Technical Mathematics Grade 12 P2 NSC November 2025 QP

Confidential basic education [SZ \\ y 8 Department: Ch), Basic Education Sa REPUBLIC OF SOUTH AFRICA TECHNICAL MATHEMATICS P2 NOVEMBER 2025 MARKS: 150 TIME: 3 hours This question paper consists of 17 pages, a 2-page information sheet and a 30-page SPECIAL ANSWER BOOK. Copyright reserved Please turn over

Technical Mathematics/P2 2 DBE/November 2025 NSC Confidential INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. 2. This question paper consists of 11 questions. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. Clearly show ALL calculations, diagrams, graphs, etc. that you used in determining your answers. Answers only will NOT necessarily be awarded full marks. If necessary, round off answers to TWO decimal places, unless stated otherwise. Diagrams are NOT necessarily drawn to scale. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. An information sheet with formulae is included at the end of the question paper. Write neatly and legibly. Copyright reserved Please turn over

Technical Mathematics/P2 3 DBE/November 2025 NSC Confidential QUESTION 1. The diagram below shows parallelogram ABDC with vertices A(0;-4), B(-2;-3), D and c( 332 ). AB ||CD and AC || BD. The angle of inclination of AC with the positive x-axis is @. 1 1,2 1.3 14 1.5 1.6 1.7 N ¥ D C( 352) js O Bi|-2; -3) A(0; -4} Write down the length of OA. qd) Determine the midpoint of AB. (2) Determine the gradient of AC. (2) Hence, determine the size of angle 0. (2) Complete the following statement: If two lines are parallel, then their gradients are ... (1) Hence, determine the equation of BD in the form y =... (3) Determine the gradient of a line that passes through point B and has an inclination of 2 a, where cosa = 2 Q) [14] Copyright reserved Please turn over

Technical Mathematics/P2 4 DBE/November 2025 . NSC Confidential QUESTION 2 2.1 In the diagram below, O is the centre of the circle which passes through points L(13;0) and N(—5;4). Tangents MN and ML touch the circle at points N and L respectively. ML || y-axis. > 3 Pa 7 eo \ i) y L(13; 0} N(-5; #)\ / NM 2.1.1 Determine the equation of the circle. (2) 2.1.2 Determine the numerical value of k. (2) 2.1.3 Hence, determine the coordinates of point M, the point of intersection of the two tangents. (5) 2.2 Sketch th h defined b aed 1 . aph de’ : ta etch the grap! ry’ 35° 4 @) [12] Copyright reserved Please tum over

Technical Mathematics/P2 5 NSC Confidential QUESTION 3 3.1 Given: A=72° and B = 30,5° Determine the numerical value of ,/sinB + secA 3.2 Given: sind => and tand <0 DBE/November 2025 Determine, without the use of a calculator, the numerical value of the following: 3.2.1 cos 0 3.2.2 cot — cosecd 3.3 Solve for x: cotx =—0,587 for x € [0°;360°] QUESTION 4 4.1 Given the expression: sin (180° + x)-sin(360° — x) + cos(2m — x)-cosx 1 sin x tan(180° + x) 4.1.1 Complete the reduction: tan(180° + x) =... 4.1.2 Complete the quotient identity in terms of sine and cosine: cotx = 4.1.3 Write down any TWO values of x for which the expression is undefined if x € [0°;360°] 4.1.4 Hence, simplify the given expression fully. sin@ —cos@-sind _ 4.2 Given: hi cos@ — (1 — sin’ 6) an@ 4.2.1 Factorise the expression: sin @ — cos@:sin@ sin@ —cos6-sin@ _ 4.2.2 Hence, show that. ——————__,__ = cos — (1 — sin“ #) Copyright reserved an @ Please turn over (3) @) (3) (4) [13] () QQ) (2) (5) () (3) [13]

Technical Mathematics/P2 6 DBE/November 2025 NSC Confidential QUESTION 5 The graph below represents the functions defined by g(x) = cos(x — p) and A(x) = sinmx for 0°< x <180° R(165° ; —0,5) is a point of intersection of g and h. 1s —— & 0 45° 90° 180°” 4 R(165°: -0,5)s ~N “AT \ -|+ = 5.1 Determine the values of p and m. (2) 5.2 Write down the period of h. qd) 5.3 Write down the maximum value of g. qd) 5.4 Use the graph above to write down the values of x for which: 5.4.1 g(x) < A(x) (2) 5.4.2 2(x)-h(x) 20 (4) 5.5 If the graph of A is shifted 1 unit downwards, write down the new equation of h. qa) [11] Copyright reserved Please turn over

