Mathematics Grade 12 P1 NSC November Mock 2026 QP (Eng)

Mock hlayiso mock exams Practice Series: Grade 12 NSC Preparation WWW.HLAYISO.COM NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS P1 MOCK 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/Mock 2026 NSC Mock 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/Mock 2026 NSC Mock QUESTION 1 1.1 Solve for x: 1.1.1 x(2x − 5) = 0 (2) 1.1.2 3x2 − 5x − 4 = 0 (correct to TWO decimal places) (4) 1.1.3 x2 + 2x − 15 ≤ 0 (3) √ 1.1.4 x+7−x = 1 (4) 1.1.5 3x+1 + 3x−1 = 90 (3) 1.2 Solve for x and y simultaneously: y = 2x − 3 x2 + xy − y2 = 5 (6) 1.3 Given: x2 − kx + k + 3 = 0 Determine the values of k for which the roots of the equation are non-real. (3) [25] QUESTION 2 2.1 The following quadratic number pattern is given: 4 ; 9 ; 18 ; 31 ; . . . 2.1.1 Write down the next term of the pattern. (1) 2.1.2 Determine the general term, Tn , of the pattern. (4) 2.1.3 Which term of the pattern is equal to 1 038 ? (3) 2.1.4 Show that no two consecutive terms of the pattern differ by 300. (2) 2.2 Consider the arithmetic series: 3 + 7 + 11 + . . . How many terms of the series must be added to obtain a sum of 1 830 ? (4) [14] Copyright reserved Please turn over

Mathematics/P1 4 Hlayiso/Mock 2026 NSC Mock QUESTION 3 3.1 Given the geometric series: 48 + 24 + 12 + . . . 3.1.1 Calculate the value of T8 . (2) 3.1.2 Calculate the sum to infinity of the series. (2) 3.1.3 Determine the smallest value of n for which S∞ − Sn < 0,01. (4) p 3.2 Determine the value of p if ∑ 2(3)k−1 = 6 560 (3) k=1 ∞ 3.3 For which values of x will the series ∑ (2x − 1)k converge? (2) k=1 [13] QUESTION 4 The graph of f (x) = bx , where b > 0 and b ̸= 1, passes through the point (−2 ; 9). 1 4.1 Show that b = . (2) 3 4.2 Determine the equation of f −1 , the inverse of f , in the form y = . . . (2) 4.3 Sketch the graphs of f and f −1 on the same system of axes. Clearly indicate the intercepts with the axes and the coordinates of ONE other point on each graph. (4) 4.4 For which values of x is f −1 (x) ≤ −2 ? (2) 4.5 The graph of g is obtained by reflecting f in the y-axis and then translating it 3 units downwards. Write down the range of g. (1) [11] Copyright reserved Please turn over

Mathematics/P1 5 Hlayiso/Mock 2026 NSC Mock QUESTION 5 a The graph of f (x) = + q is sketched below. The asymptotes of f are x = 1 and y = 2. The x+ p point (4 ; 3) lies on f . y f (4 ; 3) x O 5.1 Write down the values of p and q. (2) 5.2 Show that a = 3. (2) 5.3 Calculate the x-intercept of f . (2) 5.4 Determine the equation of the axis of symmetry of f that has a negative gradient. (2) 5.5 Determine the values of k for which the line y = −x + k will NOT intersect the graph of f . (4) [12] Copyright reserved Please turn over

Mathematics/P1 6 Hlayiso/Mock 2026 NSC Mock QUESTION 6 The graphs of f (x) = −x2 + 2x + 8 and g(x) = 2x − 1 are drawn below. A and B are the x-intercepts of f and C is the turning point of f . The graphs of f and g intersect at D and E. y C g D A B x O f E 6.1 Calculate the coordinates of A and B. (3) 6.2 Determine the coordinates of C. (2) 6.3 Show that the x-coordinates of D and E are 3 and −3 respectively. (3) 6.4 For which values of x is f (x) ≥ g(x) ? (2) 6.5 Calculate the maximum vertical distance between f and g for −3 ≤ x ≤ 3. (2) 6.6 Determine the equation of the tangent to f that is parallel to g. (3) [15] Copyright reserved Please turn over

Mathematics/P1 7 Hlayiso/Mock 2026 NSC Mock QUESTION 7 7.1 A delivery van was bought for R320 000. It depreciates at 14% p.a. on the reducing-balance method. Calculate the value of the van after 5 years. (2) 7.2 Calculate the effective annual interest rate if interest is charged at 9% p.a., compounded monthly. (3) 7.3 Thabo took out a home loan of R1 200 000. Interest is charged at 11,25% p.a., compounded monthly. The loan will be repaid over 20 years by equal monthly payments, starting one month after the loan was granted. 7.3.1 Show that Thabo’s monthly payment is R12 591,07. (3) 7.3.2 Calculate the balance outstanding on the loan immediately after the 60th payment. (3) 7.4 Lerato needs R250 000 in 6 years’ time. She opens a savings account that earns interest at 7,2% p.a., compounded monthly, and makes equal monthly deposits, starting one month from now. Calculate the value of each monthly deposit. (3) [14] QUESTION 8 8.1 Determine f ′ (x) from first principles if f (x) = 2x2 − 3x. (5) 8.2 Determine: dy √ 3 8.2.1 if y = 4 x − 2 (3) dx x x3 − 8 8.2.2 f ′ (x) if f (x) = , x ̸= 2 (3) x−2 [11] Copyright reserved Please turn over

Mathematics/P1 8 Hlayiso/Mock 2026 NSC Mock QUESTION 9 Given: f (x) = x3 − 3x2 − 9x + 27 9.1 Show that (x − 3) is a factor of f and hence calculate the x-intercepts of f . (3) 9.2 Calculate the coordinates of the turning points of f . (4) 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 x is f concave up? (2) 9.5 For which values of k will f (x) = k have three distinct real roots? (2) [14] QUESTION 10 An open box (without a lid) is made from a rectangular sheet of cardboard measuring 24 cm by 15 cm. Squares with sides of x cm are cut from each corner and the sides are folded up along the dotted lines. x x 15 cm 24 cm 10.1 Show that the volume of the box is given by V (x) = 4x3 − 78x2 + 360x. (2) 10.2 Determine the value of x for which the volume of the box will be a maximum and hence calculate the maximum volume. (4) [6] Copyright reserved Please turn over

Mathematics/P1 9 Hlayiso/Mock 2026 NSC Mock QUESTION 11 11.1 For two events, A and B, it is given that: P(A) = 0,4 P(B) = 0,35 P(A or B) = 0,61 Determine, showing ALL calculations, whether events A and B are independent. (3) 11.2 Sipho travels to school by taxi on 70% of school days. On the other days he walks. If he travels by taxi, the probability that he is late is 0,1. If he walks, the probability that he is late is 0,25. 11.2.1 Draw a tree diagram to represent the information above. Show the probabilities associated with each branch as well as the outcomes. (3) 11.2.2 Calculate the probability that Sipho is late on a randomly chosen school day. (2) 11.3 The letters of the word PROBLEMS are rearranged to form different 8-letter arrangements. (Letters may not be repeated.) 11.3.1 How many different arrangements are possible? (1) 11.3.2 How many different arrangements are possible if the vowels must be next to each other? (3) 11.3.3 Calculate the probability that an arrangement, chosen at random, starts with a consonant and ends with a vowel. (3) [15] TOTAL: 150 Copyright reserved Please turn over

Mathematics/P1 10 Hlayiso/Mock 2026 NSC Mock 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