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Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
Skills In Add Maths
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Skills In Add Maths

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  • 1. MODULE 1 ADDITIONAL MATHEMATICS SPM PRINCIPLES OF MATHEMATICS 1. Solve the equations below, (a) 2 – 4 3 (2 + x) = 1 – 2x (b) 2p3 32p − − = 5 (c) 2m = 3 – 2 m (d) 4p – 3(1 + p) = 2 5 (e) p 3 + 2 = 7 (f) p 4 + 2p 3 = 2 (g) 7x – 2 3 = 1 – x (h) 2 3m + = 3 – 2m (i) 5m – 3 = 2 – 3(1 + m) (j) 2 5x = 3 – 2x 2. Expand the following (a) (2x – 3)(3x + 5) (b) (4x + 7)(2x + 3) (c) (5p – 3) 2 (d) (4p + 3)(4p – 3) (e) (3x + 4)(2x – 4) (f) (4p – 3)(2p – 3) (g) (5k + 3)(2k – 3) (h) (3x – 7) 2 (i) (4m – 3)(2m + 5) (j) (7h – 3)(2h + 1) 3. Factorise the following completely (a) 2x 2 + x – 10 (b) 3x 2 + 10x – 8 (c) 3p 2 – 12 (d) 4p 2 – 2p (e) 5m 2 + 12m (f) 4x 2 – 12x + 5 (g) 5p 2 + p – 6 (h) x 2 – 4x (i) 5x 2 – 45xy (j) 4x 2 –15x + 9 (k) 7p 2 – 20p (l) 8p 2 – 6p + 1 (m) 3p 2 – 12p (n) 2x 2 – 18 (o) 2x 2 – 3x – 9 (p) 3x 2 – 7x – 6 (q) 4p 2 – 11p + 6 (r) p– 9p 2 (s) 2m 2 – 17m + 8 (t) 7p 2 – 16p + 4 (u) 3m 2 – 28m + 9 (v) 2m 2 – 8m (w) 3p 2 + 11p +6 (x) 1 – 16y 2 (y) 3m 2 – 13m + 10 (z) 9p 2 – 19p + 2 4. Solve the equations below,
  • 2. MODULE 1 ADDITIONAL MATHEMATICS SPM (a) (3x + 1)(2x – 1) = 4 (b) p 3 + 2p = 7 (c) 3x(x + 1) = x + 5 (d) 2x 2 = 3(x + 1) – 1 (e) p 4 + 3p = 8 (f) 5x(x – 1) = x + 8 (g) (4m + 1)(2m + 1) = 6 (h) 3x(x + 1) = 6 (i) p 3 + 3 2p = 3 (j) (2m – 1)(m – 1) = 6 (k) 2x 2 + 3 = 3(x + 1) + 2 (l) 3x 2 = 2x + 8 (m) 2x 2 = 3(x – 1) + 12 (n) 5 62 +x = x 5. Find the value of x and y which satisfy the equations below, 3x + 2y = 0 2x – 3y = 26 6. Find the value of x and y which satisfy the equations below, 4x + 5y = – 1 3x + 2y = 8 7. Find the value of x and y which satisfy the equations below, 3x + 2y = – 4 x – 3y = 17 8. Find the value of m and n which satisfy the equations below, 3m + 4n = 5 2m – 3n = 9 9. Find the value of m and n which satisfy the equations below, 4m – 7n = 23 6m + 2n = – 3 10. Find the value of p and q which satisfy the equations below, 5p + 3q = 2 5 3p – 2q = 11 SIMULTANEOUS EQUATIONS
