# Mathematics Grade 12 P1 Challenge Paper 2026 Question Paper

> Grade 12 Mathematics Paper 1 question paper from the Challenge Paper 2026, 9 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-1/4a063e9c-b511-46b3-bd42-295e7ba5a44f
- Updated: 2026-10-21
- Subject: Mathematics
- Grade: Grade 12
- Document type: Question paper
- Year: 2026
- Exam period: Challenge Paper
- Paper: 1
- Publisher: Hlayiso
- Curriculum: CAPS
- Pages: 9

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## About this document

Mathematics · Grade 12 · Hlayiso Challenge Paper · 2026. A deliberately harder practice Paper 1 (question paper), 9 pages. Read online or download the PDF.

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## Document text

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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 P1
CHALLENGE PAPER 2026
MARKS: 150
TIME: 3 hours
This question paper consists of 8 pages and 1 information sheet.

### Page 2

Mathematics/P1
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.
3.
Number the answers correctly according to the numbering system used in this question paper.
4.
Clearly show ALL calculations, diagrams, graphs, etc. that you have used in determining
your answers.
5.
Answers only will NOT necessarily be awarded full marks.
6.
You may use an approved scientific calculator (non-programmable and non-graphical), unless
stated otherwise.
7.
If necessary, round off answers to TWO decimal places, unless stated otherwise.
8.
Diagrams are NOT necessarily drawn to scale.
9.
An information sheet with formulae is included at the end of the question paper.
10.
Write neatly and legibly.
11.
This is a challenge paper: it is deliberately harder than the NSC examination.
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### Page 3

Mathematics/P1
3
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 1
1.1
Solve for x:
1.1.1
x2 + 12
x2 = 7
(4)
1.1.2
3x+1 +31−x = 10
(4)
1.2
For which values of x will
√
12+x−x2
x−2
be a real number?
(3)
1.3
Solve for x and y simultaneously:
2x ·4y = 32
and
x2 +2y2 = 11
(5)
1.4
Given the equation: (k −1)x2 −2(k +1)x+(k +3) = 0, where k̸ = 1.
1.4.1
Prove that the roots of the equation are rational for every rational value of
k (k̸ = 1).
(3)
1.4.2
Determine ALL the integer values of k for which BOTH roots of the
equation are integers.
(3)
1.5
Given: x =
√
3+
√
2 and y =
√
3−
√
2.
Without using a calculator, determine the value of x
y + y
x.
(3)
[25]
QUESTION 2
2.1
Given the quadratic pattern: −7 ; p ; 5 ; q ; 33 ; ...
2.1.1
Show that p = −3 and q = 17.
(4)
2.1.2
Determine an expression for the nth term of the pattern.
(3)
2.1.3
Prove that 2 026 is NOT a term of the pattern.
(2)
2.2
Determine the value of k if
k
∑
r=1
log2
(8r
2
)
= 100.
(4)
[13]
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### Page 4

Mathematics/P1
4
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 3
3.1
The first, third and seventh terms of an arithmetic sequence with first term a and
common difference d (d̸ = 0) are, in that order, the first three terms of a geometric
sequence.
3.1.1
Show that a = 2d.
(3)
3.1.2
The sum of the first 10 terms of the arithmetic sequence is 260. Determine
the 5th term of the geometric sequence.
(3)
3.2
Given the infinite geometric series: 12+12(1−2x)+12(1−2x)2 +...
3.2.1
For which values of x will the series converge?
(2)
3.2.2
The sum to infinity of the series is 20. Determine the smallest number of
terms of the series that must be added so that their sum differs from the sum
to infinity by less than 0,01.
(4)
[12]
QUESTION 4
Given: f(x) = 2x+1
x−1
4.1
Show that f(x) can be written as f(x) =
3
x−1 +2.
(2)
4.2
Write down the equations of the asymptotes of f.
(2)
4.3
For which values of m will the straight line y = mx NOT intersect the graph of f?
(4)
4.4
The graph of g(x) =
3
x+1 −2 is obtained by transforming the graph of f. Describe
this transformation in words.
(2)
[10]
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### Page 5

Mathematics/P1
5
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 5
The graph of h(x) = 2x+p + q is sketched below. The graph of h passes through the origin O
and through the point B(2 ; 12). The horizontal asymptote of h is shown by the dotted line.
x
y
O
B(2 ; 12)
h
5.1
Show that p = 2 and q = −4.
(4)
5.2
Determine the equation of h−1, the inverse of h, in the form y = ...
(3)
5.3
Write down the domain of h−1.
(1)
5.4
For which values of x is h−1(x) ≤2 ?
(3)
[11]
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### Page 6

Mathematics/P1
6
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 6
The graph of f(x) = a(x+1)(x−5) is sketched below. A and B are the x-intercepts of f. D, the
turning point of f, lies on the straight line y = 3x−15.
x
y
O
A
B
D
f
y = 3x−15
6.1
Show that a = 1.
(3)
6.2
The straight line y = 2x+k is a tangent to f. Determine the value of k.
(3)
6.3
For which values of k will the line y = 2x+k intersect the graph of f at two distinct
points that BOTH have positive x-coordinates?
(3)
6.4
The graph of g is defined by g(x) = f(x −t). For which values of t will BOTH
x-intercepts of g be positive?
(2)
6.5
For which values of x is f(x)· f ′(x) > 0 ?
(3)
[14]
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### Page 7

Mathematics/P1
7
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 7
7.1
Mpho invests R50 000. For the first 3 years the investment earns interest at 8% p.a.,
compounded quarterly. Thereafter the interest rate changes to 9% p.a., compounded
monthly. After how many months, measured from the date of the original investment,
will the investment first be worth more than R100 000?
(5)
7.2
Thandi takes out a loan of R420 000 to buy a car. Interest is charged at 12% p.a.,
compounded monthly. She will repay the loan with 72 equal monthly instalments.
Her first instalment is paid 3 months after the loan is granted.
7.2.1
Show that her monthly instalment is R8 376,12.
(4)
7.2.2
Immediately after paying her 30th instalment, Thandi pays an additional
amount of R60 000 into the loan account. She continues to pay instalments
of R8 376,12 per month. How many more FULL instalments of R8 376,12
will she pay?
(6)
[15]
QUESTION 8
8.1
Determine f ′(x) from first principles if f(x) = 2x2 −1
x.
(5)
8.2
Determine dy
dx if y = (x+3)2 −9
3√x
.
(3)
8.3
The tangent to the curve y = x3 −3x at the point P(p ; p3 −3p) passes through the
point (0 ; 2), which does NOT lie on the curve. Determine the coordinates of P.
(4)
[12]
QUESTION 9
Given: f(x) = x3 + px2 +qx+16. The graph of f touches the x-axis at x = 2.
9.1
Show that p = 0 and q = −12.
(4)
9.2
Determine the coordinates of the turning points of f.
(3)
9.3
Sketch the graph of f. Clearly indicate the intercepts with the axes and the turning
points.
(3)
9.4
For which values of k will the equation f(x) = k have two distinct POSITIVE roots?
(3)
9.5
The function g is defined by g(x) = f(x−1). For which values of x is g decreasing?
(2)
[15]
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### Page 8

Mathematics/P1
8
NSC Challenge
Hlayiso/Challenge 2026
QUESTION 10
A closed cylindrical tin must hold 500π cm3. The tin has radius r cm and height h cm. The metal
for the top and the bottom of the tin costs R0,04 per cm2 and the metal for the curved side costs
R0,02 per cm2.
r
h
10.1
Show that the cost of the metal, in rand, is given by C(r) = 0,08πr2 + 20π
r .
(3)
10.2
Determine the radius for which the cost of the metal is a minimum, and calculate this
minimum cost.
(5)
[8]
QUESTION 11
11.1
A and B are independent events. P(A) = x, P(B) = x+0,2 and P(A or B) = 0,76.
11.1.1
Calculate the value of x.
(4)
11.1.2
Calculate the probability that EXACTLY ONE of the events A and B
occurs.
(2)
11.2
Seven learners, three girls (one of them is Lerato) and four boys (one of them is
Thabo), sit at random in a row of seven chairs.
11.2.1
In how many different ways can they be seated if the three girls must sit
next to each other?
(3)
11.2.2
Calculate the probability that no two girls sit next to each other.
(4)
11.2.3
Calculate the probability that Thabo and Lerato do NOT sit next to each
other.
(2)
[15]
TOTAL:
150
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### Page 9

Mathematics/P1
9
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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