Technical Mathematics Grade 12 P1 Gauteng Mock 2025 QP (Eng)

PREPARATORY EXAMINATION 2025 11091 TECHNICAL MATHEMATICS (PAPER 1) NAME OF SCHOOL: SURNAME: NAME: DATE: 2 0 2 5 - M M - D D MARKER MODERATOR QUESTION MARKER’S MODERATOR’S MARKS MARKS INITIALS INITIALS 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 TOTAL TIME: 3 hours MARKS: 150 29 pages + a 2-page information sheet P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 2 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 spaces 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. If necessary, round-off answers to TWO decimal places, unless stated otherwise. 6. Diagrams are NOT necessarily drawn to scale. 7. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 8. A 2-page INFORMATION SHEET with formulae is included at the end of the question paper. 9. Candidates may NOT retain a question paper or remove it from the examination room. Question papers must be returned to the invigilator at the end of the examination session. 10. Answers must be written in blue/black ink as distinctly as possible. Do NOT write in the margins. 11. Indicate the questions you have answered by drawing a circle around the relevant numbers on the front cover of the question paper where marks are to be recorded. 12. Draw a neat line through any work/rough work that must NOT be marked. 13. If you use the additional space provided: 13.1 Write down the number of the question. 13.2 Leave a line and rule off after your answer. 14. Write neatly and legibly. P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 3 QUESTION 1 1.1 Solve for 𝑥: 1.1.1 −𝑥 (2 − 4𝑥) = 0 (2) 1.1.2 𝑥 − 3 = 4 − 5𝑥 (Correct to TWO decimal places) (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 4 1.1.3 𝑥 + 5𝑥 ≤ 6 (3) 1.2 Solve for 𝑥 and 𝑦 simultaneously if: 𝑥 + 𝑦 = 3 and 𝑥 + 𝑦 = 89 (6) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 5 1.3 The relationship between the temperatures of the hot and cold reservoirs and the engine’s efficiency is used to determine the temperature of the cold reservoir in a thermodynamic system like a Carnot engine. The efficiency (𝜂) of an ideal Carnot engine is given by the formula: 𝑇 𝜂 = 1− 𝑇 Where: 𝜂 = efficiency (in decimal form) 𝑇 = temperature of the cold reservoir (in Kelvin) 𝑇 = temperature of the hot reservoir (in Kelvin) 1.3.1 Make 𝑇 the subject of the formula. (2) 1.3.2 Hence, or otherwise, calculate the temperature of the cold reservoir in Kelvin if the efficiency of a Carnot engine is 0,4 and the temperature of the hot reservoir is 600 K. (2) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 6 1.4 Evaluate the following expression and leave your answer in binary form. (Show ALL calculations.) (11101 + 13) − 111 (4) [23] QUESTION 2 Given: 𝑥 − 𝑝𝑥 + 𝑝 + 3 = 0 2.1 Express ∆ (the discriminant) of the given equation in terms of p. (2) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 7 2.2 Hence, determine the value(s) of p for which the roots of the equation will be real and equal. (3) [5] QUESTION 3 3.1 Simplify the following without the use of a calculator. (Show ALL calculations.) 3.1.1 √25𝑥 − 16𝑥 (2) 3.1.2 (log 32)(log 4) (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 8 3.2 Solve for A if: 2 ×3 𝐴= 2 + 3. 2 (4) ° 3.3 Given: 𝑧 = 2 𝑐𝑖𝑠 315 3.3.1 Express 𝑧 in the form 𝑧 = 𝑎 + 𝑏𝑖. (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 9 3.3.2 In which quadrant of the complex plane does 𝑧 lie? (1) 3.3.3 Write down the value of the modulus of 𝑧. (1) 3.4 Solve for 𝑥 𝑎𝑛𝑑 𝑦 if: (𝑥 − 𝑦𝑖) = 𝑖(4𝑖 − 3) (4) [18] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 10 QUESTION 4 4.1 Given: 𝑓(𝑥) = 2 − 8 4.1.1 Write down the equation of the asymptote of 𝑓 . (1) 4.1.2 Determine the x- and y-intercepts of f. (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 11 4.1.3 Sketch the graph of 𝑓 on the set of axes provided. Show ALL intercepts with the axes and the asymptotes. (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 12 4.2 The graphs below represent functions defined by: 𝑔(𝑥) = −2𝑥 + 𝑑 and ℎ(𝑥) = − 𝑟 − 𝑥  C and 𝑆(−2 ; 0) are the intercepts of g.  A, B, and C are the intercepts of h 4.2.1 Determine the numerical value of d. (2) 4.2.2 Write down: (a) The coordinates of C (2) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 13 (b) The equation of h (1) (c) The domain of h (2) 4.3 Sketched below are the graphs of the functions defined by: 𝑓(𝑥) = ( 𝑥 + 6)(𝑥 − 2) and t(𝑥) = − 𝑞  K, L, and M are the intercepts of 𝑓.  K is also the 𝑥-intercept of 𝑡.  N is the turning point of f and lies on the horizontal asymptote of t. 4.3.1 Determine the coordinates of K and L. (2) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 14 4.3.2 Determine the coordinates of N. (4) 4.3.3 Write down: (a) The coordinates of M (2) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 15 (b) The range of f (1) (c) The equation(s) of the asymptotes of t (2) 4.3.4 Determine the numerical value of 𝑎. (3) 4.3.5 Determine the value(s) of x for which 𝑡(𝑥) > 0. (2) 4.3.6 State the value(s) of 𝑥 for which 𝑓(𝑥) > 𝑡(𝑥). (2) [34] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 16 QUESTION 5 5.1 John borrowed R150 000 to start a business. He must repay the loan in 2 years at 18% interest, compounded monthly. 5.1.1 Determine the amount of money that John needs to pay back. (4) 5.1.2 Determine the effective interest rate on the loan. (3) 5.2 School ABC buys sport equipment to the value of R100 000. The value of the equipment decreases by 2% per year on the straight-line method. Calculate the value of the equipment 10 years after it was bought. (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 17 5.3 Mr Smith deposited R6 000 into a savings account to use for a holiday. The interest rate is 9,6%, compounded quarterly. After 2 years the interest rate changed to 10,6%, compounded semi-annually. What is the amount of money in the account after 6 years? (5) [15] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 18 QUESTION 6 6.1 Determine 𝑓 (𝑥) using FIRST PRINCIPLES if 𝑓(𝑥) = −3𝑥 + 5. (4) 6.2 Determine the following: 6.2.1 𝑓 (𝑥) if 𝑓(𝑥) = 4√𝑥 + 2(3𝑥 ) (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 19 (𝑥 + 2 ) 6.2.2 𝐷 𝑥 (3) 𝑑 1 6.2.3 7𝑥 + 3𝜋 − 𝑑𝑥 𝑥 (2) 6.3 Given: 𝑔(𝑥) = 𝑥 − 8𝑥 + 20 Determine the coordinates of the point on the curve of 𝑔 where the gradient of the curve is 4. (4) [16] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 20 KEEP THIS PAGE BLANK. P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 21 QUESTION 7 The function 𝑓(𝑥) = 3𝑥 + 7𝑥 − 4 is given. It is also given that (3𝑥 − 2) and (𝑥 + 2) are factors of 𝑓 . 7.1 Determine the third factor of 𝑓 . (1) 7.2 Hence, or otherwise, determine the 𝑥-intercepts of 𝑓 . (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 22 7.3 Determine the coordinates of the 𝑦-intercept of 𝑓 . (2) 7.4 Determine the coordinates of the turning points of 𝑓 . (5) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 23 7.5 Sketch the graph of 𝑓 below. Clearly show all the turning points and the intercepts with the axes. (4) 7.6 Hence, give the values of 𝑥 where the graph of 𝑓 is decreasing. (2) [17] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 24 QUESTION 8 A soft drink company produces aluminium cans in the shape of a cylinder. Each can is designed to hold 864 ml of liquid. Let the radius of the cylinder be 𝑟. 8.1 Determine ℎ in terms of 𝑟, where 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 ℎ. (1) 8.2 Show that the total surface area of the cylinder (S) is given by 𝑆= + 2𝜋𝑟 , where 𝑆𝐴 = 2𝜋𝑟 + 2𝜋𝑟ℎ. (2) 8.3 Calculate r (in terms of 𝜋) if the total surface area of aluminium sheeting needs to be minimised. (5) [8] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 25 QUESTION 9 Determine the following integrals: 9.1.1 𝑑𝑥 (2) 2 3 9.1.2 𝑥 𝑥+ + 𝑑𝑥 𝑥 𝑥 (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 26 9.2 The sketch below represents the graph of the function defined by 𝑓(𝑥) = −𝑥 . 9.2.1 Determine: (− 𝑥 )𝑑𝑥 (1) P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 27 9.2.2 Hence, determine the total area enclosed between the curve 𝑓(𝑥) = − 𝑥 and the 𝑥-axis over the interval − 2 ≤ 𝑥 ≤ 2. (7) [14] P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 28 ADDITIONAL SPACE P.T.O.

