MATHEMATICS PAPER 1 COLLECTABLE MARKS NW 2020 1 1 1

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https://ultradeep.co.za Mathematics/P1 2 Collectable Marks/ NW2020 ALGEBRAIC EXPRESSIONS, EQUATIONS AND INEQUALITIES ATP: GARDE 10 ATP: GRADE 11 Rational and irrational numbers Theory of numbers Linear equations Quadratic equations Solve quadratic equations by Factorisation Taking square roots Completing the square Using quadratic formula Literal equations (changing subject of the formula) Linear inequalities (interpret Quadratic inequalities (interpret solutions solutions graphically) graphically) System of linear equations Equations in two unknowns One equation linear and the other quadratic Nature of roots Word problems Word problems (modelling) Exponential laws and simplify using Simplify expressions using laws of exponents (rational exponents) Add, subtract, multiply and divide surds Exponential equations Solve equations using laws of exponents Solve simple equations involving surds VOCABULARY Expression It is made up of constants, variables and number operations It only be SIMPLIFIED (write it in simple form) by grouping, adding, subtracting like terms or factorising. Equations Expressions with equal sign. Main question is SOLVE When solving equations, you need to find the unknown values Quadratic formula b  b 2  4ac x 2a Discriminant 2   b 2  4ac where y  ax  bx  c Perfect square A number that is the square of an integer Standard form Linear: ax  by  c 2 Quadratic: ax  bx  c  0 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 3 Collectable Marks/ NW2020 QUESTION 1 1.1 Solve for x : 1.1.1 (3) x  x  1  6 1.1.2 (4) 3x 2  4 x  8 (correct to TWO decimal places.) 1.1.3 (4) 4x2  1  5x 1.1.4 (4) 2x  3  x  0 1.2 Solve for x and y simultaneously: (6) 2 2 y  2 x  3  0 and x  y  x  y [21] QUESTION2 NW SEP 2015 2.1 Solve for x: 2.1.1 (4) 2.1.2 (Leave your answer correct to TWO decimal places.) (4) 2.1.3 (5) 2.2 Solve for x and y simultaneously: (7) 2.3 (2) For which values of m will be a factor of ? [22] QUESTION 3 NW SEP 2016 3.1 Solve for x : 3.1.1 (2) 7 x  2 x  1  0 3.1.2 (4) 2 x2  x  4 (Leave your answer correct to TWO decimal places.) Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 4 Collectable Marks/ NW2020 3.1.3 (3)  x  4  x  5  0 3.1.4 2 1 (4) 3x 5  5 x 5  2  0 3.2 Solve for x and y simultaneously: 2x (6)  1; y  1 1 y and  3 x  y  x  y   0 3.3 3 f ( x)  x 2 Given: x  2 and g ( x)  3 . Explain why f ( x )  g ( x) will have (3) only ONE root. Motivate your answer. [22] QUESTION 4 NW SEP 2017 4.1 Solve for x : 4.1.1 (3) x2  5x  6 4.1.2 (4) 2 x 2  8x  3  0 (correct to TWO decimal places.) 4.1.3 (2) x 2  64  0 4.1.4 (4) 4 x  8.2 x  0 4.2 Solve for x and y simultaneously: (7) x y 2 2 2  4 and x  52  y (West Coast ED – Sep 2014) 4.3 2 Given: 2mx  3x  8, where m  0. Determinethe value of m if the roots of the given equation are non-real. (4) 4.4 If i  1, show WITHOUT THE USE OF A CALCULATOR, that 24 1  i   4096. (4) [28] QUESTION 5 NW SEP 2018 5.1 Solve for x : 5.1.1 (2) 2 x  5 x  3  0 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 5 Collectable Marks/ NW2020 5.1.2 (4)  x2  4  5 x (Leave your answer correct to TWO decimal places.) 5.1.3 15 (5) x6 2  x6 5.1.4 (2)  x  2   x  3  0 2 5.2 Solve for x and y simultaneously: (6) 2 2 x  2 y  3 and 3x  4 xy  9 y  16  0 5.3 2022 (4) Determine the sum of the digits of: 2 .52018 [23] QUESTION 6 NW SEP 2019 6.1 Solve for x : 6.1.1 (3) 3x2  18 x  0 6.1.2 (4) 7 x2  4 x  5 (Leave your answer correct to TWO decimal places.) 6.1.3 (2)  x  5 x  2  0 6.1.4 2 (6)  26  52 x  5x  6  6.2 Solve for x and y simultaneously: (6) 2 2 x  4 y  5 and 3x  5xy  2 y  25 6.3 (4) Solve for x if: x  12  12  12  12  ... [25] PATTERNS, SEQUENCE AND SERIES ATP: GRADE 10 ATP: GARDE 11 ATP: GRADE 12 Linear patterns Linear patterns Number patterns dealt with in grade 10 and 11 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 6 Collectable Marks/ NW2020 Determine general term Determine general term using Patterns, including arithmetic sequences and series using Tn  dn  q Tn  dn  q Exponential patterns Sigma notation Number patterns leading to Derivation and application for the sum of those where there is a arithmetic series constant difference between S n  a   a  d    a  2 d  ...a   n  1  d consecutive terms, and the general term is therefore   quadratic. n  S n   2 a   n  1  d  Tn  an 2  bn  c 2 Number patterns, including geometric a  b  c  First term of sequences and series quadratic sequence Derivation and application of the formula for 3a  b  First term from the the sum of geometric series row of first difference S n  a  ar  ar 2  ..  ar n 1    Sn   a r n 1  r 1 Sigma notation 2a  Second difference Infinite geometric series Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za VOCABULARY DEFINITION/ CLARIFICATION First term or Leading term a or T1 Common d  Tn  Tn 1 Test T2  T1  T3  T2 difference Which term or How many terms n or n th term Arithmetic Each term is the result of adding the same number (called the common sequence difference) to the previous term. Tn  a   n  1 d Common ratio Tn T2 T3 r  Tn 1 Test T1 T2 Arithmetic series n n Sn   2a   n  1 d  Sn   a    2 or 2 where   Tn Geometric Each term of G. P is found by multiplying the previous term by a fixed sequence n 1 number (called common ratio). Tn  a.r Geometric series Sn  a ( r n  1) Sn   a 1 r n  r 1 or 1 r Infinite series a S  1  r , 1  r  1 Quadratic 2 sequence The sequence whose second difference is constant: Tn  an  bn  c Sigma notation n T n The sum of k 1 Find Tn given Sn Tn  Sn  Sn 1 QUESTION 7 7.1 The fifth term of an arithmetic sequence is zero and the thirteenth term is equal to 16. Determine: 7.1.1 The common difference (4) 7.1.2 The first term (2) 7.1.3 The sum of the first 21 terms (4) 7.2 1 3; 1; ; ... (4) Determine the twelfth term of the geometric sequence 3 7.3 Cells are continually dividing and thus increasing in number. A cell divides and becomes two new cells. The process repeats itself forming a geometric sequence. The following sketch represents this cell division. Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 8 Collectable Marks/ NW2020 How many cells will there be altogether after twenty stages? (4) [18] QUESTION 8 8.1 The following arithmetic sequence is given: – 1; 6; 13; ... Determine: 8.1.1 The 49th term (3) 8.1.2 The sum of the first 87 terms (3) 8.2 20; 16; ... is a geometric sequence. (4) Calculate the sum of the first ten terms. 8.3 The following are the consecutive terms of a geometric sequence: (6) 3x  2; 2 x  2; 4 x  1 ( x is a natural number) Calculate the value of x. 8.4 The tiles are arranged as shown below. The first arrangement has 5 tiles, the second arrangement has 9 tiles, the third arrangement has 13 tiles and the fourth arrangement has 17 tiles. The arrangements continue in this pattern th Derive, in terms of n, a formula for one of tiles in the n arrangement. (3) [19] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 9 Collectable Marks/ NW2020 QUESTION 9 9.1 The following arithmetic series is given: 5  9  13  ...  401. Calculate: 9.1.1 The number of terms in the series (4) 9.1.2 The sum of the terms in the series (3) 9.2 Given the sequence: 2; x; 18; ... (4) Calculate x if this sequence is: 9.2.1 An arithmetic sequence (3) 9.2.2 A geometric sequence (4) 9.3 10  32  1 k Given: k 1 9.3.1 Write down the first three terms of the series (3) 9.3.2 Determine the sum of the series (5) [22] QUESTION 10 10.1 The sequence 3; 9; 17; 27; ... is quadratic. 10.1.1 (4) Determine an expression for the n  th term of the sequence. 10.1.2 What is the value of the first term of the sequence that is greater than (4) 269? 10.2 2 (3) The sum of n terms in a sequence is given by S n   n  5. Determine the 23rd term. 10.3 (NW SEP 2016) 54; x; 6 are the first three terms of a geometric sequence. 10.3.1 (2) Calculate x 10.2.2 Is this geometric sequence convergent? Motivate your answer by (3) clearly showing all your calculations. 10.4 Determine the value of k for which: 60 5 (5)   3r  4   k r 5 p 2 [21] QUESTION 11NW SEP 2017 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 10 Collectable Marks/ NW2020 11.1 69; 0;  63; ... forms a quadratic sequence. 11.1.1 Write down the value of the next term in the sequence (1) 11.1.2 th (4) Calculate an expression for the n term of the sequence. 11.1.3 Determine the value of the smallest term in this sequence (4) 11.2 A finite arithmetic series of 16 terms has a sum of 632. The eleventh term if 47. 11.2.1 Calculate the 1st term of the series (5) 11.2.2 Determine the fifth term of the series (1) 11.2.3 Express the sum of the last five terms of this series in sigma notation. (2) [17] QUESTION 12 12.1 NW SEP 2018 If x  1; x  1; 2 x  5; ... are the first 3 terms of a geometric series, calculate: 12.1.1 (5) The value(s) of x 12.1.2 (4) For which value of x in QUESTION 12.1.1 will the series converge? 12.1.3 (2) The sum to infinity if x  3 12.2 NW SEP 2019 (4) Write 4  3  10  ...  486 in sigma notation. 12.3 This arithmetic sequence 11;  4; 3; ... forms the first three first difference of a quadratic sequence. Which term in this quadratic sequence will be the (5) smallest? Show all your calculations. [20] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 11 Collectable Marks/ NW2020 FUNCTIONS AND GRAPHS GRADE 10 GRADE 11 y  ax  q - linear y  ax  q - linear parabolic y  ax2  q - parabolic 2 y  a  x  p  q y  ax2  bx  c y  a  x  x1  x  x2  a a y q y q x - hyperbolic x p - hyperbolic y  ab x  q; b  0, b  1 - exponential y  a.b x  p  q; b  0, b  1 - exponential Point-by-point plotting Generalise effects of parameters a Generalise effect of a, p and q on graphs and q on graphs Drawing the graphs using parameters, Drawing the graphs using parameters, x  x  intercept(s) and intercept(s) and y  intercept and asymptotes y  intercept and asymptotes Finding equations of the functions Finding equations of the functions and and interpretation interpretation Solving problems involving all Solving problems involving all functions functions Average gradient between two points on a curve ALGEBRAIC FUNCTIONS AND INVERSES GRADE 12 Determine and sketch graphs of the Focus on the following characteristics inverses of the functions defined by Domain and range y  ax  q Intercepts with the axes y  ax 2 Turning point Minima and maxima y  b x  b  0; b  1 Horizontal and vertical asymptotes y  b x  0  b  1 Shape and symmetry Average gradient (average rate of change) y  log b x  x  b y  b  0; b  1 Interval on which graph decreases/increases y  log b x  x  b y  0  b  1; b  1 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 12 Collectable Marks/ NW2020 VOCABULARY Function A relationship that assigns one output for each input value Gradient of a line y y m 2 1 x2  x1 Average gradient f ( x2 )  f ( x1 ) m x2  x1 Line of symmetry A line that divides a figure so that the two parts of the figure are congruent. It divides a figure into mirror images Line symmetry When a figure can be folded on line so that its two parts are congruent Reflection When a figure is flipped across a line Asymptote The line that a curve will get closer and closer without ever reaching it. Axis of symmetry (A. S) b x 2a Turning point of f  b  b    ; f    2a  2a   