  • 3. MODULE 1 ADDITIONAL MATHEMATICS SPM 1. Solve the following pairs of simultaneous equations, (a) 5p – 3q = 2 (b) 4p – 3q = 2p 2 – 3q 2 = 5 5p 2 – p– q 2 = 3 (c) y + 2x = 4 (d) 2p – 3q = 4 y 2 + 2x – x 2 = 5 4p 2 + q – 2q 2 + 2 = 0 (e) 3p + 2q = 1 (f) 3p + 4q = 7 9p 2 - 3p + 2q 2 = 8 9p 2 + 3p – 4q 2 = 8 (g) x y + x = 5 – y (h) 2m – 3n = 5 3y – 2x = 2 4m 2 + 4mn + n 2 = 1 (i) 2x – 3y = 5 (j) 2p – q = 2p 2 – q 2 = 1 2x 2 -3xy + y 2 = 6 ( SPM CLONE (SIMULTANEOUS EQUATIONS) 1. Calculate the coordinates of the point of intersection of the line x + y = 4 and the curve x 2 + y = 10. 2. Solve the simultaneous equations x – y = 2 and x 2 + 2y = 8. Give your answers correct to three decimal places. 3. Solve the following pairs of simultaneous equations. Give your answers correct to two decimal places. (a) 2x + y = 9, x(1 – y) = 3x 2 + 1 (b) 2x + y = x – 2y + 1 = x 2 + 3(x – y) (c) y + 2x = 1 (d) 2p – 3q = 4 y 2 + 2x – x 2 = 3 4p 2 + q – 2q 2 +5 = 0
  • 4. MODULE 1 ADDITIONAL MATHEMATICS SPM (e) 3p + 2q = 1 (f) 3p + 4q = 1 9p 2 - 3p + 2q 2 = 9 9p 2 + 3p + 2q 2 = 2 (g) x y + x = 4 – y (h) 2m – 3n = 2 3y – 2x = 2 m 2 + 4mn – 7n 2 = 5 (i) 2x – 3y = 5 (j) 2p –3q = 2p 2 – q 2 = 3 2x 2 -3xy + y 2 = 7 INDICES Solve each of the following equations, 1. 9 x = 27 1 – x 2. 3 x = 3 x + 1 – 6 3. 14k 2 5− = 16 7 4. 3(25 x ) = 75 5. 3 x + 2 = 3 x + 72 6. 3 x 2 x = 5 x + 2 7. 2 x + 1 = 5 x - 2 8. 2x 2 3 = 16 9. 36 y = 6 2 – y 10. 3 x + 2 + 3 x + 4 = 90 11. 3 x + 1 = 2 x 7 x + 1 12. 9 3 – 2x = 243 13. 5 x + 1 = 5 x – 1 + 24 14. 2x 2 3− = 54 15. 2 x – 1 = 5 x + 1 16. 2 x + 2 = 2 x + 1 + 16 17. 3y 3 5 = 96 18. 3 x = 5 x + 2 19. 5 x = 5 x – 2 + 24 20. 2p 2 5− = 16 1 21. 7 p + 2 + 7 p = 350 22. 5 x – 1 – 5 x – 2 = 100 23. x 3 5 = 243 1 24. 3 p + 3 = 3 p + 1 + 72
  • 5. MODULE 1 ADDITIONAL MATHEMATICS SPM 25. 2 x (3 x + 1 ) = 7 26. 5 x + 1 = 5 x – 2 + 124 27. 3p 2 3 = 24 28. 3 x + 1 = 108 – 3 x SPM CLONE (INDICES) 1. Find the value of x for each of the following, (a) 81 3 4 = − x (b) 5 2 40 3 2 = − x (c) 5 4 5 1 9 3 − = x x (d) 5 2 5 4 12 3 x x = − (e) 8 13 =− x (f) 8 7 4 1 32 xx = 2. Solve the following equations, (a) ( ) 1832 3 =x (b) 0 32 1 8 1 =−− x x (c) 1 2x 24 1 16 + + × = xx (d) mm 323 6482 =× − (e) 153 0 102 327 7 9 −+ − × = yy y (f) 21972 =× xx (g) 100853 =× xx (h) 153 5 1 =×      −x x (i) ( ) 4 32 1 8 y y =× (j) p8p4 29216 ×= LOGARITHMS A. Solve each of the following equations, (a) log 2 x – log 2 (1 – x) = 4 (b) log 3 3y = 2 + log 3 (1 – y) (c) log x 3 = 2 3 - log x 9 (d) 2 x3log = 8
