TECH NATHS P2 QP ENG JUNE 2025

PROVINCIAL ASSESSMENT GRADE 12 TECHNICAL MATHEMATICS P2 JUNE 2025 MARKS: 150 TIME: 3 hours This question paper consists of 15 pages and a 2-page information sheet. Copyright reserved Please turn over

Technical Mathematics/P2 2 NW/June/2025 Grade 12 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 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. 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. 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/P2 3 NW/June/2025 Grade 12 QUESTION 1 The diagram below shows quadrilateral ABCD with vertices 𝐴(−2; 4), 𝐵(−3; −3), 𝐶(3; −4) and 𝐷(𝑥; 3). The angle of inclination of the line DC with the 𝑥- axis is 𝜃. Determine: 1.1 the gradient of BC (2) 1.2 the equation of BC (2) 1 1.3 the x-coordinate of D if the gradient of AD is − 7 (2) 1.4 whether BC is parallel to AD, give a reason for your answer (2) 1.5 the midpoint of AB (2) 1.6 the equation of the line perpendicular to BC and passing through(−1; −4) (3) 1.7 the value of 𝜃 (3) [16] Copyright reserved Please turn over

Technical Mathematics/P2 4 NW/June/2025 Grade 12 QUESTION 2 2.1 In the diagram below, O is the centre of the circle. OC is the radius, ED is a tangent to the circle at point C. Determine the: 2.1.1 equation of the circle (2) 2.1.2 coordinates of B (2) 2.1.3 gradient of OC (1) 2.1.4 equation of ED (3) 2.1.5 coordinates of D, the x-intercept of the tangent. (2) 2.2 𝑥2 𝑦2 Given: + 2=1 (√7)2 3 Hence, sketch the graph defined by the equation given above. (3) [13] Copyright reserved Please turn over

Technical Mathematics/P2 5 NW/June/2025 Grade 12 QUESTION 3 3 𝜋 3.1 Determine the following if cos 𝛽 = 5 ; 𝛽 ∈ (0°; 90°) and 𝛼 = 6 3.1.1 𝛽 (Round off to the nearest whole number) (2) 3.1.2 Convert 𝛼 to degrees. (1) 3.1.3 sin(2β) − sec α (3) 𝟒 3.2 Given: tan 𝜃 = − 𝟓 and , 𝜃 𝜖 [0°; 180°] 3.2.1 Draw a diagram to illustrate the above ratio. (1) 3.2.2 Hence, use the diagram to determine 𝑐𝑜𝑠 2 𝜃 + 𝑠𝑖𝑛2 𝜃 without the use of a calculator. (3) 3.3 Determine the value of x if 8 cos 𝑥 − 2 = 2 for 𝑥 ∈ [0°; 360°] (3) [13] QUESTION 4 4.1 Simplify the following: 𝑐𝑜𝑠(𝜋 + 𝜃). 𝑡𝑎𝑛(180° + 𝜃). 𝑠𝑖𝑛2 (180° − 𝜃) 1 − 𝑡𝑎𝑛(180° − 𝜃) . 𝑠𝑖𝑛 𝜃 . 𝑐𝑜𝑠(180 − 𝜃) . 𝑠𝑒𝑐 𝜃 (7) 4.2 𝑠𝑖𝑛2 𝜃 𝑐𝑜𝑠2 𝜃 1 Prove that: + = (4) 𝑐𝑜𝑠2 𝜃 𝑐𝑜𝑠2 𝜃 𝑐𝑜𝑠2 𝜃 [11] Copyright reserved Please turn over

Technical Mathematics/P2 6 NW/June/2025 Grade 12 QUESTION 5 The graph below represents the function defined by 𝑔(𝑥) = tan 𝑥 for 0° ≤ 𝑥 ≤ 360° 5.1 Use the graph above to determine the following: 5.1.1 the equations of the asymptotes. (2) 5.1.2 the period of g (1) 5.1.3 the value of the x-coordinate at point C (1) 5.1.4 the coordinates of A (2) Copyright reserved Please turn over

