Mathematics Grade 12 P1 NSC November Prediction 2026 QP

Prediction hlayiso mock exams Practice Series: Grade 12 NSC Preparation WWW.HLAYISO.COM NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P1 PREDICTION EXAMINATION 2026 MARKS: 150 TIME: 3 hours This question paper consists of 9 pages and 1 information sheet. Copyright reserved Please turn over

Mathematics/P1 2 Hlayiso/Prediction 2026 NSC Prediction 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. Copyright reserved Please turn over

Mathematics/P1 3 Hlayiso/Prediction 2026 NSC Prediction QUESTION 1 1.1 Solve for x: 1.1.1 (3x − 2)(x + 4) = 0 (2) 1.1.2 3x2 = 6x + 5 (correct to TWO decimal places) (3) 1.1.3 (x + 1)(x − 3) > 5 (4) √ 1.1.4 x − 2x + 7 = 4 (4) 1.1.5 4x − 3 · 2x+1 − 16 = 0 (4) 1.2 A rectangular vegetable garden has a perimeter of 34 m and a diagonal of 13 m. Calculate the length and the breadth of the garden. (5) 22026 + 22024 1.3 Without using a calculator, determine the value of 2025 . (3) 2 − 22023 [25] QUESTION 2 2.1 Given the arithmetic series: 2 + 9 + 16 + . . . + 345 2.1.1 How many terms are there in the series? (2) 50 2.1.2 Calculate the value of ∑ (7n − 5). (3) n=21 2.1.3 Calculate the sum of all the terms of the series that are EVEN numbers. (4) 2.2 Consider the quadratic pattern: 3 ; 8 ; 17 ; 30 ; . . . 2.2.1 Determine an expression for the nth term of the pattern. (3) 2.2.2 Between which two consecutive terms of the pattern is the first difference equal to 397? (3) [15] QUESTION 3 A pendulum is released and swings to and fro. On its first swing the bob travels along an arc of length 80 cm. On every following swing the length of the arc is 90% of the length of the previous swing. 3.1 Calculate the length of the arc on the 5th swing. (2) 3.2 On which swing will the length of the arc be less than 10 cm for the first time? (3) 3.3 Calculate the total distance travelled by the bob during the first 15 swings. (2) 3.4 Thabo claims that if the pendulum is left to swing forever, the bob will travel more than 8 m in total. Is Thabo correct? Justify your answer with calculations. (3) [10] Copyright reserved Please turn over

Mathematics/P1 4 Hlayiso/Prediction 2026 NSC Prediction QUESTION 4 ( ) Given: f (x) = bx , where b > 0 and b ̸= 1. The graph of f passes through the point −1 ; 31 . 4.1 Show that b = 3. (1) 4.2 The graph of g is the reflection of f in the line y = x. Determine the equation of g in the form y = . . . (2) 4.3 Sketch the graphs of f and g on the same system of axes. Clearly indicate the intercepts with the axes and ONE other point on each graph. (3) 4.4 The graph of h is obtained by translating f 1 unit to the right and 3 units down. Write down the equation of h and calculate the x-intercept of h. (2) 4.5 For which values of x is g(x) · h(x) ≥ 0 ? (3) [11] QUESTION 5 a The graph of f (x) = + q is sketched below. The asymptotes of f intersect at (−1 ; 3). x+ p The graph of f passes through (1 ; 1) and cuts the x-axis at A. y f A (1 ; 1) x O 5.1 Write down the values of p and q. (2) 5.2 Show that a = −4. (2) 5.3 Determine the equation of the axis of symmetry of f that has a positive gradient. (2) 5.4 For which values of x is x · f (x) ≤ 0 ? (3) 4 5.5 The graph of g(x) = + 3 is obtained by reflecting f in the line y = 3. Describe x+1 a DIFFERENT single transformation of f that will also give the graph of g. (2) [11] Copyright reserved Please turn over

Mathematics/P1 5 Hlayiso/Prediction 2026 NSC Prediction QUESTION 6 The graphs of f (x) = −x2 + 6x + 7 and the straight line g are sketched below. A and B are the x-intercepts of f and D is the turning point of f . The graphs of f and g intersect at A and at C(6 ; 7). M is a point on f and N is a point on g such that MN is parallel to the y-axis, with M and N between A and C. y D M g C(6 ; 7) A N x O B f 6.1 Calculate the coordinates of D. (2) 6.2 Write down the range of f . (1) 6.3 Determine the equation of g. (2) 6.4 For which values of x is f (x) ≥ g(x) AND f increasing? (2) 6.5 Calculate the maximum length of MN. (4) 6.6 Hence, or otherwise, determine the equation of the tangent to f that is parallel to g. (2) [13] Copyright reserved Please turn over

Mathematics/P1 6 Hlayiso/Prediction 2026 NSC Prediction QUESTION 7 7.1 A taxi owner bought a new minibus taxi for R680 000. The taxi depreciates according to the reducing-balance method. After 5 years its book value is R301 719,61. Calculate the annual rate of depreciation. (3) 7.2 The owner plans to replace the taxi at the end of the 5 years. The price of a new taxi, currently R680 000, increases at 6,2% p.a., compounded annually. The old taxi will be traded in at its book value. To pay the difference, the owner makes monthly deposits into a sinking fund that earns 9,6% p.a., compounded monthly. The first deposit is made one month from now and the last deposit at the end of the 5 years. Calculate the monthly deposit. (4) 7.3 Sipho takes out a home loan of R950 000. Interest is charged at 11,4% p.a., compounded monthly. The loan will be repaid over 20 years by equal monthly instalments, starting one month after the loan was granted. 7.3.1 Calculate Sipho’s monthly instalment. (3) 7.3.2 Immediately after paying the 60th instalment, Sipho increases his monthly instalment to R12 500. How many months sooner will the loan be paid off? (5) [15] QUESTION 8 8.1 Determine f ′ (x) from first principles if f (x) = x2 − 4x. (5) 8.2 Determine: dy x3 − 8 8.2.1 if y = (3) dx x−2 [( ) ] √ 2 2 8.2.2 Dx x− (3) x [11] Copyright reserved Please turn over

Mathematics/P1 7 Hlayiso/Prediction 2026 NSC Prediction QUESTION 9 The graph of y = f ′ (x), the derivative of f (x) = ax3 + bx2 + cx + d, is sketched below. The graph of f ′ cuts the x-axis at 1 and 3 and the y-axis at 9. y 9 f′ 1 3 x O 9.1 Show that a = 1, b = −6 and c = 9. (3) 9.2 It is further given that f (0) = −4. 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. (4) 9.4 Determine the coordinates of the point of inflection of f . (2) 9.5 Determine the equation of the tangent to f at its point of inflection and calculate the angle of inclination of this tangent. (4) 9.6 For which values of k will f (x) = k have three distinct real roots? (2) [18] Copyright reserved Please turn over

Mathematics/P1 8 Hlayiso/Prediction 2026 NSC Prediction QUESTION 10 A right circular cylinder with radius r cm and height h cm stands inside a right circular cone with base radius 6 cm and perpendicular height 12 cm. The base of the cylinder lies on the base of the cone and the top edge of the cylinder touches the curved surface of the cone. The diagram shows a vertical cross-section through the axis of the cone. 12 h r 6 10.1 Show that the volume of the cylinder is V (r) = 12π r2 − 2π r3 . (3) 10.2 Calculate the maximum volume of the cylinder. (3) [6] Copyright reserved Please turn over

Mathematics/P1 9 Hlayiso/Prediction 2026 NSC Prediction QUESTION 11 11.1 120 learners were asked which of the apps TikTok (T), WhatsApp (W) and Instagram (I) they use. The following information was obtained: – 60 learners use T, 70 use W and 40 use I. – 12 learners use all three apps. – 10 learners use W and I, but not T. – 8 learners use none of the three apps. – The events uses T and uses I are independent. 11.1.1 Show that 20 learners use both T and I. (2) 11.1.2 Draw a Venn diagram to represent the information. Hence, determine the number of learners who use T and W only. (4) 11.1.3 Calculate the probability that a learner chosen at random uses exactly ONE of the three apps. (2) 11.2 Five-digit numbers are formed using the digits 1 ; 2 ; 3 ; 4 ; 5 ; 6 and 7. A digit may NOT be repeated in a number. 11.2.1 How many different five-digit numbers can be formed? (1) 11.2.2 How many of these numbers are even AND greater than 50 000? (3) 11.2.3 Calculate the probability that a number chosen at random from these five-digit numbers contains the digits 1 and 2 next to each other. (3) [15] TOTAL: 150 Copyright reserved Please turn over

Mathematics/P1 10 Hlayiso/Prediction 2026 NSC Prediction INFORMATION SHEET √ −b ± b2 − 4ac x= 2a A = P(1 + ni) A = P(1 − ni) A = P(1 − i)n A = P(1 + i)n n[ ] Tn = a + (n − 1)d Sn = 2a + (n − 1)d 2 a (r − 1) n a Tn = ar n−1 Sn = ; r ̸= 1 S∞ = ; −1 < r < 1 r−1 1−r x [(1 + i)n − 1] x [1 − (1 + i)−n ] F= P= i i f (x + h) − f (x) f ′ (x) = lim h→0 h √ ( ) x1 + x2 y1 + y2 d = (x2 − x1 ) + (y2 − y1 ) 2 2 M ; 2 2 y2 − y1 y = mx + c y − y1 = m(x − x1 ) m= m = tan θ x2 − x1 (x − a)2 + (y − b)2 = r2 a b c In △ABC : = = a2 = b2 + c2 − 2bc. cos A sin A sin B sinC area △ABC = 12 ab. sinC sin(α + β ) = sin α . cos β + cos α . sin β sin(α − β ) = sin α . cos β − cos α . sin β cos(α + β ) = cos α . cos β − sin α . sin β cos(α − β ) = cos α . cos β + sin α . sin β   cos α − sin α 2 2 cos 2α = 1 − 2 sin2 α sin 2α = 2 sin α . cos α   2 cos2 α − 1 2 ∑x ∑ni=1 (xi − x̄) x̄ = σ2 = n n n(A) P(A) = P(A or B) = P(A) + P(B) − P(A and B) n(S) ∑(x − x̄)(y − ȳ) ŷ = a + bx b= ∑(x − x̄)2 Copyright reserved