Technical Mathematics Grade 12 P1 2026 QP (Eng)

Prediction hlayiso mock exams Practice Series: Grade 12 NSC Preparation WWW.HLAYISO.COM NATIONAL SENIOR CERTIFICATE GRADE 12 TECHNICAL MATHEMATICS P1 PREDICTION EXAMINATION 2026 MARKS: 150 TIME: 3 hours This question paper consists of 8 pages, a 2-page information sheet and a 21-page SPECIAL ANSWER BOOK. Copyright reserved Please turn over

Technical 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 NINE 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. Copyright reserved Please turn over

Technical Mathematics/P1 3 Hlayiso/Prediction 2026 NSC Prediction QUESTION 1 1.1 Solve for x: 1.1.1 (5 − 2x)(x + 3) = 0 (2) 1.1.2 x(3x − 2) = 7 (correct to TWO decimal places) (4) 1.1.3 x(x + 1) > 12 (3) 1.2 Given: y − 2x = 3 and x2 + xy − y2 = −1 1.2.1 Make y the subject of the equation y − 2x = 3. (1) 1.2.2 Hence, solve for x and y simultaneously. (5) 1.3 The kinetic energy (Ek ) of a moving forklift is given by the formula: 1 Ek = mv2 2 Where: Ek = kinetic energy (in joules, J) m = mass of the forklift and its load (in kg) v = speed (in m/s) 1.3.1 Make v the subject of the formula. (2) 1.3.2 Hence, calculate the numerical value of v if Ek = 2,4 × 105 J and m = 1,5 × 103 kg. (2) 1.4 Express 1101012 as a decimal number. (1) 1.5 Evaluate 1101012 + 27 and leave your answer as a binary number. (2) [22] QUESTION 2 √ 4 ± 9 − 3m 2.1 Given: x = . Determine the numerical value(s) of m if x is: 2m + 6 2.1.1 Undefined (1) 2.1.2 Non-real (2) 2.2 Determine the value(s) of p for which the equation 2x2 − 5x + p = 3x − 4 will have non-real roots. (4) [7] Copyright reserved Please turn over

Technical Mathematics/P1 4 Hlayiso/Prediction 2026 NSC Prediction QUESTION 3 3.1 Simplify the following, showing ALL calculations, where applicable: 52x+1 · 251−x 3.1.1 (3) 125 √ √ √ 3.1.2 50 − 18 + 8 (2) log2 32 + log 0,01 3.1.3 (3) log 1 000 3.2 Solve for x: log3 (x + 1) + log3 (x − 5) = 3 (4) 3.3 Given the complex numbers: z1 = 2 − 3i and z2 = −4 + i 3.3.1 Express z = z1 + z2 in the form a + bi. (1) 3.3.2 Represent z graphically on the Argand diagram provided in the SPECIAL ANSWER BOOK. (2) 3.3.3 Hence, express z in the form r cis θ (θ in degrees). (4) 3.4 Determine the numerical values of m and n if m + ni = (3 − 2i)2 − 4i (3) [22] QUESTION 4 √ 6 4.1 Given the functions f and g defined by f (x) = 16 − x2 and g(x) = − − 2 x respectively. 4.1.1 Write down the domain of f . (2) 4.1.2 Write down the equations of the asymptotes of g. (2) 4.1.3 Determine the x-intercept of g. (2) 4.1.4 Sketch the graphs of f and g on the same set of axes provided in the SPECIAL ANSWER BOOK. Clearly show ALL the intercepts with the axes and the asymptotes. (5) 4.1.5 Use your graphs to determine the values of x for which f (x) · g(x) > 0 (2) Copyright reserved Please turn over

Technical Mathematics/P1 5 Hlayiso/Prediction 2026 NSC Prediction 4.2 The graphs below represent the functions h and k defined by h(x) = ax2 + bx + c and k(x) = −x + 5. • A(−1 ; 0) and B(5 ; 0) are the x-intercepts of h. • C(0 ; 5) is the y-intercept of both h and k. The graphs of h and k intersect at B and C. • D is the turning point of h. E is a point on k such that DE is perpendicular to the x-axis. y D C(0 ; 5) E A(−1 ; 0) x O B(5 ; 0) k h 4.2.1 Show that h(x) = −x2 + 4x + 5 (3) 4.2.2 Determine the coordinates of D. (2) 4.2.3 Determine the length of DE. (2) 4.2.4 Write down the values of x for which h(x) ≥ k(x) (2) 4.3 Sketch, on the set of axes provided in the SPECIAL ANSWER BOOK, the graph of the function defined by p(x) = ax + q that satisfies ALL the following conditions: • a>1 • The range of p is y > −4 • p(0) = −3 • p(2) = 0 Clearly show the intercepts with the axes and the asymptote. (3) [25] Copyright reserved Please turn over

