Technical Mathematics Grade 12 P1 Gauteng Mock 2018 QP

Downloaded from hlayiso.com GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION 2018 11091 TECHNICAL MATHEMATICS PAPER 1 TIME: 3 hours MARKS: 150 8 pages, + 2 information sheets and 1 answer sheet P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 2 (Paper 1) 11091/18 GAUTENG DEPARTMENT OF EDUCATION PREPARATORY EXAMINATION TECHNICAL MATHEMATICS (Paper 1) TIME: 3 hours MARKS: 150 INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 9 questions. 2. Answer ALL the questions. 3. Answer QUESTION 4.2.2 and QUESTION 7.4 on the ANSWER SHEET provided. 4. Number the answers correctly according to the numbering system used in this question paper. 5. Clearly show ALL calculations, diagrams, graphs, etc. that you used in determining your answers. 6. Answers only will not necessarily be awarded full marks. 7. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. 8. If necessary, round-off answers to TWO decimal places, unless stated otherwise. 9. Diagrams are NOT necessarily drawn to scale. 10. Write neatly and legibly. P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 3 (Paper 1) 11091/18 QUESTION 1 1.1 Given: 𝑓(𝑥) = (𝑥 − 3)(𝑥 + 5) and 𝑔(𝑥) = 𝑥 2 − 36 Solve for x if: 1.1.1 𝑔(𝑥) = 0. (3) 1.1.2 f (x) ≤ 0 and represent the solution on a number line. (3) 1.1.3 f (x) = 5, rounded-off to TWO decimal places. (5) (2,4 × 103 )(3 ×10−2 ) 1.2 Write the simplified value of in scientific notation (4,8 × 104 ) without any rounding. (1) 1.3 Solve algebraically for x and y simultaneously if: x+y = 4 and 2x2 + y = 4 + 5x (6) 1.4 An approximate relationship between the number of teeth, T on a milling cutter, the diameter of the cutter D, and the depth of the cutter d is given by: 12,5𝐷 T= 𝐷+4𝑑 1.4.1 Make d the subject of the formula. (3) 1.4.2 Hence, calculate the value of d if T = 10 and D = 32. (2) 1.5 Given: 𝑚2 + 𝑛2 = 41 and 𝑚𝑛 = 9 1.5.1 Show that the numerical value of (𝑚 + 𝑛)2 = 59. (2) 1.5.2 Convert 59 to a binary number. (2) [27] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 4 (Paper 1) 11091/18 QUESTION 2 √25−2𝑥 2.1 Given: M = 𝑥 Determine the value(s) of x for which M will be: 2.1.1 Undefined (1) 2.1.2 Real (2) 2.1.3 Rational, where 𝑥 ∈ ℕ (1) 2.2 Determine the nature of the roots of 6𝑥 2 − 6𝑥 + 1 = 0 without solving the equation. (3) [7] QUESTION 3 3.1 Simplify the following expressions fully without using a calculator. 3.1.1 2 log 2 + log 25 (3) 52006 − 52004 + 24 3.1.2 (3) 52004 + 1 3.2 Solve for x: 1 3.2.1 2𝑥 4 = 10 (2) 3.2.2 log 7 (5𝑥 + 2) – 2 log 7 𝑥 = log 7 3 (5) 3.3 Given the complex number: 𝑧 = −5 + 12𝑖 3.3.1 Determine the modulus of z. (2) 3.3.2 Determine the argument of z. (3) 3.3.3 Hence, or otherwise, express z in trigonometric (polar) form. (1) 3.4 Solve for x and y if x + 2i – y = yi – 3xi. (4) [23] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 5 (Paper 1) 11091/18 QUESTION 4 𝑎 4.1 The graphs of the circle x2 + y2 = 10 and g(x) = + k are shown in the 𝑥 figure below. B(–3 ; 3) is a point on g. B(–3 ; 3) g • y=2 g 4.1.1 Determine the values of a and k. (3) 4.1.2 Write down the radius of the circle in surd form. (1) 4.1.3 Write down the domain of the circle. (2) 4.1.4 Write down the range of g. (2) 1 3 4.2 Consider the functions defined by f (x) = – 𝑥 2 + 𝑥 + and g(x) = 3x – 1. 2 2 4.2.1 Determine: (a) The y – intercept of f (1) (b) The x – intercepts of f (4) (c) The coordinates of the turning point of f (4) (d) The equation of the asymptote of g (1) (e) The x – intercept of g (2) 4.2.2 On the ANSWER SHEET provided, draw neat sketch graphs of f and g. Clearly show all salient points. (6) 4.2.3 On the graphs that you have sketched, shade the region represented by (1) f (x) ≥ g (x). [27] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 6 (Paper 1) 11091/18 QUESTION 5 5.1 The nominal interest rate charged on an investment is 8,95% compounded quarterly. Calculate the annual effective interest rate for the investment. (3) 5.2 Sipho bought new equipment for his business three years ago for R350 000. This equipment depreciated to R179 200 on the reducing balance method. Calculate the annual rate of depreciation. (3) 5.3 Zanele deposited R20 000 into a savings account at 9,75% p.a. compounded quarterly. Eighteen (18) months later, she deposited a further R10 000 into the same account. Three (3) months after the second deposit was made, the interest rate changed to 9,95% p.a. compounded monthly. Determine the balance in Zanele’s account three years after the account was opened. (7) [13] QUESTION 6 6.1 Determine the average gradient of p(x) = 3x2 – 2x between x = –3 and x = 3. (4) 6.2 Determine 𝑓 ′ (x) from FIRST PRINCIPLES if f (x) = 2x + 1. (4) 𝑑𝑦 5 6.3 Determine if 𝑦 = (𝑥 − 3)2 − (5) 𝑑𝑥 𝑥 6.4 The tangent to the curve of y = x2 – 4x is perpendicular to the line −1 y = 𝑥 + 4. Determine the x – coordinate of the point of tangency. 2 (The point where the tangent and the curve touch) (4) [17] QUESTION 7 Given: f (x) = x3 – 2x2 – 4x + 8 7.1 Show that (x – 2) is a factor of f. (2) 7.2 Hence, or otherwise, factorise f completely. (3) 7.3 Determine the coordinates of the turning points of f. (5) 7.4 Sketch the graph of f on the ANSWER SHEET provided. Clearly show all the intercepts with the axes and the turning points. (4) [14] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 7 (Paper 1) 11091/18 QUESTION 8 Mpho and Isaac were given 30 m of masking tape to use for marking the floor to indicate the dance area at their matric farewell. They wanted the dance floor to be as large as possible. Relevant formulae for Question 8: Area of rectangle = l × b Perimeter of rectangle = 2l + 2b Area of circle = 𝜋𝜋𝑟𝑟 2 Circumference of circle = 2𝜋𝜋 𝜋𝜋 8.1 Mpho was of the opinion that the dance floor should have the shape of a square as it would be the rectangle with the largest area. 8.1.1 If the length of the rectangle is x, show that the breadth y, can be written as y = 15 – x. (2) 8.1.2 Write an expression in terms of x for the area of the dance floor. (2) 8.1.3 Determine the dimensions of the dance floor with the largest possible area. (3) 8.1.4 Hence, or otherwise, determine the maximum area of the dance floor. (1) 8.2 Isaac, on the other hand, maintained that a circular dance floor would ensure the largest possible area. He had written the following formulae down: Circumference of circle = 2𝜋𝜋𝑟𝑟 ; Area of circle = 𝜋𝜋𝑟𝑟 2 . 8.2.1 If r is the radius of a circle with a circumference of 30 m, determine the value of r in terms of 𝜋𝜋. (2) 8.2.2 Hence, or otherwise calculate, correct to two decimal places, the area of a circle with a circumference of 30 m. (Use 𝜋𝜋 = 3,14) (2) 8.3 Based on your calculations above, which shape of dance floor has the greatest area, given the 30 m masking tape? (1) [13] P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 8 (Paper 1) 11091/18 QUESTION 9 9.1 Use the rules of integration to determine the following integral: 5𝑥 2 +1 ∫� 𝑥 �dx (4) 9.2 The velocity v of an object after moving for a certain time , t seconds, is given by v = (2t2 + 5) m/s. The graph below shows the object’s movement for t = 0 to t = 4 seconds. Determine, using integration, the distance travelled in the interval t = 0 to t = 4 seconds, to the nearest integer. (5) [9] TOTAL: 150 P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 9 (Paper 1) 11091/18 INFORMATION SHEET: TECHNICAL MATHEMATICS − b ± b 2 − 4ac b 4ac − b 2 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 m  im  = ieff  1 +  −1  m   f ( x + h) − f ( x) f / ( x ) = lim h →0 h  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 θ x 2 − x1 x2 y2 + =1 a2 b2 a b c In ∆ABC: = = a 2 = b 2 + c 2 − 2bc. cos A sin A sin B sin C area of ∆ ABC = 1 ab ⋅ sin C 2 sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ cot 2 θ + 1 = cos ec 2θ f ( x + h) − f ( x) x n +1 f / ( x ) = lim ∫ x= n dx +C , n ≠ −1 h →0 h n +1 1 ax ∫ x dx = ∫ x ln( x) + C , x>0 a dx =+ C , a>0 ln a P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 10 (Paper 1) 11091/18 π rad = 180° Angular velocity= ω= 2π n= 3600 n where n = rotation frequency Circumferencial velocity= ν= π Dn where D = diameter and n = rotation frequency s = rθ where r = radius and θ = central angle in radians rs r 2θ = Area of a sector = where r = radius, s = arc length and θ = central angle in 2 2 radians 4h 2 − 4dh + x 2 = 0 where h = height of segment, d = diameter of circle and x = length of chord o +o  = A T a  1 n + o2 + o3 + o4 + . . . + on −1  where a = equal parts, oi = ith ordinate and  2  n = number of ordinates OR o +o A T= a ( m1 + m2 + m3 + . . . + mn ) where a = equal parts, m1 = 1 2 2 and n = number of ordinates P.T.O.

Downloaded from hlayiso.com TECHNICAL MATHEMATICS 11 (Paper 1) 11091/18 ANSWER SHEET Name of learner School Grade QUESTION 4.2.2 QUESTION 7.4 END