Technical Mathematics/P2 7 DBE/November 2025 NSC Confidential QUESTION 6 In the diagram below, SRPM is a trapezium with PM = 20m, SR = 30m, P = 120° and RMP= 30°. SR || MP 30m Ss R M 30° 12%, 20m 6.1 Write down the size of MRP. (1) 6.2 What type of triangle is AMRP? qd) 63 Determine the length of MR. (Leave your answer in simplified surd form.) (3) 6.4 Write down the size of MRS, Give a reason for your answer. (2) 6.5 Hence, determine whether AMRS is a right-angled triangle. (5) [12] Copyright reserved Please turn over

Technical Mathematics/P2 8 DBE/November 2025 NSC Confidential Give reasons for your statements in QUESTIONS 7, 8 and 9. QUESTION 7 7A Below is drawn a circle with centre O. A, B, C, D and E are on the circle. CA is a diameter. AB, CD and DE are drawn. BC | AE Ai = 20° WAL Give a reason why B=90°. qd) 7A2 Determine, with reasons, the size of D. (6) Copyright reserved Please turn over

Technical Mathematics/P2 9 DBE/November 2025 NSC Confidential 7.2 In the diagram below, A, B, C and D are four points on the circumference of the circle. Chords BD and AC intersect at point E. Chords DC and AB are drawn. D A 7.2.1 Complete the statement of the following theorem: Angles subtended by a chord of a circle, on the same side of the chord, are... Q) 7.2.2 Write reasons for the statements given below: STATEMENT REASON (a) CDB=CAB dd) (b) Ei =E: q) (c) ADEC|||AAEB q) [8] Copyright reserved Please turn over

Technical Mathematics/P2 10 DBE/November 2025 NSC Confidential QUESTION 8 8.1 In the diagram below, G, B, A and D are points on a circle with centre C. GD, GA and GB are drawn. AB = AD ACB = 40° G B D=———— A 8.1.1 Write down TWO angles that will be equal if AB = AD. (1) 8.1.2 Determine, with a reason, the size of G : (2) 8.1.3 Give a reason why A = Bi. qd) 8.1.4 Determine, with a reason, the size of A Le (2) 8.1.5 Determine, with a reason, the size of DAC. @) Copyright reserved Please turn over

Technical Mathematics/P2 ll DBE/November 2025 NSC Confidential 8.2 Complete the statement of the following theorem by filling in the missing information: The (8.2.1) ... angle of a cyclic quadrilateral is equal to the (8.2.2) ... opposite angle. (2) 8.3 In the diagram below, A, B, C and D lie on the circumference of the circle. E is the point of intersection of chords BD and AC. Chord BC is produced to F. BAD = 79° and Bz = 30° 8.3.1 Determine, with reasons, the size of the following angles: @ @) (b) De (2) (©) Di ifitis given that AB || CD (3) 8.3.2 Show that CE = DE (4) [22] Copyright reserved Please turn over

Technical Mathematics/P2 12 NSC Confidential DBE/November 2025 QUESTION 9 9.1 AABC with BC||DE is drawn below. AE = 4 units BE = 8 units CD = 12 units A 4 units D E 12 units 8 units Cc ; B 9.1.1 Complete the following statement and reason: STATEMENT REASON * = AE Prop. theorem; ... || ... ” Q) 91.2 © Hence, or otherwise, determine the length of AD. (2) Copyright reserved Please turn over

Technical Mathematics/P2 13 DBE/November 2025 NSC Confidential 9.2 Given: ADEF with EF = 4 units, DE = 3 units and DF = 5 units, AGHI with HI = 10 units, GH = 7,5 units and GI = 12,5 units G 12,5 units 7,5 units oN E 4units F H 10 units i Prove that ADEF|||AGHI. (4) [8] Copyright reserved Please turn over

Technical Mathematics/P2 14 DBE/November 2025 NSC Confidential QUESTION 10 10.1 The diagram below models the rotating blade of a lawnmower, as shown in the picture alongside it. The circular body of the lawnmower has a radius BD = 310 mm. The length of the rotating blade from the centre of the circular body, D, to its edge, F, is 300 mm. e The blade rotates at an angular velocity of 377 radians per second. Picture of a rotating blade Diagram of the rotating blade Body of lawnmower Blade of lawnmower Determine: 10.1.1 The rotational frequency of the blade, in revolutions per second (3) 10.1.2 The circumferential velocity of the rotating blade in millimetres per second (3) 10.1.3 The height (A) of the minor segment of chord AB if CD 1 AB and AB = 130 mm (5) Copyright reserved Please turn over

Technical Mathematics/P2 15 DBE/November 2025 NSC Confidential 10.2 The picture and the diagram below show a sector in a circular garden with a radius of 4,5 m. The size of obtuse angle BOC = 115° PICTURE DIAGRAM B Cc 10.2.1 Convert 115° to radians. qd) 10.2.2 Determine the area of minor sector BOC. (3) 10.2.3 The gardener has to plant flower seedlings in the shaded area BDEC, as shown in the diagram below. Theratioof BD : DO=2: 3. Determine the number of seedlings needed to be planted in the shaded area if 4 seedlings are planted per square metre. (5) [20] Copyright reserved Please turn over

Technical Mathematics/P2 16 DBE/November 2025 NSC Confidential QUESTION 11 11.1 The diagram below shows an irregular shape with one straight side of length 120 cm, divided into 4 equal parts. e The ordinates dividing these parts are xem, ycm, 50cm, 60cm and 2xcm respectively. oo | a 5 g 6 : 3 7 o 2 a + > nm Noy 120 cm 11.1.1 Write down the length of each equal part. (i) 11.1.2 Write down the value of y, the 2 ordinate, if it is 20 cm longer than the middle ordinate. (1) 11.1.3 Use the mid-ordinate rule to determine the value of x if the area of the irregular shape is 7 200 cm?. (4) Copyright reserved Please turn over

Technical Mathematics/P2 17 DBE/November 2025 NSC Confidential 11.2 A company melts cylindrical steel rods to manufacture solid steel ball bearings, as shown in the picture below. The diagram below each picture gives the dimensions of each shape. e The cylindrical steel rod has a diameter of 20 cm and a height of 0,5 m. e The steel ball bearing has a diameter of 40 mm. Solid cylindrical steel rod Solid steel ball bearing Dimensions of solid cylindrical steel rod Dimensions of steel ball bearing The following formulae may be used: Total surface area of cylinder = 277? + 2arh Volume of cylinder = zr7h 4 Total surface area of sphere = 477? Volume of sphere = 3 11.2.1 Convert 0,5 m to centimetres. (1) 11.2.2 Calculate the length of the radius of the steel ball bearing in centimetres. (2) 11.2.3 Determine the total surface area of the cylindrical steel rod in cm”. (2) 11.2.4 Determine whether more than 400 steel ball bearings can be manufactured from one melted cylindrical rod if there is 18% loss of steel during the melting process. (6) [17] TOTAL: 150 Copyright reserved

Technical Mathematics/P2 DBE/November 2025 NSC Confidential INFORMATION SHEET: TECHNICAL MATHEMATICS -b +b? —4ac b _ 4ac—b? x= x=—-— 2a 2a » 4a a*=bax=log,b, a>0,a4land b>0 A=P(1+ni) A=P(1-ni) A=P(1+i)" A=P(1-i)" ieg =(14 5) 1 fie jim, fies) f(x) n+l x fax’ de =k- +C ,n¥-l1 n+] [Ade=e-mx+e, x>0 x nx a fka™ de =k- +C ,a>0 nina “ +X, + d= —HP +0, -nY u(2 een) 2 2 yomete yy, =m(x - x,) m =22— tan 0= m yh 2 2 x St+oy=l a b a 5 ig a=b +c -2bc-cosA snA sinB~ sinC Area of A ABC =5 ab, sin C sin? @ + cos*@=1 1+ tan? @=sec’@ 1+ cot?6 = cosec’@ Copyright reserved

Technical Mathematics/P2 DBE/November 2025 NSC Confidential mrad = 180° Angular velocity =@ = 27n where n = rotation frequency Circumferential velocity = v = zDn where D = diameter and n = rotation frequency Circumferential velocity = v = wr where @ = angular velocity and r = radius Arc length = s =r0 where r = radius and @ = central angle in radians Area ofasector = > where r = radius, s = arc length Area of asector = ve where ¢ = radius and @ = central angle in radians 4h? -4dh+x? =0 where h = height of segment, d = diameter of circle and x = length of chord A,=a(m,+m,+m,+...+m,) where a = width of equal parts, m, = ao o, =n" ordinate and n = number of ordinates n OR 0, +9, . ; A= ! 5 24 0, +0; ++ ons) where a = width of equal parts, 0, = n” ordinate and m = number of ordinates Copyright reserved