Technical Mathematics Grade 12 P2 Gauteng Mock 2025 QP (Eng)

PREPARATORY EXAMINATION 2025 11092 TECHNICAL MATHEMATICS (PAPER 2) SCHOOL: SURNAME: NAME: DATE: 2 0 2 5 - M M - D D MARKER MODERATOR MARKER’S MODERATOR’S QUESTION 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 10 0 0 11 0 0 12 0 0 TOTAL TIME: 3 hours MARKS: 150 25 pages + a 2-page information sheet P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 2 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of TWELVE 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 that 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 2) 11092/25 3 KEEP THIS PAGE BLANK. P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 4 QUESTION 1 In the diagram below, ∆ABC with vertices A (2 ; 1); B (8 ; 5) and C (4 ; 8) is given. Point M (𝑥 ; 𝑦) lies on the y-axis such that MC // AB. 1.1 Determine the length of BC. (2) 1.2 Determine the gradient of AB. (2) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 5 1.3 Determine the equation of CM. (3) 1.4 Determine the coordinates of M. (2) 1.5 Determine the value of 𝑡 if E(𝑡 ; 6 ) is a point on straight line CM. (3) [12] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 6 QUESTION 2 2.1 In the diagram below, the circle with the origin as its centre cuts the straight line 𝑓 at points P(0 ; 5) and R. The point M(2 ; 4) is the midpoint of PR, and 𝛼 is the size of the acute angle that 𝑓 makes with the 𝑥-axis. 2.1.1 Determine the equation of the circle. (2) 2.1.2 Write down the coordinates of R. (3) 2.1.3 Show, by using analytical methods, that OM ⊥ PR. (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 7 2.1.4 Calculate the size of 𝛼 correct to one decimal place. (2) 2.2 Draw the following ellipse and clearly indicate ALL intercepts with axes. x2 y2  1 4 25 (3) [14] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 8 QUESTION 3 3.1 In the diagram below, OP = 17 units and point P(𝑚 ; − 4) is given. XOP = 𝜃 3.1.1 Determine the value of 𝑚. (Leave your answer in simplified surd form.) (3) 3.1.2 Determine the value of 𝑠𝑒𝑐 𝜃 − 1, without the use of a calculator. (2) 3.1.3 Determine the magnitude of 𝜃. (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 9 3.2 Calculate the value of cosec 2𝜋 − 𝜋 , correct to three decimal places. (3) [12] QUESTION 4 4.1 Complete: 1 − cos 𝜃 = … (1) 4.2 Simplify the following, without the use of a calculator, to a numerical value: sin(180° − 𝜃). cos(2𝜋 − 𝜃). tan (180° + 𝜃) 1 + cos𝜃. cos(180° − 𝜃) (7) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 10 4.3 Solve for 𝑥: tan2𝑥 = sin 41, 3° + sin 153, 6° for 𝑥 ∈ [0 ; 90 ] (4) [12] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 11 QUESTION 5 Given: The functions defined by 𝑓(𝑥) = sin2𝑥 and 𝑔(𝑥) = 2 cos𝑥 for 𝑥 ∈ [0° ; 360° ]. 5.1 Draw a sketch graph of 𝑓 and 𝑔 on the same set of axes. Clearly show ALL turning points and intercepts with the axes. (6) 5.2 Write down the period of 𝑓. (1) 5.3 Determine the range of 𝑔. 5.4 Use the graph in QUESTION 5.1 to write down the value(s) of 𝑥 for which: (2) 5.4.1 sin2𝑥 = 2cos𝑥 (2) 5.4.2 2cos𝑥 − sin2𝑥 = 2 (2) [13] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 12 QUESTION 6 6.1 Complete for any ∆ABC: 1 Area of ∆ABC = × 𝑎𝑐 × … 2 (1) 6.2 ° ° ° In the diagram below E = 40 , EFH = 60 , GFH = 50 , EH = 40 𝑚𝑚 and FG = HG = 𝑘 6.2.1 Determine the length of FH using ∆EHF. (3) 6.2.2 Calculate the size of G. (1) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 13 6.2.3 Hence, use the cosine rule to show that HF = 2𝑘 (1 − cos 80° ). (2) 6.2.4 Hence, calculate the value of 𝑘 correct to the nearest mm. (3) 6.2.5 Determine the area of ∆HFG (correct to two decimal places). (3) [13] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 14 QUESTION 7 7.1 Fill in the missing word(s) in the following theorem statement: The tangent to a circle is ... to the radius of the circle at the point of contact. (1) 7.2 In the diagram below, K, L, N and P are points on a circle with centre O. LM is a tangent to the circle. LM is parallel to KN. MNP is a straight line that cuts the circle at N and P. LMN = 26° and KOL = 134°. Determine, stating reasons, the size of the following angles: 7.2.1 K (3) 7.2.2 N (2) 7.2.3 L (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 15 7.2.4 N (2) [12] QUESTION 8 8.1 Complete the following: Two tangents drawn to a circle from the same point outside the circle are … (1) 8.2 In the diagram below, P, Q, R, and S are points on the circle with centre O. RP is a diameter. PT and ST are tangents to the circle at P and S respectively. PQS = 68°. 