Horizontal number line x  axis First number in an ordered pair x  coordinate x  intercept(s) x  coordinate of the point where a curve crosses the x  axis Vertical number line y  axis Second number in an ordered pair y  coordinate y  intercept y  coordinate of the point where a curve crosses the y  axis Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 13 Collectable Marks/ NW2020 2 THE COORDINATES OF THE TURNING POINT OF f ( x)  ax  bx  c USING DEFFERENTIATION 2 If f ( x)  ax  bx  c then the gradient at any point is f ( x) That is f ( x )  2ax  b At the turning point (either maximum or minimum) f ( x)  0 That is 2ax  b  0 and therefore b x 2a  b  f  The corresponding y  value is  2a  Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Function Equation Orientation Linear y  ax  q a  0 y  2 x  3 a  0 y  3x  2 Parabolic y  ax2  bx  c 2 a  0 y  x  x  3 2 a  0 y  2x  4x  3 2 2 y  a  x  p  q  1  13 y   x    2  2 4 y  2  x  1  5 Hyperbolic a 1 3 y q y 2 y 1 x p a0 x 3 a0 x2 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 15 Collectable Marks/ NW2020 Exponential x2 x 1 y  a.b x  p  q b  1 y  2.  3 1 1 y  4 0  b 1 2 Logarithmic y  log 1 ( x) y  logb k ( x  p) b  1 y  log 2 x 0  b 1 3 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za QUESTION 13 NW SEP 2015 The sketch below represents the graphs of: . The point A(−3 ; 2) is the point of intersection of the asymptotes off. The graph off intersects the x-axis at (−1 ; 0). The graph of g intersects the y-axis at (0 ; −2). 13.1 Write down the equations of the asymptotes off. (2) 13.2 Determine the equation of f. (3) 13.3 Write down the equation of the axes of symmetry of fin the form y = mx +cifm< 0. (2) 13.4 Write down the domain of 5f(x – 1). (2) 13.5 Write down the equation of k, the reflection off about the y-axis. Leave your (2) answer in the form . 13.6 (2) For which value(s) of x is ? 13.7 Determine the equation of g. (3) 13.8 Determine the equation of if . Leave your answer in (3) the formy = ... 13.9 The inverse of his not a function. Restrict the domain of h such that is a (2) function. Sketch the restricted graph of h and on the same system of axes. [21] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 17 Collectable Marks/ NW2020 QUESTION 14 The graphsof are represented in the sketch below. D, the turning point of f, is also a point of intersection of g andf. The asymptote of g passes through C,the y-intercept off. 14.1 Write down the coordinates of C. (2) 14.2 Calculate the values of a and q. (3) 14.3 Write down the range of g. (2) 14.4 Write down the coordinates of D', if D is reflected about the line y = 8. (1) 14.5 (3) If , describe the transformation from f tok. 14.6 (1) Write down the equation of in the form y = ... 14.7 (2) Determine the minimum value of . [14] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 18 Collectable Marks/ NW2020 QUESTION 15 NW SEP 2016 15.1 3 f ( x)  1 Given: x2 15.1 (2) Calculate the coordinates of the y  intercept of f . 15.2 (2) Calculate the coordinates of the x  intercept of f . 15.3 (3) Sketch the graph of f in your ANSWER BOOK, clearly showing the asymptotes and the intercepts with the axes. 15.4 (2) Write down the range of f . 15.5 Another function h, is formed by translating f , 3 units to the right and 4 (2) units down. Write down the equation of h. 15.6 (3) For which value(s) of x is h( x)  4 ? 15.7 3x  5 (3) k ( x)  . Determine the equations of the asymptotes of x 1 [17] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 19 Collectable Marks/ NW2020 QUESTION 16 NW SEP 2016 x 1 2 g ( x)    The graphs of f ( x )  2   x  1  8 and 2 are represented in the sketch below. P and Q are the x  intercepts of f and R is the turning point of f . The point A  2;4  is a point in the graph of g. 16.1 (1) Write down the equation of the axis of symmetry of f . 16.2 (1) Write down the coordinates of the turning point of f . 16.3 Determine the length of PQ. (4) 16.4 (3) Write down the equation of k , if k is the reflection of f in the y  axis. Give 2 your answer in the form y  ax  bx  c. 16.5 1 (1) Write down the equation of g , the inverse of g , in the form y  ... 16.6 1 Sketch the graph of g in your ANSWER BOOK, clearly showing the intercept (3) 1 with the axis as well as ONE other point on the graph of g . 16.7 For which value(s) of x will: 16.7.1 (2) g 1 ( x)  2 16.7.2 (4) x. f ( x )  0 [19] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 20 Collectable Marks/ NW2020 QUESTION 17 NW SEP 2017 17.1 4 h( x )  1 Given: 2 x 17.1.1 (2) Write down the equation of the asymptotes of h. 17.1.2 Sketch the graph of h. Show clearly all asymptotes and intercepts with (4) the axes. 17.1.3 x2 k ( x)  . Given: x  2 How will you use h to sketch k ? Show all your (3) calculations. 17.2 2 f ( x)   x 2  bx  c The sketch below shows the graphs of 3 and g , a straight line. Q  2;6  is the turning point of f . P and R are the x  intercepts of f . D is the y  intercept of g. The graphs of f and g intersect at P and T. 17.2.1 (3) Calculate the values of b and c. 17.2.2 13 PD  (2) Calculate the coordinates of D if 2 units. 17.2.3 (2) For which value(s) of x is x.g ( x )  0 ? 17.2.4 1 (4) Determine the equation of g in the form y  ... [20] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 21 Collectable Marks/ NW2020 QUESTION 18 NW SEP 2017 x f ( x)   The graphs of 5 for x  0 and g ( x)  log5 x are shown below. T  20;  2  is apoint on f . A is the x  intercept of g and B is a point f . AB is parallel to the y  axis. 18.1 (1) Write down the range of g. 18.2 Write down the coordinates of A. (1) 18.3 Calculate the length of AB. (3) 18.4 1 (3) It is given that h is obtained when g is shifted two units to the left. Write down the equation of h. 18.5 1 (2) Determine the value(s) of x for which f ( x)  20. [10] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 22 Collectable Marks/ NW2020 QUESTION 19 NW SEP 2018 19.1 a f ( x)   q. Given: x p ● The point B  1;0  is an x  intercept of f . ● The domain of f is real numbers, but x  2. ● The range of f is real numbers, but y  3. ● f is a decreasing function. 19.1.1 (3) Determine the equation of f . 19.1.2 (2) Determine the coordinates of the y  intercept of f . 19.1.3 Sketch the graph of f in your ANSWER BOOK, clearly showing the asymptotes and the intercepts with the axes. (3) Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 23 Collectable Marks/ NW2020 19.2 2 2 g ( x)  3 The graphs of f ( x )  a  x  p   q and x 1 are sketched below. P is the y  intercept of f and g. The horizontal asymptote of g is also a tangent to f at the turning point of f . 19.2.1 (1) Write down the equation of the vertical asymptote of g. 19.2.2 (2) Determine the coordinates of P. 19.2.3 (3) Determine the equation of f . 19.2.4 One of the axes of symmetry of g is a decreasing function. Write (2) down the equation of this axis of symmetry, h( x ). 19.2.5 For which values of k will g ( x )  h( x)  k have TWO real roots that (2) are of opposite signs? 19.2.6 (3) Give the domain of m( x ) is m( x)  g (2 x )  5. [21] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 24 Collectable Marks/ NW2020 QUESTION 20 NW SEP 2018 1 2 f ( x)  x , x The diagram below shows the curves of 4 where x  2 and g ( x)  a , where a  0. The point A  1;3 lies on the graph of g. P is the y  intercept of f and g. The horizontal asymptote of g is also a tangent to f at the turning point of f . 20.1 x (1) 1 g ( x)    . Show that  3 20.2 (2) For which value(s) of x is the graph of f strictly decreasing? 