Technical Mathematics/P1 6 Hlayiso/Prediction 2026 NSC Prediction QUESTION 5 5.1 A bank offers a nominal interest rate of 10,8% per annum, compounded monthly. Calculate the annual effective interest rate. (3) 5.2 An engineering workshop bought an industrial air compressor for R185 000. The compressor depreciates at a rate of 18% per annum on the reducing-balance method. 5.2.1 Calculate the book value of the compressor at the end of 3 years. (2) 5.2.2 The compressor will be replaced as soon as its book value is less than R40 000. Determine, showing ALL calculations, after how many years the compressor will be replaced. (4) 5.3 Lerato invested R50 000 in a savings account to pay a deposit of R80 000 on a bakkie at the end of 6 years. • For the first 2 years the interest rate was 7,2% per annum, compounded monthly. • At the end of the 2nd year the interest rate changed to 8,4% per annum, compounded quarterly, and remained unchanged for the rest of the period. • At the end of the 3rd year she withdrew R8 000 to repair her car. • At the end of the 4th year she deposited an additional R10 000 into the account. Determine, showing ALL calculations, whether Lerato will have enough money in the account at the end of the 6th year to pay the deposit. (6) [15] QUESTION 6 6.1 Determine f ′ (x) using FIRST PRINCIPLES if f (x) = 8 − 3x (5) 6.2 Determine: dy 3 6.2.1 if y = 2x5 − + 7 (3) dx x [ 3 ] x − 4x 6.2.2 Dx (3) 2x √ f ′ (x) if f (x) = 4 x2 3 6.2.3 (2) 6.3 Given: g(x) = x3 − 2x2 + 3 6.3.1 Determine the average gradient of g between the points where x = −1 and x = 2. (3) 6.3.2 Determine the equation of the tangent to g at the point where x = 2. Write the answer in the form y = mx + c. (4) [20] Copyright reserved Please turn over

Technical Mathematics/P1 7 Hlayiso/Prediction 2026 NSC Prediction QUESTION 7 Given: f (x) = x3 − 12x + k, where (x − 4) is a factor of f . 7.1 Show that k = −16. (2) 7.2 Write down the y-intercept of f . (1) 7.3 Determine the x-intercepts of f . (3) 7.4 Determine the coordinates of the turning points of f . (5) 7.5 Sketch the graph of f on the system of axes provided in the SPECIAL ANSWER BOOK. Clearly show ALL the intercepts with the axes and the turning points. (4) 7.6 Use your graph to determine the values of x for which f is decreasing. (2) [17] QUESTION 8 A company manufactures closed cardboard boxes with a square base to package electrical fittings. Each box has a base of x cm by x cm and a height of h cm, as shown in the diagram below. Exactly 600 cm2 of cardboard is used for each box (ignore any overlaps). h cm x cm x cm The following formulae may be used: Total surface area of a rectangular prism = 2(lb + lh + bh) Volume of a rectangular prism = l × b × h 300 − x2 8.1 Show that the height of the box can be expressed as h = (2) 2x 1 8.2 Hence, show that the volume of the box can be expressed as V (x) = 150x − x3 (2) 2 8.3 Determine the dimensions of the box that will have the maximum volume. (4) [8] Copyright reserved Please turn over

Technical Mathematics/P1 8 Hlayiso/Prediction 2026 NSC Prediction QUESTION 9 9.1 Determine the following integrals: ∫ ( ) 6 9.1.1 − 3 dx x (3) x ∫ (√ ) 9.1.2 x x − 4 dx (3) 9.2 The sketch below shows the graph of f defined by f (x) = x2 − 4x. • Region A is bounded by the graph of f and the x-axis between the x-intercepts of f . • Region B is bounded by the graph of f , the x-axis and the line x = 6. y f x=6 B x O A 9.2.1 Determine the x-intercepts of f . (2) Thabo claims that the total area of the shaded regions A and B is 0 square units, ∫ 6( ) because x2 − 4x dx = 0. 0 ∫ 6( ) 9.2.2 Show that x2 − 4x dx = 0 (2) 0 9.2.3 Explain why Thabo’s claim is NOT valid. (1) 9.2.4 Determine the correct total area of the shaded regions A and B. (3) [14] TOTAL: 150 Copyright reserved Please turn over

Technical Mathematics/P1 9 Hlayiso/Prediction 2026 NSC Prediction INFORMATION SHEET: TECHNICAL MATHEMATICS √ −b ± b2 − 4ac b 4ac − b2 x= x=− y= 2a 2a 4a a = b ⇔ x = loga b , a > 0 , a ̸= 1 and b > 0 x A = P(1 + ni) A = P(1 − ni) A = P(1 + i)n A = P(1 − i)n ( ) i m ieff = 1 + −1 m f (x + h) − f (x) f ′ (x) = lim h→0 h ∫ ∫ xn+1 k kxn dx = k · +C , n ̸= −1 dx = k · ln x +C , x>0 n+1 x ∫ anx k anx dx = k · +C , a>0 n ln a √ ( ) x2 + x1 y2 + y1 d = (x2 − x1 )2 + (y2 − y1 )2 M ; 2 2 y2 − y1 y = mx + c y − y1 = m(x − x1 ) m= tan θ = m x2 − x1 x2 y2 + =1 a2 b2 a b c In △ABC : = = a2 = b2 + c2 − 2bc · cos A sin A sin B sinC 1 Area of △ABC = ab · sinC 2 sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = cosec2 θ Copyright reserved Please turn over

Technical Mathematics/P1 10 Hlayiso/Prediction 2026 NSC Prediction π rad = 180◦ Angular velocity = ω = 2π n where n = rotation frequency Angular velocity = ω = 360◦ n where n = rotation frequency Circumferential velocity = v = π Dn where D = diameter and n = rotation frequency Circumferential velocity = v = ω r where ω = angular velocity and r = radius Arc length = s = rθ where r = radius and θ = central angle in radians rs Area of a sector = where r = radius and s = arc length 2 r2 θ Area of a sector = where r = radius and θ = central angle in radians 2 4h2 − 4dh + x2 = 0 where h = height of segment, d = diameter of circle and x = length of chord o1 + o2 AT = a (m1 + m2 + m3 + . . . + mn ) where a = width of equal parts, m1 = , 2 on = nth ordinate and n = number of ordinates OR ( ) o1 + on AT = a + o2 + o3 + . . . + on−1 where a = width of equal parts, on = nth ordinate and n 2 = number of ordinates Copyright reserved