  • 6. MODULE 1 ADDITIONAL MATHEMATICS SPM (e) log 2 (x + 2) + log 2 (x + 2) = 4 (f) log 2 y = 4 + log 2 (3 – 2y) (g) 2 log +y 3 + 2 log +y 9 = 2 3 (h) log 3 3x + log 3 x = 3 (i) log 6 p = 1 + log 6 (12 – p) (j) 4)(2log 5 −y = 5 1 (k) log x 8 + log x 4 = 5 (l) log 3 (m – 2) = log 3 m – 2 (m) y2log 3 = 9 (n) log 3 y = 2 + log 3 (1 – y) (o) 1 log +y 4 + 1 log +y 8= 3 5 (p) 2)(3log 5 −p = 25 (q) log 5 p = 1 – log 5 (3p – 2) (r) log 3 (x – 6) = 4 - log 3 3x (s) log 5 5x + log 5x = -1 (t) 2)(3log 2 −y = 2 1 (u) log 2 x + log 4 x = 2 3 (v) log 5 (4m – 1) = 1 – log 5 m B. Express y in terms of x in each of the following, (a) log 2 x + log 2 y = 3 (b) log 3 y + log 9 x = 3 (c) log 4 x + log 2 y = 1 (d) log 4 x = log 2 y + 1 (e) log 3 y – log 3 (x – y) = 2 (f) log 4 y = 2 – log 2 x (g) log 3 y = 2 + log 9 x (h) log 5 (y – 1) + 2 = log 5 x (i) log 2 (y – x) – log 2 x = 2 (j) log 2 y + 3 = log 2 (y + x) C. (a) Given that log 2 m = p, express the following in terms of p, (i) log 8m 4m 2 (ii) log 4m 8 m (iii) log 16m 3 8m 2
  • 7. MODULE 1 ADDITIONAL MATHEMATICS SPM (b) Given that log 3 p = t, express the following in terms of t, (i) p27 log 9p 3 (ii) log 3 81 p (iii) log 27 9p (c) Given that log 7 k = m, express the following in terms of m. (i) log 49 k (ii) log k7 49k 3 (iii) log 7 k7 (d) Given that log 3 p = k, express the following in terms of k, (i) log p 27 (ii) log 9 3 p (iii) log 3p 81p 2 (e) Given that log 2 k = n, express the following in terms of n, (i) log 4 k (ii) log k4 8k 2 (iii) log k 16k 3 (f) Given that log 5 x = m, express the following in terms of m, (i) log x 5 (ii) log 25 x5 (iii) log x5 125x 2 D. Without using four-figure tables or calculator, calculate, (a) log 8 27 . log 81 16 (b) log 5 50 – log 5 500 + log 5 10 (c) log 3 8 – log 3 108 – log 3 6 (d) log 25 4 . log 8 36 . log 216 125 (e) log 5 10 – log 5 4 + log 5 250 (f) log 2 3 + log 2 12 – log 2 24 – log 2 48 (g) log 27 8 . log 125 9 . log 4 25 (h) log 3 6 + log 3 162 – log 3 108 (i) log 4 49 . log 125 8 . log 7 25 (j) log 3 8 – log 3 6 – log 3 108 E. Find the value of, (a) 3 43log (b) 9 23log (c) 25 4 5 log (d) 5 9 25 log (e) 8 52log (f) 2 8116log
  • 8. MODULE 1 ADDITIONAL MATHEMATICS SPM (g) 16 32log (h) 3 169log (i) 9 43log (j) 49 37log (k) 6 936log (l) 5 25log F. Solve each of the following equations, (a) 5 2)-( 5 log x = 2 (b) 5 2 1)-( log t = 2 (c) m 3 )-(8 log m = 3 (d) 7 3 2)-( log k = 3 (e) (3m – 1) 3 )-(4 log m = 3 (f) 2 31)-( 2 log k = 8 (g) (m – 1) 3 )-(4 log m = 3 (h) 7 2)-( 7 log k = 5 (i) 3 2 5)-( log x = 2 (j) 4 3 )(2 log k+ = 3 G. (a) Given that log 2 m = k and log 2 y = p, express the following in terms of k and/or p (i) log 8 my (ii) log 8m m y (iii) log my 4m 3 (b) Given that log 3 m = k and log 2 m = p, express the following in terms of k and/or p (i) log 3 27 m (ii) log 2m 8m 2 (iii) log 12m m (c) Given that log 2 m = p and log 3 m = t. express logm 36 in terms of p and/or t.