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 29 ADDITIONAL SPACE TOTAL: 150 END

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 1 INFORMATION SHEET: TECHNICAL MATHEMATICS −𝑏 ± √𝑏 − 4𝑎𝑐 𝑏 4𝑎𝑐 − 𝑏 𝑥= 𝑥=− 𝑦= 2𝑎 2𝑎 4𝑎 𝑎 = 𝑏 ⇔ 𝑥 = 𝑙𝑜𝑔 𝑏, 𝑎 > 0, 𝑎 ≠ 1 and 𝑏 > 0 𝐴 = 𝑃(1 + 𝑛𝑖) 𝐴 = 𝑃(1 − 𝑛𝑖) 𝐴 = 𝑃(1 + 𝑖) 𝐴 = 𝑃(1 − 𝑖) 𝑖 𝑖 = 1+ −1 𝑚 f  x  h  f  x  f /  x   lim h0 h 𝑥 𝑥 𝑥 𝑑𝑥 = + 𝐶, 𝑛 ≠ −1 𝑘𝑥 𝑑𝑥 = 𝑘. + 𝐶, 𝑛 ≠ −1 𝑛+1 𝑛+1 1 𝑘 𝑑𝑥 = ln 𝑥 + 𝐶, 𝑥 > 0 𝑑𝑥 = 𝑘 𝑙𝑛 𝑥 + 𝐶, 𝑥 > 0 𝑥 𝑥 𝑎 𝑎 𝑎 𝑑𝑥 = + 𝐶, 𝑎 > 0 𝑘𝑎 𝑑𝑥 = 𝑘. + 𝐶, 𝑎 > 0 ln 𝑎 𝑛 ln 𝑎 𝑑= (𝑥 − 𝑥 ) + (𝑦 − 𝑦 ) M ; 𝑦 −𝑦 𝑦 = 𝑚𝑥 + 𝑐 𝑦 − 𝑦 = 𝑚(𝑥 − 𝑥 ) 𝑚= 𝑡𝑎𝑛 𝜃 = 𝑚 𝑥 −𝑥 𝑥 𝑦 + =1 𝑎 𝑏 𝑎 𝑏 𝑐 In ABC: = = 𝑎 = 𝑏 + 𝑐 − 2𝑏𝑐. 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶 Area of ∆𝐴𝐵𝐶 = 𝑎𝑏. 𝑠𝑖𝑛𝐶 𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 = 1 1 + 𝑡𝑎𝑛 𝜃 = 𝑠𝑒𝑐 𝜃 1 + 𝑐𝑜𝑡 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃

TECHNICAL MATHEMATICS (PAPER 1) 11091/25 2 𝜋𝑟𝑎𝑑 = 180° Angular velocity = 𝜔 = 2𝜋𝑛 where n = rotation frequency Angular velocity = 𝜔 = 360°𝑛 where n = rotation frequency Circumferential velocity = 𝑣 = 𝜋𝐷𝑛 where D = diameter and n = rotation frequency Circumferential velocity = 𝑣 = 𝜔𝑟 where 𝜔 = angular velocity and r = radius Arc length = 𝑠 = 𝑟𝜃 where r = radius and  = central angle in radians Area of a sector = where r = radius, s = arc length Area of a sector = where r = radius and  = central angle in radians where h = height of segment, d = diameter of circle 4ℎ − 4𝑑ℎ + 𝑥 = 0 and x = length of chord 𝐴 = 𝑎(𝑚 + 𝑚 + 𝑚 + ⋯ + 𝑚 ) where 𝑎 = width of the equal parts, 𝑚 = On = nth ordinate and n = number of ordinates OR 𝑜 +𝑜 𝐴 =𝑎 + 𝑜 + 𝑜 + ⋯+ 𝑜 where 𝑎 = width of the equal parts, 2 𝑜 =𝑛 ordinate and 𝑛 =number of ordinates

PREPARATORY EXAMINATION 2025 11091 TECHNICAL MATHEMATICS (PAPER 1)