + _______ + �̂�2 + �̂�3 = ________ ( _________________________________________________)

so ______ + _______ = �̂�2 + �̂�3

but, �̂� = �̂�2 (proven)

so ______ = ______

∴ FB is a tangent. (___________________________________________________)

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Question 10: [5 marks]

The function 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 has the following properties:

• The range of f is (−∞ ; 4].

• 𝑏 = 2𝑎.

• The distance between the 𝑥-intercepts is 2 units.

Draw a neat sketch of 𝑓(𝑥) on the set of axes below, showing ALL relevant information.

Question 11: [5 marks]

Determine the values of 𝑚 for which 𝑦 = 𝑚𝑥 − 1 is a tangent to 𝑦 = 𝑥2 − 2𝑥 + 3.

END OF PAPER [150]

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EXTRA WORKING SPACE

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EXTRA WORKING SPACE: