Mathematics Grade 12 P1 Challenge Paper 2026 Question Paper

Copyright reserved Please turn over 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.

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. Copyright reserved Please turn over

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] Copyright reserved Please turn over

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] Copyright reserved Please turn over

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] Copyright reserved Please turn over

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] Copyright reserved Please turn over

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] Copyright reserved Please turn over

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 Copyright reserved Please turn over

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 Copyright reserved