Centre Number Candidate Number Candidate Name
NAMIBIA SENIOR SECONDARY CERTIFICATE
MATHEMATICS HIGHER LEVEL 8323/2
PAPER 2 3 hours
Marks 120 2017
Additional Materials: Geometrical instruments
Non programmable calculator
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INSTRUCTIONS AND INFORMATION TO CANDIDATES
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• Candidates answer on the Question Paper in the spaces provided.
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• Write your Centre Number, Candidate Number and Name in the spaces at the top of this page.
• Write in dark blue or black pen.
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• You may use a soft pencil for any diagrams or graphs.
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• Do not use correction fluid.
• Do not write in the margin For Examiner’s Use.
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• Answer all questions.
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• If working is needed for any question it must be shown below, or where working is indicated.
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• The number of marks is given in brackets [ ] at the end of each question or part question.
• Non-programmable calculators may be used.
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• If the degree of accuracy is not specified in the question, and if the answer is not exact, give the
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answer to three significant figures. Give answers for angle sizes to one decimal place but angles in
radians to three significant figures.
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• For � either use your calculator value, or use 3.142.
For Examiner's Use
Marker
Checker
This document consists of 18 printed pages and 2 blank pages.
Republic of Namibia
MINISTRY OF EDUCATION, ARTS AND CULTURE
© MoEAC/DNEA 8323/2/17 [Turn over
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1 It is given that f(x) = 2x3 – 7x2 – 24x + 45. For
Examiner’s
(a) Show that (x + 3) is a factor of f(x). Use
Answer (a).................................... [2]
(b) Given that f(x) can also be written as (x + 3)(ax2 + bx + c),
find the values of a, b and c.
Answer (b) a =............................. b =........................ c =.............................. [3]
(c) Hence, solve the equation f(x) = 0.
Answer (c) x =.............................. or ........................... or............................. [3]
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2 Differentiate For
Examiner’s
2 Use
(a) 3 − 6,
√x4
Answer (a) ................................... [2]
(b) 2x + x .
3
x2
Answer (b) ................................... [3]
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3 For
Examiner’s
A Use
NOT TO SCALE
D G
B p E F p C
In the diagram triangle ABC is an equilateral triangle with sides 30 cm in length,
The rectangle DEFG touches the sides AB, BC and AC as shown.
and BE = FC = p.
(a) Show that DE = √3p .
Answer (a)
[2]
(b) Given that p can vary, find the value of p for which the area of the rectangle
DEFG will be a maximum.
Answer (b) p =.............................. [4]
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4 (a) The expression 2x3 + ax2 + bx −30 is divisible by (x + 2) and leaves a For
Examiner’s
reminder of −35 when divided by 2x − 1. Use
Show that a = 5 and b = – 13.
Answer (a)
[4]
3 2
(b) Factorise 2x + 2x −13x – 30 completely, and hence solve the equation
3y+1 y+ y
2 + 22 1−13 × 2 – 30 = 0, giving your answer correct to 2 significant figures.
Answer (b) y = ............................. [5]
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5 The line y = k(4x – 3), where k is a constant, intersects the curve For
Examiner’s
y = 4x2 + 8x – 8 at 2 distinct points. Use
Find the set of values of k.
Answer ......................................... [5]
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6 (a) The function f is such that f(x) = x2 – 4x + 5 for the domain – 3 ≤ x ≤ 5. For
Examiner’s
(i) Express f(x) in the form a(x + B)² + C, where a, B and C are constants. Use
Answer (a) (i) a = ..........B = ......... C = ............... [3]
(ii) Find the range of f(x) when the domain is – 3 ≤ x ≤ 5.
Answer (a) (ii) .............................. [2]
(iii) Determine, with a reason, whether f – 1 (x) is a function, when the domain
is – 3 ≤ x ≤ 5.
Answer (a) (iii)........................................................................................
Reason....................................................................................................
................................................................................................................ [2]
(b) The function f is also defined for the domain x ≥ 2.
(i) Write down the range of f(x) when the domain is x ≥ 2.
Answer (b) (i) ............................... [1]
(ii) Determine, with a reason, whether f – 1 (x) is a function when the domain
is x ≥ 2.
Answer (b) (ii) ........................................................................................
Reason....................................................................................................
................................................................................................................ [2]
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7 For
Examiner’s
NOT TO SCALE Use
F
B C
9 cm
E
k j
A O i D
4 cm
The diagram shows a cylinder with a diameter of 4 cm and a height of 9 cm.
O is the centre of the circular base and AD is a diameter.
The horizontal circular base has centre O and AD is a diameter.
The point E lie on the circumference of the base and DOE is 90°.
Points B, C and F are vertically above A, D and E respectively.
Unit vectors i, j and k are parallel to OD, OE and DC respectively.
2
Q is a point on AB such that AQ = AB .
3
→ →
(a) Express vectors QO and QF in terms of some or all of i, j and k.
→ →
Answer (a) QO =.................................... QF =....................................... [3]
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(b) Find angle OQF. For
Examiner’s
Use
Answer (b) Angle OQF = ............ ° [4]
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8 For
Examiner’s
Use
y NOT TO SCALE
O A x
The diagram shows part of the graph y = e2x – 2.
The graph cuts the x-axis at A and the y-axis at B.
(a) Calculate the coordinates of A and B.
Answer (a) A ......................... B........................... [2]
(b) Calculate the equation of the tangent to the curve at B.
Answer (b) ................................... [3]
(c) Find the volume generated when the region bounded by the curve, the x-axis
and the line x = 1 and x = 2 is rotated through 360° about the x-axis.
Answer (c) ................................... [4]
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9 When a dam is full, a sluice is opened. The depth of water, D m, at a given point For
Examiner’s
P is given by Use
1 1
D =32 − t − t3 ,
16 8
where t is the time in hours after the sluice has been opened.
(a) Find the depth of water at P after 4 hours.
Answer (a) ................................ m [2]
(b) Find the rate of decrease of the depth of water at P when t = 2.
Answer (b) ............................ m/h [3]
(c) Find the time it takes for the rate of decrease of the depth of water at P to
55
reach metres per hour.
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Answer (c) ................................ h [3]
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10 (a) Solve the equation 3|1− x| = 9x. For
Examiner’s
Use
Answer (a) ................................... [3]
(b) The diagram shows the graph of y = a x + b + c .
y
(–3, 2)
NOT TO SCALE
(– 4, 0) (–2, 0)
(0, – 4)
The salient point (vertex) of the graph is (– 3, 2), the x-intercepts are (– 4, 0)
and (– 2, 0) and the y-intercept is (0, – 4).
Find the values of a, b and c.
Answer (b) a = .......... b = .......... c = .................. [4]
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11 (a) Differentiate (2x – 3)5. For
Examiner’s
Use
Answer (a) ................................... [2]
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(b) Given that ∫ 3p × √3 x dx = 360
0
, find the value of the constant p.
Answer (b) ................................... [3]
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For
12 (a) The number 0.5 can also be written as Examiner’s
Use
0.5 = 0.5 + 0.05 + 0.005 + ...
5
Show that this number can be written as the vulgar fraction .
9
Answer (a)
[3]
24
(b) Given ∑ [2(2n + 1) +1] = 1272 .
n=1
r
(i) Write down the term in x .
Answer (b) (i) ............................... [1]
(ii) Find the value of r for which the rth term is 63.
Answer (b) (ii) .............................. [2]
(iii) Write down the sum of the first 24 terms.
Answer (b) (iii) ............................. [1]
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13 The diagram shows the graph of y = a tan bx + c for 0° ≤ x ≤ 180° with an For
Examiner’s
asymptote at x = 180°. Use
x
3
NOT TO SCALE
0 x
90° 180°
(a) Find the values of a, b and c.
Answer (a) a = ............ b = ............ c = ............... [3]
(b) Write down the period of y = a tan bx + c.
Answer (b) ................................... [1]
(c) Determine the range of y = a tan bx + c for 0° ≤ x ≤ 180°.
Answer (c) ................................... [1]
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For
Examiner’s
cot x + 1 1 + tan x Use
14 (a) Prove the identity ≡
cot x − 1 1 − tan x
Answer (a)
[4]
(b) Solve the equation cosec 2x = 3 for – π ≤ x ≤ π.
Answer (b) ................................... [4]
(c) Solve for 2 sin²x + 3 cosx = 0 for 0° ≤ x ≤ 360°.
Answer (c) ................................... [5]
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15 (a) Write down the range of values of x for which log3x is defined. For
Examiner’s
Use
Answer (a) x =.............................. [1]
(b) Solve the equation 2 log3x – 1 = 6 logx3.
Answer (b) ................................... [6]
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16 A bird starts from rest on a tree, T, and flies in a straight line until it comes to rest For
Examiner’s
on the roof of a house, R. Its velocity, v m/s, at time t seconds after leaving T, is Use
given by
v = 3t – t².
(a) Find, in terms of t,
(i) the acceleration of the bird at time t,
Answer (a) (i) ........................ m/s2 [1]
(ii) the displacement of the bird at time t.
Answer (a) (ii) ........................... m [2]
(b) How long does the bird take to reach R?
Answer (b) ..................................s [2]
(c) Find the distance between T and R.
Answer (c) ................................ m [2]
(d) Find the greatest speed of the bird between T and R.
Answer (d) ..............................m/s [2]
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