Technical Mathematics/P2 7 NW/June/2025 Grade 12 5.2 The graph below represents the function defined by: 𝑓(𝑥) = cos2 𝑥 for 0° ≤ 𝑥 ≤ 360° 5.2.1 What is the amplitude of f ? (1) 5.2.2 What is the period of f ? (1) 5.2.3 Determine the coordinates of the turning point at A. (2) 5.3 Use the graph above to determine the following: 5.3.1 the value(s) of x for which f is increasing if 𝑥 ∈ (0°; 180°) (2) 5.3.2 the value(s) of x for which 𝑓(𝑥) < 0 if 𝑥 ∈ (0°; 180°) (2) 5.4 Write down the range of f. (2) [16] Copyright reserved Please turn over

Technical Mathematics/P2 8 NW/June/2025 Grade 12 QUESTION 6 The sketch below shows AB, a light house perpendicular to the horizontal level at B. The light house is 110 m high. C and D represent the position of two boats respectively. 𝐴̂3 = 63° and 𝐹𝐴̂𝐶 = 70° 6.1 Determine with reason the sizes of: 6.1.1 𝐶̂1 (2) 6.1.2 𝐴̂2 (1) 6.2 Determine the length of AC. (2) 6.3 Determine the distance between the two boats, CD. (3) 6.4 Use the cosine rule to determine the length of AD. (2) [10] Copyright reserved Please turn over

Technical Mathematics/P2 9 NW/June/2025 Grade 12 QUESTION 7 7.1 In the diagram below, HG is a tangent to the circle at D. DF̂E = 65° and 𝐷 ̂1 = 35° A, E, C, D and F are on the circumference of the circle. CD = DF and AC║FD 7.1.1 Write down, stating reasons, THREE other angles equal to 35°. (6) Determine, with reason(s), the size of the following angles: 7.1.2 ̂H AD (1) 7.1.3 Ĉ (2) 7.1.4 ̂F CD (2) 7.1.5 ̂3 D (2) 7.1.6 Prove that ∆JAB ⫴ ∆JDF (3) Copyright reserved Please turn over

Technical Mathematics/P2 10 NW/June/2025 Grade 12 7.2 In the diagram below, O is the centre of the circle. PLMN is cyclic quadrilateral. 𝐿̂1 = 80° and 𝑀 ̂ = 120°. MN is extended and forms a straight line. Determine, by giving reason(s), the following: 7.2.1 ̂1 𝑁 (2) 7.2.2 𝑃̂1 (2) 7.2.3 ̂ 𝐾 (2) 7.2.4 Solve for x if KL║PN (2) [24] Copyright reserved Please turn over

Technical Mathematics/P2 11 NW/June/2025 Grade 12 QUESTION 8 8.1 Complete the following theorem: Two tangents from the same point outside the circle are … (1) 8.2 In the diagram below, O is the centre of the circle. AC and BC are tangents to the circle. D is a point on the circumference of the circle and forms chords AD and BD. ̂ = 40° 𝐷 Determine, by stating reason(s), the size of the following angles: 8.2.1 𝑂̂1 (2) 8.2.2 𝐴̂1 (2) 8.2.3 𝐴̂2 (2) 8.2.4 𝐴̂3 (1) 8.2.5 𝐶̂ (3) [11] Copyright reserved Please turn over

Technical Mathematics/P2 12 NW/June/2025 Grade 12 QUESTION 9 9.1 Complete the following theorem: The line through the midpoint of one side of a triangle and parallel to another side of a triangle is … of the third side. (1) 9.2 In the diagram below, ST ║PR, 𝑅̂ = 45°, QS = 6 ; PR = 20 and QT = c Write down, stating reasons, the values of the following: 9.2.1 a (2) 9.2.2 b (1) 9.2.3 x (2) 9.2.4 Determine the length of QR in terms of c. (1) [7] Copyright reserved Please turn over

Technical Mathematics/P2 13 NW/June/2025 Grade 12 QUESTION 10 10.1 The picture below shows a small 6 blade wind turbine for residents to generate their own electricity. The diameter of the turbine is 1,2 m. The turbine is rotating at 20 revolutions per minute. The diagram next to the picture represents the 6 blades. The angles between the blades are 60° each. 10.1.1 Determine the angular velocity in radians per second. (4) 10.1.2 Determine the circumferential velocity of the turbine. (4) 10.1.3 Calculate the area of sector EFG. (4) 10.2 In the diagram below, the length of chord 𝐴𝐵 = 18 𝑐𝑚 and 𝑂𝐸 = 15 𝑐𝑚. Determine the length of DC. (4) [16] Copyright reserved Please turn over

Technical Mathematics/P2 14 NW/June/2025 Grade 12 QUESTION 11 11.1 Determine the area of a part of a golf green, by using the mid-ordinate rule. The length of 24 m is divided into 6 equal parts. The value of x is equal to the sum of the 1st and 2nd ordinate divided by 2. (5) Copyright reserved Please turn over

Technical Mathematics/P2 15 June2025/NW Grade 12 11.2 In the diagram below is a solid two-step wooden stair with an open basis. The stair is build out of two rectangular wooden boxes that is glued together. The following formulae may be used: 𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒑𝒓𝒊𝒔𝒎 = 𝟐𝒍𝒃 + 𝟐𝒍𝒉 + 𝟐𝒃𝒉 𝑽𝒐𝒍𝒖𝒎𝒆 = 𝒍 × 𝒃 × 𝒉 11.2.1 Determine the volume of the wooden stair. (4) 11.2.2 Calculate the surface area of each of the two rectangular parts separately. (3) 11.2.3 Determine the total surface area of the wooden stair. (1) [13] TOTAL: 150 Copyright reserved

Technical Mathematics/P2 June2025/NW Grade 12 FORMATION SHEET: TECHNICAL MATHEMATICS  b  b 2  4ac b 4ac  b2 x x y 2a 2a 4a a x  b  x  log a b , a  0 , a  1 and b  0 A  P(1  ni) A  P(1  ni) A  P(1  i ) n A  P(1  i ) n   1 m ieff  1  i m f  x  h  f  x f /  x   lim h 0 h x n 1  x dx  n C , n  1 n 1 ax  1 dx  ln x  C , x0  a dx  x x C , a0 ln a  x  x2 y1  y 2  d  ( x2  x1 ) 2  ( y 2  y1 ) 2 M  1 ;   2 2  y 2  y1 y  mx  c y  y1  m( x  x1 ) m m  tan  x 2  x1 x2 y2  1 a2 b2 a b c In ABC:   sin A sin B sin C a2 = b2 + c2 – 2bccos A area of ABC  1 ab  sin C 2 sin 2   cos 2   1 1  tan 2   sec 2  cot 2   1  cosec 2 Copyright reserved

Technical Mathematics/P2 June2025/NW Grade 12  rad  180 Angular velocity    2 n Where n = rotation frequency Angular velocity  360n where n = rotation frequency Circumferencial velocity     Dn where D = diameter and n = rotation frequency Cirfumferential velocity  v   r where  = angular velocity and r = radius Arc length= s  r where r = radius and  = central angle in radians rs Area of sector = where r = radius, s = arc length 2 r 2 Area of a sector = where r = radius, s = arc length 2  = central angle in radians 4h 2  4dh  x 2  0 where h = height of segment, d = diameter of circle and x = length of chord o o AT  a  m1  m2  m3  . . .  mn  where a = width of equal parts, m1  1 2 and 2 n = number of ordinates OR o o  AT  a  1 n  o2  o3  o4  . . .  on 1  where a = width of equal parts, on = nth  2  ordinate and n = number of ordinate Copyright reserved