8.2.1 State, with reasons, THREE angles that are equal to 90° . (5) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 16 8.2.2 Determine, stating reasons, the size of the following angles: (a) R (2) (b) T (4) 8.3 Complete the following statement. The … angle of a cyclic quadrilateral is equal to the interior opposite angle. (1) 8.4 In the diagram below, A, B, C, D and E are points on a circle. DE = CD, DAE = 20° and AD̂C = 70° . AE is produced to F. 8.4.1 Give a reason why CAD = 20°. (1) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 17 8.4.2 Show that AD is the diameter of the circle. (3) 8.4.3 Hence, determine with reason(s), the size of 𝐷𝐸 𝐹. (2) [19] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 18 QUESTION 9 In the diagram below, FG is the diameter of the larger circle. S is the centre of both the smaller circle and the larger circle. DTG is a tangent to the smaller circle at T. FS = 33 cm 9.1 Write down the length of GS. Give a reason for the answer. (2) 9.2 State, with reasons, why TS is parallel to DF. (3) 9.3 Show that ∆GDF ||| ∆GTS. (3) [8] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 19 QUESTION 10 10.1 The picture below shows a tractor pulling a disc plough. Alongside is the disc plough and beneath it is a diagram of one of the discs (not drawn to scale) represented as a circle with a radius of 2,25 m and a central angle of 45°. The disc is connected to the disc plough by a rod marked AB. The disc rotates in an anticlockwise direction at 12 revolutions per minute. 10.1.1 Determine the reflex angle AOD in radians. (2) 10.1.2 Hence, calculate the length of the major arc AD. (3) 10.1.3 Determine the circumferential velocity of the disc in metres per second. (4) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 20 10.1.4 Calculate the angular velocity in radians per second. (3) 10.2 The major sector of the disc is covered by mud after a day in the field. Calculate the area of the major sector AOD. (3) [15] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 21 QUESTION 11 11.1 In the diagram, ABCDEFGH is a water trough, with base BCHG. The cross-section ABCD is an isosceles trapezium with AD parallel to BC. AD = 2,5 m, BC = 1,5 m and AB = DC. The trough is 4 m long. The water trough is filled from a cylindrical reservoir with a radius of 15 m. 𝒔𝒖𝒎 𝒐𝒇 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍 𝒔𝒊𝒅𝒆𝒔 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝒕𝒓𝒂𝒑𝒆𝒛𝒊𝒖𝒎 = ×𝒉 𝟐 𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒍𝒆 = 𝒍𝒆𝒏𝒈𝒕𝒉 × 𝒃𝒓𝒆𝒂𝒅𝒕𝒉 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂 𝒕𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 = 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒃𝒂𝒔𝒆 × 𝒉𝒆𝒊𝒈𝒉𝒕 𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂 𝒄𝒚𝒍𝒊𝒏𝒅𝒆𝒓 = 𝝅𝒓𝟐 𝒉 Given: The area of the isosceles trapezium ABCD is 8 m2. Calculate: 11.1.1 The height ℎ (2) 11.1.2 The length of AB (3) 11.2 Calculate the cost of painting the interior of the trough, if paint costs R70 per m2. (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 22 11.3 Calculate, the capacity of the trough, in litres. (3) 11.4 The trough was filled with water from the cylindrical reservoir. Calculate the drop in the water level (height) of the cylindrical reservoir, giving your answer in centimetres. (π = ). (3) [14] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 23 QUESTION 12 The picture below shows a corroded metal sheet. The students of a welding class have been tasked to clean this piece of metal sheet before it is galvanised. 12.1. If the metal sheet is divided into 6 equal parts of 8 cm each, and the ordinates are 12 cm; 6,5 cm; 7,2 cm; 8,1 cm; 7,9 cm; 6,7 cm and 9 cm, calculate the area of the corroded part using the mid-ordinate rule. (3) P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 24 12.2 If the width of the metal sheet is 2,2 m, calculate the total area that needs to be galvanised in m2. (3) [6] P.T.O.

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 25 ADDITIONAL SPACE TOTAL: 150 END

TECHNICAL MATHEMATICS (PAPER 2) 11092/25 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 2) 11092/25 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 +⋯ 2 where 𝑎 = width of the equal parts, +𝑜 𝑜 =𝑛 ordinate and 𝑛 =number of ordinates

PREPARATORY EXAMINATION 2025 11092 TECHNICAL MATHEMATICS (PAPER 2)