20.3 (2) Determine the inverse of f in the form y  ... 20.4 1 (2) Sketch the graph of f in your ANSWER BOOK. 20.5 1 (2) Write down the range of f . 20.6 (2) Determine the inverse of g in the form y  ... 20.7 1 (2) For which values of x will g ( x)  1? [13] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 25 Collectable Marks/ NW2020 QUESTION 21 NW SEP 2019 1 2 a g ( x)   x  2  9 h( x )  q The graphs of 2 and x p are sketched below. The axis of symmetry of graph g is the vertical asymptote of graph h. The line f is an axis of symmetry of graph h. B is the y  intercept of h, g and f . 21.1 (2) Write down the coordinates of C, the turning point of g. 21.2 Determine the coordinates of B. (2) 21.3 (2) Write down the equation of f . 21.4 (5) Determine the equation of h. 21.5 Write down the equationsof the vertical and horizontal asymptotes of (2) k ( x)  3h( x)  2. 21.6 (3) Determine the x  intercept of h. 21.7 (2) For which values of x will: 21.7.1 g ( x) (3) 0 h( x) 21.7.2 (4) f 1 ( x  1)  2 21.8 Calculate the value(s) of k for which g ( x)  f ( x)  k has two unequal positive (6) roots. [29] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 26 Collectable Marks/ NW2020 QUESTION 22 NW SEP 2019 22.1 5 x f ( x)    Consider the function 6 22.1.1 (1) Write down the equation of h, the reflection of f in the y  axis. 22.1.2 1 (2) Write down the equation of f ( x) in the form y  ... 22.1.3 1 (3) For which value(s) of x will f ( x)  0? 22.2 2 The function defined as f ( x)  ax  bx  c has the following properties: ● f ( 2,5)  0 ● f (1)  0 ● b 2  4ac  0 (4) ● f (2,5)  6 Draw a neat sketch graph of f . Clearly show all x  intercepts and turning point. [9] Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 27 Collectable Marks/ NW2020 POLYNOMIALS Remainder theorem f ( x)   ax  b  .Q( x)  R If a polynomial f ( x) is divided by  b  f  R x  k , then the remainder is r  f ( k )  a  Factor theorem  b  f  0  a  Factorise third degree polynomials Factorisation by Inspection A polynomial f ( x) has a factor x  k Using factor theorem if and only if f ( k )  0 - Long division method - Synthetic method CALCULUS Concept Formula Average gradient f ( x  h)  f ( x ) h Limit concept f ( a  h)  f ( a ) lim h 0 h Definition of derivative (first f ( x  h)  f ( x ) principles) f ( x)  lim h 0 h Power rule d n dx  x   nx n 1 Derivative of a curve dy Gradient function of a curve dx Dx  y  y f ( x) Second derivative of f f ( x ) Gradient of the tangent to a curve mtan  f ( x) Equation of tangent line y  y1  f ( x1 ).  x  x1  x  intercept(s) of f Set f ( x )  0 Turning point of a curve (local maximum or minimum point) Set f ( x)  0 Point of inflection Set f ( x )  0 VOCABULARY First principles Use definition of derivative Local maximum The y  coordinate of a turning point is higher than all nearby points Local minimum The y  coordinate of a turning point is lower than all nearby points Stationary points Maximum, minimum or point of inflection Rules of differentiation Apply power rule Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 28 Collectable Marks/ NW2020 QUESTION 23 NW SEP 2015 - 2019 23.1 Determine from FIRST PRINCIPLES the derivative of: 23.1.1 (5) f ( x)  3x  x 2 23.1.2 3 (5) g ( x)  x 23.1.3 (5) h( x)  x 2  6 x 23.1.4 (5) u ( x)  2 x 2  5 x  3 23.1.5 (6) v( x )   x 2  3 x  7 23.2 Determine: 23.2.1 dy 2 1 (4) y  2x  dx if 3x 23.2.2 2 x  5x 2 (3) dy y dx if x 23.2.3 dy 3x 1 (4) y 2 dx if 5x 2 x 23.2.4 2 x 2 2 x3  1 (5) dy y  dx if 3 x x3 23.2.5 (3) D x  3 x  3 x  23.2.6  7 x5  3x  (2) Dx    4x  23.2.7  3x 7  x  (6) D x 15 3 x 4    4 x3  Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Orientation of sketch Function graph Concave down-then- Concave up y   x 3  3x  2 a  0 a0 y  3x3  6 x 2  3x  6 Concave up-then- Concave down y  2 x3  3x 2  1 a  0 a0 y  x3  5 x2  8 x  12 Copyright reserved Please turn over https://tvetpapers.co.za

https://ultradeep.co.za Mathematics/P1 9 NW/2020 NSC QUESTION 24 NW SEP 2015 24.1 The graph of is sketched below. A and B are turning points of f 24.1.1 Determine the coordinates of A and B. (6) 24.1.2 Determine the x-coordinate of the point of inflection of f. (2) 24.1.3 Determine the equation of the tangent tof at x = 2 in the form y = mx + c. (4) 24.1.4 Determine the value(s) of k for which will have only ONE real root. (2) 24.2 The graph of y = g/(x) is sketched below, with x-intercepts atA(− 2; 0) and B( 3; 0). The y-intercept ofthe sketched graph is (0; 12). 24.2.1 (1) Determine the gradient of at x = 0. 24.2.2 For which value of x will the gradient of be the same as the (1) gradient in QUESTION 7.2.1? 24.2.3 Draw a sketch graph of . Show the x-values of the stationary points and the point of inflection on your sketch. It is not (3) necessary to indicate the intercepts with the axes. [19] Copyright reserved https://tvetpapers.co.za

Mathematics/P1 31 Collectable Marks/ NW2020 QUESTION 25 NW SEP 2016 The function defined by f ( x)  x  px  qx  12 is sketched below. A  4;36  and B 3 2 are turning points of f . g is a tangent to the graph of f at D. 25.1 (6) Show that p  5 and q  8. 25.2 (4) If C  1;0  is an x  intercept of f , calculate the other x  intercept of f . 25.3 (4) Determine the equation of g , the tangent to f at D 1; 14  . 25.4 (2) For which values of k will f ( x)  g ( x)  k have TWO positive roots? [16] Copyright reserved Please turn over

Mathematics/P1 32 Collectable Marks/ NW2020 QUESTION 26 NW SEP 2017 26.1 The sketch below shows the graph of y  f ( x ), where f is a cubic function. 26.1.1 (1) Write down the x  coordinate(s) of the stationery point(s) of f. 26.1.2 (1) For which value(s) of x is f decreasing? 26.1.3 (4) It is further given that f (0)  5 and f ( 2)  0. draw a sketch graph of f in your ANSWER BOOK. 26.2 The graph h( x)  ax  px passes through the point  3;  2  . The gradient of the 3 tangent to h at  0;0  is 3. 26.2.1 (4) Determine the values of a and p. 26.2.2 (2) Determine the gradient of the tangent to h at x  3. [12] QUESTION 27 NW SEP 2018 27.1 Given: f ( x)  2 x3  5x 2  4 x  3 27.1.1 (4) Calculate the coordinates of the x  intercepts of f if f (3)  0. Show all calculations. 27.1.2 (4) Calculate the x  values of the stationary points of f . 27.1.3 (2) For which values of x is f concave up? Copyright reserved Please turn over

Mathematics/P1 33 Collectable Marks/ NW2020 3 2 27.2 The function g , defined by g ( x)  ax  bx  cx  d has the following properties: ● g ( 2)  g (4)  0 ● The graph of g ( x) is concave up. ● The graph of g ( x) has x  intercepts at x  0 and x  4 and a turning point at x  2. 27.2.1 Use this information to draw a neat sketch graph of g without actually solving for a, b, c and d . Clearly show all x  intercepts, (4) x  values of the turning points and x  value of the point of inflection on your sketch. 27.2.2 (3) For which values of x will g ( x).g ( x)  0? [17] QUESTION 28 NW SEP 2019 2 The graph of f ( x)  x  bx  c, where f is a cubic function, is sketched below. The derivative function f  cuts the x  axis at x  3 and x  2. 28.1 (2) For which values of x is graph f increasing? 28.2 (1) At which value of x does graph f have a local maximum value? 28.3 (2) Determine the equation of f ( x ). 28.4 1 1 (4) 3 2 p , q If f ( x )  px  qx  rx  10, show that 3 2 and r  6. 28.5 (3) For which value(s) of x is graph f concave down? [12] Copyright reserved Please turn over

Mathematics/P1 34 Collectable Marks/ NW2020 INFORMATION SHEET ; ; ; M In ΔABC: P(A or B) = P(A) + P(B) – P(A and B) Copyright reserved Please turn over