Technical Mathematics Grade 12 P1 Gauteng Mock 2020 QP (Eng)

GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION 2020 TECHNICAL MATHEMATICS PAPER 1 11091 TIME: 3 hours MARKS: 150 11 pages + 2 information sheets P.T.O.

TECHNICAL MATHEMATICS 2 (Paper 1) 11091/20 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. 3. Number the answers correctly according to the numbering system used in this question paper. 4. Clearly show ALL calculations, diagrams, graphs, et cetera which you have used in determining your answers. 5. Answers only will not necessarily be awarded full marks. 6. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise. 7. If necessary, round-off answers to TWO decimal places, unless stated otherwise. 8. Diagrams are NOT necessarily drawn to scale. 9. Write neatly and legibly. P.T.O.

TECHNICAL MATHEMATICS 3 (Paper 1) 11091/20 QUESTION 1 1.1 Solve for 𝑥: 1.1.1 2( x + 3)(x −1) = 0 (2) 1.1.2 x + 7 = −3x 2 ; x {Complex numbers} (5) 1.1.3 x 2 + 3x − 4  0 (3) 1.2 Solve for x and y simultaneously in the following equations: 3 y + x = 2 and y 2 + x = xy + y (5)  a − b  log a + log b 1.3 Show that a2 − 6ab + b2 = 0 , if log  = . (5)  2  2 1.4 The formula for the focal length f, of a convex lens is given by: 1 1 1 = + f u v 1.4.1 Make v the subject of the formula. (3) 1.4.2 Hence, determine the value of v, if u = 6 and f = 5. (2) [25] QUESTION 2 p Given: 3x 2 = 2x − 4 2.1 Calculate the discriminant in terms of p. (3) 2.2 Hence, calculate the value(s) of p if: 2.2.1 The roots are rational. (2) 2.2.2 The roots are non-real. (2) [7] P.T.O.

TECHNICAL MATHEMATICS 4 (Paper 1) 11091/20 QUESTION 3 3.1 Subtract 1011012 from 1101012. (2) 3.2 Solve for x without the use of a calculator: 1 3.2.1 3x = (2) 27 5 3.2.2 log( x + 2) + log( 3x −1) = 3 log 2 + log 4 (6) [10] QUESTION 4 The given Argand diagram shows the point Z (−4, −3) . T is a point on the real axis with reflex angle TÔZ =  . y  T Z(–4 ; –3) 4.1 Calculate: 4.1.1 The Modulus (1) 4.1.2 The Argument (3) 4.2 Express Z in polar form. (2) 4.3 Solve for p and q: 3 p + 2qi = (4 − 5i) 2 + −121 (5) [11] P.T.O.

TECHNICAL MATHEMATICS 5 (Paper 1) 11091/20 QUESTION 5 2 Consider 𝑓 (𝑥) = 𝑥 + 1, where x < 0 and g (x) = 2.3x + 1 5.1 Write down the asymptotes of f. (2) 5.2 Calculate the x-intercepts of f. (2) 5.3 Write down the range of g. (2) 5.4 Calculate the y-intercepts of g. (2) 5.5 Write down the domain of g. (2) 5.6 Is the graph of g an increasing or decreasing function? (1) 5.7 Draw on the same set of axes, the graphs of f and g . Clearly show the asymptotes and the intercepts with the axes. (6) [17] P.T.O.

TECHNICAL MATHEMATICS 6 (Paper 1) 11091/20 QUESTION6 Given: f ( x) = − x 2 − 2 x + 8 and g ( x) = mx + c. Both f and g intersect at point D(−2,5 ; t ). A(−4 ; 0) and B are the x-intercepts of f. The angle of inclination of the straight line is 135°. y D (–2,5 ; t) 6.1 Determine the co-ordinates of C, the turning point of f. (3) 6.2 Hence, express f in the form 𝑓(𝑥) = 𝑎(𝑥 + 𝑝)2 + 𝑞. (1) 6.3 Determine the value of t. (2) 6.4 Determine the equation of g. (4) 6.5 Determine the co-ordinates of H, if CE is parallel to the y-axis. (2) 6.6 Write down the length of EB. (3) [15] P.T.O.

TECHNICAL MATHEMATICS 7 (Paper 1) 11091/20 QUESTION 7 7.1 Calculate how long it will take a fully charged car battery measuring 12,6 V to lose its charge to 4 V if it loses 5% of its initial charge per day. (4) 7.2 The following advertisement appeared in one of the local newspapers with regards to buying a generator on a hire purchase agreement option. Purchase price: R6 000 Deposit: R600 Loan term: Only 18 months at 9% p.a. simple interest 7.2.1 Calculate the monthly amount that the buyer must budget for, in order to pay for the generator. (4) 7.2.2 How much interest does the buyer have to pay over the full term of the loan? (2) 7.3 Isaac made an initial deposit of R15 000 into an account that paid interest at 9,6% p.a. compounded quarterly. Six months later, he withdrew R5 000 from the account. Two years after the initial deposit he deposited a further R3 500 into this account. How much money does he have in the account three years after his initial deposit was made? (4) [14] P.T.O.

TECHNICAL MATHEMATICS 8 (Paper 1) 11091/20 QUESTION 8 8.1 Given: f ( x) = −2 + 3x Determine the derivative of f, from First Principles. (5) 8.2 Determine: dy x3 − x2 +  8.2.1 if y = (4) dx x x 1 8.2.2 f ' ( x) if f ( x) = − 3 (4) 2 6x 8.3 Calculate the average speed between the points t = 2s en t = 5s if the relationship between the distance (in metres) and time (in seconds) is given by s(t ) = 3t 2 − 4t − 6. (4) [17] P.T.O.

TECHNICAL MATHEMATICS 9 (Paper 1) 11091/20 QUESTION 9 The sketch below shows the graph of f ( x) = x 3 + 5x 2 − 8x − 12. The point A is a local maximum and point B is a local minimum, and point C is the y-intercept of f. 9.1 Determine the co-ordinates of C. (2) 9.2 Determine the co-ordinates of the turning points A and B. (6) 9.3 Calculate the value(s) of x for which f strictly increases as x increases. (2) 9.4 If h( x) = f ' ( x), determine for which value(s) of x will h(x)  0. (2) 9.5 Determine the equation of the tangent to f at C. (4) [16] P.T.O.

TECHNICAL MATHEMATICS 10 (Paper 1) 11091/20 QUESTION 10 A wheel which turns through an angle of  radians in time t seconds is given by  (t ) = 256t − 24t 2 . 10.1 Calculate the instantaneous angular velocity after 3 seconds. (4) 10.2 Calculate the instantaneous angular acceleration after 3 seconds if:  ' (t) = −48t + 256 (2) [6] P.T.O.

TECHNICAL MATHEMATICS 11 (Paper 1) 11091/20 QUESTION 11 11.1 Use the rules of integration to determine: 11.1.1 7 ∫ (2𝑥 3 − 3𝑥 2 − ) 𝑑𝑥 𝑥 (4) 11.1.2 3 ∫ (3𝑥 + ) 𝑑𝑥 𝜋 (3) 11.2 The diagram below shows the shaded area bounded by the curve defined by: f ( x) = x 3 − 6 x 2 + 11x − 6. Determine the area of the shaded region of the graph of f bounded by the graph and the x-axis between x = 2 and x = 3. (5) [12] TOTAL: 150 END

TECHNICAL MATHEMATICS 12 (Paper 1) 11091/20 INFORMATION SHEET: TECHNICAL MATHEMATICS − b  b 2 − 4ac b 4ac − b2 x= x=− y= 2a 2a 4a 𝑎 𝑥 = 𝑏 ⇔ 𝑥 = 𝑙𝑜𝑔𝑎 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 m  im  ieff = 1 +  − 1  m   f ( x + h) − f ( x ) f / ( x ) = lim h→0 h xn +1  x dx = +C , n  −1 n n +1 1 𝑎𝑥 ∫ 𝑥 𝑑𝑥 = 𝑙𝑛( 𝑥) + 𝐶, 𝑥 > 0 ∫ 𝑎 𝑥 𝑑𝑥 = 𝑙𝑛 𝑎 + 𝐶, 𝑎 > 0  x + x2 y1 + y2  d = ( x2 − x1 ) 2 + ( y2 − 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: = = a 2 = b2 + c 2 − 2bc. cosA sin A sin B sin C 1 area of ΔABC = 2 𝑎𝑏 ⋅ sin C sin2 θ + cos2 θ = 1 1 + tan2 θ = sec 2 θ cot 2 θ + 1 = cosec 2 𝜃

TECHNICAL MATHEMATICS 13 (Paper 1) 11091/20  rad = 180 Angular velocity = 𝜔 = 2𝜋𝑛 = 3600 𝑛 where n = rotation frequency Circumferential velocity = 𝜈 = 𝜋𝐷𝑛 where D = diameter and n = rotation frequency s = r where r = radius and 𝜃 = central angle in radians 𝑟𝑠 𝑟2𝜃 Area of 𝑎 sector = 2 = 2 where r = radius, s = arc length and 𝜃 = central angle in radians 4ℎ2 − 4𝑑ℎ + 𝑥 2 = 0 where h = height of segment, d = diameter of circle and x = length of chord  o +o  AT = a  1 n + o2 + o3 + o4 + . . . + on−1  where a = equal parts, oi = ith ordinate and  2  n = number of ordinates OR o +o AT = a ( m1 + m2 + m3 + . . . + mn ) where a = equal parts, m1 = 1 2 and 2 n = number of ordinates