  • 9. MODULE 1 ADDITIONAL MATHEMATICS SPM (d) Given that log 2 x = p and log 2 y = t, express the following in terms of p and/or t (i) log 2 xy 2 (ii) log 8x 32 y (e) Given that log 2 k = a and log 2 m = b, express the following in terms of a and/or b (i) log km 4 (ii) log 8 mk 2 (iii) log 4 k m (f) By using log m 2 = 1.5 and log m 3 = 2.5. find the value of, (i) log 6 m (ii) log 3m 12 m SPM CLONE (LOGARITHMS) 1. Solve the following equations (a) log 8 x = 3 1 − (b) log 4 3 2 8 = m (c) 0.4771 = log 10 (4 – 2x) (d) 64 1 log n = 3 (e) log 10 15 + log 10 5 3 ─ x = 2 log 10 0.3 2. Solve log 2 5x – log 2 (x + 3) = 1
  • 10. MODULE 1 ADDITIONAL MATHEMATICS SPM 3. Solve log 3 (2x + 7) – log 3 x = 2 4. Solve log 2 (3x – 2) = 1 – log 2 (x + 1) 5. Solve log 4 (x + 4) – log 4 (3x + 1) = 2 1 6. Solve log 2 5x – log 2 (1 + 8x) = – 1 COORDINATE GEOMETRY BASIC Find the equation of the straight line, (a) which passes through the points (3, 4) and (7, 10) (b) which passes through the point (1, 8) and is parallel to the straight line 2y = -3x + 7. (c) which is perpendicular to the straight line 3y – 2x = 7 and passes through the point (-1, 4) (d) which passes through the points (7, 3) and (2, 13) TIADA KEJAYAAN YANG BOLEH DIPEROLEHI TANPA USAHA YANG GIGIH
  • 11. MODULE 1 ADDITIONAL MATHEMATICS SPM (e) which passes through the point (2, 5) and is parallel to the straight line 3y – 2x – 7 = 0. (f) which is perpendicular to the straight line 2y + 3x = 7 and passes through the point (2, 2) (g) which passes through the points (2, 5) and (7, 1) (h) which is perpendicular to the straight line 2y = – 3x – 7 and passes through the point (3, 2) (i) which passes through the point (–1, 3) and is parallel to the straight line 3y = – 2x – 5. (j) which is perpendicular to the straight line 4y – 3x – 7 = 0 and passes through the point (2, 1) CLONE SPM QUESTIONS (Paper 2) 1. (a) Write down the equation of AB in the form of intercept. (b) Given that AC : CB = 1 : 2, find the coordinates of C. (c) Given that CD is perpendicular to AB. Find the x-intercept of CD. 2. (a) Find, (i) the equation of the straight line AB, (ii) the coordinates of B. (b) The straight line AB is extended to a point D such that AB : BD = 2 : 5. Find the coordinates of D. (c) A point P moves such that 2AP = PB. Find the equation of the locus of P. 3 (a) Calculate the area, in unit 2 , of triangle OPQ. (b) Given that PR : RQ = 1 : 2, A (0, 9) D (─ 4, 2) C B (6, 0) x y O A (1, 5) 2y = 3x – 18 C B x y O P (5, 7) R Q (─ 4, ─ 2) x y O
  • 12. MODULE 1 ADDITIONAL MATHEMATICS SPM find the coordinates of R. (c) A point T moves such that PT = 2TQ. (i) Find the equation of the locus T (ii) Hence, determine whether or not this locus intercept the x-axis. 4 (a) Find, (i) the equation of PR, (ii) the equation of QR, (iii) the coordinates of Q. (b) The straight line QR extended to a point S, such that QR : RS = 2 : 3. Find the coordinates of S. 5 The diagram shows a trapezium ABCD where AB is perpendicular to the straight line 2y + 3x = 0. Find, (a) the equation of AB (b) the coordinates of A (c) the equation of AD (d) the equation of CD (e) the coordinates of D . . 6 The diagram shows a rhombus ABCD, with CD parallel to the straight line 2y = x . Given that BD is perpendicular to the straight line 3y = – x. Find, (a) the equation of CD (b) the equation of BD (c) the coordinates of D (d) the area of the rhombus 7 A B (12, 12) C (25, 12) D x y O A C (10, 6 ) B (6, 14) D x y O A D (8, 8 ) B (– 4, 4) C x y O 3y + 2x = 17 P (1, 5) R (8, 9) x y O Q
  • 13. MODULE 1 ADDITIONAL MATHEMATICS SPM The diagram shows a parallelogram ABCD with AC is parallel to the straight line 3y = 11x + 2. Find, (a) the equation of AC (b) the equation of BD (c) the equation of AB (d) the equation of CD (e) the coordinates of C (f) the area of ABCD. 8 The diagram shows a kite ABCD. Given that the equation of AD is 6y = – 7x + 18. Find, (a) the coordinates of A (b) the equation of AC (c) the equation of BD (d) the coordinates of D (e) the area of ABCD. 9. Given that AB is perpendicular to the straight line 2y = – x. Find, (a) the equation of AB (b) the coordinates of A (c) the equation of BC (d) the coordinates of B (e) the area of triangle ABC 10. The diagram shows a rectangle PQRS A D C (8, 5) B (3, 8) x y O A C (8, 10) B x y O 4 P (3, 14) R (12, 1) QS x y O
  • 14. MODULE 1 ADDITIONAL MATHEMATICS SPM where PQ is perpendicular to the straight line y = 3x. Find, (a) the equation of PQ (b) the equation of QR (c) the coordinates of Q (d) the area of PQRS 11 The diagram shows a rhombus PQRS where SR is parallel to the straight line 12y – x = 0. Given that PR is perpendicular to the straight line 2y – 5x = 4. Find, (a) the equation of RS (b) the equation of PR (c) the equation of QS (d) the coordinates of S (e) the area of PQRS 12 The diagram shows a rhombus PQRS The equation of PS is 3y + 7x = 33. Given that PR is parallel to the straight line y = – x. Find, (a) the coordinates of P (b) the equation of PR (c) the equation of QS (d) the coordinates of S (e) the area of PQRS EQUATION OF LOCUS Q (7, 14) R (15, 5) P S x y O Q (7, 8) R P S x y O
  • 15. MODULE 1 ADDITIONAL MATHEMATICS SPM 1. Given that A(3, 2), B(5, 7) and C(8, 9) are collinear. (a) Find the distance of BC (b) A point X moves such that AX = 2XB. Find the equation of locus X. 2. A point P moves along the arc of a circle with centre A(2, 1). The arc of the circle passes through Q(5, 5). Find the equation of locus P. 3. Given that A(4, 0) and B(– 1, 2). Find the equation of the locus of a moving point X such that AX : XB = 2 : 3. 4. Given that A(2, 1) and B(5, 3). Find the equation of locus P if it moves such that AP is perpendicular to PB. 5. Find the locus of moving point Q such that its distances from L(4, 1) and M(2, 3) are equal. 6. Given that A(2, 3) and B(4, 0). A point X moves such that AX : XB = 2 : 1. Find the equation of locus X. 7. A point Y moves along the arc of a circle with centre A(2, 5) The arc of the circle passes through B(5, 7). Find the equation of locus Y. DIVISION OF LINE SEGMENTS 1. AB : BC = 1 : 2 Find the coordinates of point B. 2. AB : BC = 3 : 4 Find the coordinates of point B. 3. AB : BC = 3 : 2 Find the coordinates of point C. A(2, 1) C(8, 13) B A(– 1, 2) C(6, 23) B A(2, 3) B(8, 12) C
  • 16. MODULE 1 ADDITIONAL MATHEMATICS SPM 4. AB : BC = 3 : 4 Find the coordinates of point C. 5. AB : BC = 2 : 3 Find the coordinates of point B. 6. PQ : QR = 2 : 5 Find the coordinates of point Q. 7. AB : BC = 3 : 2 Find the coordinates of point C. 8. AB : BC = 2 : 3 Find the coordinates of point C. 9. Find (a) the ratio AB : BC (b) the value of k 10. Find (a) the ratio PQ : QR (b) the value of k 11. Find (a) the ratio PQ : QR (b) the value of k A(5, 8) B(11, 20) C A(1, 8) C(21, 3) B P(3, 2) R(10, – 12 ) Q A(1, 2) CB(7, – 1) A(– 3, 0) B(3, 2) C A(3, 2) B(k, 6) C(23, 12) P(3, 2) Q(7, k) R(17, 16) P(k, 3) Q(9, 9) R(21, 18)
  • 17. MODULE 1 ADDITIONAL MATHEMATICS SPM QUADRATIC EQUATIONS 1. Given the equation 2x 2 + x = 6 has the roots of m and n Form a quadratic equation which has roots m – 2 and n – 2. 2. Given the equation x 2 – mx + 18 = 0 has the positive roots and one of the roots is two times of the other root. (a) Find the roots of the equation (b) Find the value of m 3. Given the equation 3x 2 = 7x – 4 has the roots of p and q Form a quadratic equation with roots 2p + 1 and 2q + 1. 4. Given the equation x 2 – (b + 2)x + b = 14 has one of the root of the equation is negative of the other. Find, (a) the value of b (b) the roots of the equation 5. Given the equation 3x 2 = 7x + 6 has the roots of m and n. Form a quadratic equation with roots m + 2 and n + 2. 6. