# Mathematics Grade 12 Paper 2 Challenge Paper 2026 Question Paper

> Grade 12 Mathematics Paper 2 question paper from the Challenge Paper 2026, 13 pages. Read online or get the free PDF download, memo linked when available.

- Page: https://www.hlayiso.com/library/grade-12-mathematics-2026-challenge-paper-question-paper-paper-2/b088d17f-40ca-427c-a293-5cdf38ffb211
- Updated: 2026-10-24
- Subject: Mathematics
- Grade: Grade 12
- Document type: Question paper
- Year: 2026
- Exam period: Challenge Paper
- Paper: 2
- Publisher: Hlayiso
- Curriculum: CAPS
- Pages: 13

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## About this document

Mathematics · Grade 12 · Hlayiso Challenge Paper · 2026. A deliberately harder practice Paper 2 (question paper), 13 pages. Read online or download the PDF.

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### Page 1

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Challenge
hlayiso mock exams
Practice Series:
Grade 12 NSC Preparation
WWW.HLAYISO.COM
NATIONAL
SENIOR CERTIFICATE
GRADE 12
MATHEMATICS P2
CHALLENGE PAPER 2026
MARKS: 150
TIME: 3 hours
This question paper consists of 12 pages, 1 information sheet and a SPECIAL ANSWER BOOK.

### Page 2

Mathematics/P2
2
NSC Challenge
Hlayiso/Challenge 2026
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully before answering the questions.
1.
This question paper consists of 11 questions.
2.
Answer ALL the questions in the SPECIAL ANSWER BOOK provided.
3.
Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining
your answers.
4.
Answers only will NOT necessarily be awarded full marks.
5.
You may use an approved scientific calculator (non-programmable and non-graphical), unless
stated otherwise.
6.
If necessary, round off answers to TWO decimal places, unless stated otherwise.
7.
Diagrams are NOT necessarily drawn to scale.
8.
An information sheet with formulae is included at the end of the question paper.
9.
Write neatly and legibly.
10.
This is a challenge paper: it is deliberately harder than the NSC examination.
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### Page 3

Mathematics/P2
3
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 1
The travelling times (in minutes) of 11 learners from a school in Soweto were recorded one
morning. The data, arranged in ascending order, is given below. x and y are whole numbers.
25
28
33
37
x
45
48
52
56
y
71
The mean travelling time is 46 minutes and y is 25 more than x.
1.1
Calculate the values of x and y.
(3)
1.2
Draw a box and whisker diagram of the data on the number line provided in the
ANSWER BOOK.
(3)
1.3
Calculate the standard deviation of the travelling times.
(2)
1.4
How many learners have a travelling time that lies within ONE standard deviation of
the mean?
(2)
1.5
Because of road works, every learner’s travelling time increases by 10%. A new taxi
route then reduces every learner’s (increased) travelling time by 5 minutes. Write
down the new mean and the new standard deviation of the travelling times.
(2)
[12]
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### Page 4

Mathematics/P2
4
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 2
The owner of a tuck shop in Polokwane recorded the maximum temperature (x, in ◦C) and the
number of cold drinks sold (y) on 8 school days. The data is shown in the table and on the scatter
plot below.
MAXIMUM
TEMPERATURE
(IN ◦C) (x)
18
21
23
25
26
28
30
33
NUMBER OF
COLD DRINKS
SOLD (y)
34
45
49
58
55
66
70
79
15
20
25
30
35
30
40
50
60
70
80
90
Maximum temperature (in ◦C)
Number of cold drinks sold
SCATTER PLOT
2.1
Determine the equation of the least squares regression line for the data.
(3)
2.2
Write down the value of the correlation coefficient and describe the relationship
between the temperature and the number of cold drinks sold.
(2)
2.3
Predict the number of cold drinks that will be sold on a day when the maximum
temperature is 27 ◦C.
(2)
2.4
On a ninth day the maximum temperature was 35 ◦C. When the data for this day is
included, the mean number of cold drinks sold per day increases by 3. Calculate the
number of cold drinks sold on the ninth day.
(3)
[10]
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### Page 5

Mathematics/P2
5
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 3
In the diagram, A(1 ; 6), B(−2 ; 0) and C(3 ; 0) are the vertices of △ABC. D is a point on AC
such that BD ⊥AC.
x
y
O
A(1 ; 6)
B(−2 ; 0)
C(3 ; 0)
D
3.1
Calculate the size of B ˆAC.
(4)
3.2
Determine the equation of BD in the form y = mx+c.
(3)
3.3
Determine the coordinates of D.
(3)
3.4
Prove that △ABD is isosceles.
(2)
3.5
Calculate the area of △BDC.
(3)
3.6
P(x ; y) is a variable point above the x-axis which moves such that BˆPC = B ˆAC at all
times. Determine the equation of the path along which P moves. Write your answer
in the form (x−a)2 +(y−b)2 = r2.
(4)
[19]
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### Page 6

Mathematics/P2
6
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 4
In the diagram, M is the centre of the circle with equation x2 +y2 −4x+6y−12 = 0. P(6 ; 0) is
a point on the circle. The tangent to the circle at P cuts the y-axis at T. TQ is the second tangent
from T to the circle, touching the circle at Q.
x
y
O
M
P(6 ; 0)
T
Q
4.1
Determine the coordinates of M and the length of the radius.
(3)
4.2
Determine the equation of the tangent TP in the form y = mx+c.
(4)
4.3
Write down the coordinates of T and hence calculate the length of TP.
(3)
4.4
Determine the coordinates of Q.
(5)
4.5
Determine the equation of the circle that passes through M, P, T and Q.
(3)
4.6
For which values of k will the line y = −4
3x + k cut the circle with centre M at two
distinct points?
(3)
[21]
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### Page 7

Mathematics/P2
7
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 5
5.1
Given: sin(A+B) = 4
5 and sin(A−B) = 1
5.
Without using a calculator, determine the value of tanA
tanB.
(4)
5.2
Without using a calculator, simplify the following expression to a single number:
cos(540◦−2x)
sin(x−450◦).cos(−x) +tan(x−180◦).tan(540◦+x)
(6)
5.3
Without using a calculator, determine the value of cos15◦−sin15◦. Leave your
answer in simplest surd form.
(3)
[13]
QUESTION 6
Given: f(x) = 1−cos2x+sin2x
1+cos2x+sin2x
6.1
Prove that f(x) = tanx for all values of x for which f(x) is defined.
(4)
6.2
Determine the values of x in the interval x ∈[−180◦; 180◦] for which f(x) is
undefined.
(3)
6.3
Hence, determine the general solution of the equation [ f(x)]2 = 2 f(x)+3.
(6)
[13]
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### Page 8

Mathematics/P2
8
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 7
In the diagram, the graphs of f(x) = asinbx and g(x) = cos(x + p) are drawn for the interval
x ∈[−180◦; 180◦], where a > 0, b > 0 and −180◦< p < 180◦. The point (45◦; 1) lies on f
and (30◦; 1) is a turning point of g.
x
y
0
(45◦; 1)
(30◦; 1)
f
g
−180◦
−150◦
−120◦
−90◦
−60◦
−30◦
30◦
60◦
90◦
120◦
150◦
180◦
−1
−0,5
0,5
1
7.1
Write down the values of a, b and p.
(3)
7.2
Determine, without using a calculator, the values of x in the interval x ∈
[−180◦; 180◦] for which f(x) = g(x).
(4)
7.3
For which values of x in the interval x ∈[0◦; 180◦] is f(x) ≥g(x)?
(3)
7.4
Describe a horizontal translation of the graph of g that will produce the graph of
y = −sinx.
(2)
[12]
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### Page 9

Mathematics/P2
9
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 8
In the diagram, PQ is a vertical tower of height h metres. Q, R and S are points in the same
horizontal plane. The angles of elevation of P from R and from S are both equal to α. R ˆQS = 2θ
and RS = d metres. M is the midpoint of RS.
α
α
2θ
R
S
Q
P
M
h
d
2
d
2
8.1
Show that h = d tanα
2sinθ
(5)
8.2
Calculate the height of the tower if d = 80 m, α = 32◦and θ = 50◦.
(2)
8.3
Using the values in QUESTION 8.2, calculate the angle of elevation of P from M.
(3)
[10]
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### Page 10

Mathematics/P2
10
NSC Challenge
Hlayiso/Challenge 2026
Give reasons for your statements in QUESTIONS 9, 10 and 11.
QUESTION 9
9.1
In the diagram, O is the centre of the circle and A, B and C are points on the circle.
TAS is a tangent to the circle at A. Chords AB, BC and AC are drawn.
O
A
B
C
T
S
Use the diagram to prove the theorem which states that the angle between a tangent
to a circle and a chord drawn from the point of contact is equal to the angle in the
alternate segment, that is, prove that B ˆAS = A ˆCB.
(6)
9.2
In the diagram, O is the centre of the circle and ABCD is a cyclic quadrilateral. PA
is a tangent to the circle at A. AD = DC and P ˆAD = 35◦. BD, AC, OA and OC are
drawn.
O
35◦
A
B
C
D
P
Calculate, with reasons, the size of:
9.2.1
A ˆBD
(2)
9.2.2
A ˆBC
(2)
9.2.3
O ˆAC
(3)
[13]
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### Page 11

Mathematics/P2
11
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 10
In the diagram, D is a point on side BC of △ABC such that BD : DC = 3 : 2. E is the midpoint
of AD. BE is produced to meet AC at F.
A
B
C
D
E
F
10.1
Determine, with reasons, the ratio AF : FC.
(5)
10.2
Determine, with reasons, the ratio BE : EF.
(4)
10.3
Calculate the value of area of △AEF
area of △ABC
(4)
[13]
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### Page 12

Mathematics/P2
12
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 11
In the diagram, O is the centre of the circle and AB is a diameter. AB is produced to D. DC is a
tangent to the circle at C. DC is produced to E such that AE ⊥DE. AC, BC and OC are drawn.
Let B ˆCD = x.
x
O
A
B
D
C
E
11.1
Name, with reasons, TWO other angles that are each equal to x.
(4)
11.2
Prove that AC bisects E ˆAB.
(3)
11.3
Prove that △DCB ||| △DAC.
(3)
11.4
If DB = 4 units and DC = 8 units, calculate the length of AE.
(4)
[14]
TOTAL:
150
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### Page 13

Mathematics/P2
13
NSC Challenge
Hlayiso/Challenge 2026
INFORMATION SHEET
x = −b±
√
b2 −4ac
2a
A = P(1+ni)
A = P(1−ni)
A = P(1−i)n
A = P(1+i)n
Tn = a+(n−1)d
Sn = n
2
[
2a+(n−1)d
]
Tn = arn−1
Sn = a(rn −1)
r −1
; r̸ = 1
S∞=
a
1−r ; −1 < r < 1
F = x[(1+i)n −1]
i
P = x[1−(1+i)−n]
i
f ′(x) = lim
h→0
f(x+h)−f(x)
h
d =
√
(x2 −x1)2 +(y2 −y1)2
M
(x1 +x2
2
; y1 +y2
2
)
y = mx+c
y−y1 = m(x−x1)
m = y2 −y1
x2 −x1
m = tanθ
(x−a)2 +(y−b)2 = r2
In △ABC :
a
sinA =
b
sinB =
c
sinC
a2 = b2 +c2 −2bc.cosA
area △ABC = 1
2ab.sinC
sin(α +β) = sinα.cosβ +cosα.sinβ
sin(α −β) = sinα.cosβ −cosα.sinβ
cos(α +β) = cosα.cosβ −sinα.sinβ
cos(α −β) = cosα.cosβ +sinα.sinβ
cos2α =





cos2 α −sin2 α
1−2sin2 α
2cos2 α −1
sin2α = 2sinα.cosα
¯x = ∑x
n
σ2 = ∑n
i=1 (xi −¯x)2
n
P(A) = n(A)
n(S)
P(A or B) = P(A)+P(B)−P(A and B)
ˆy = a+bx
b = ∑(x−¯x)(y−¯y)
∑(x−¯x)2
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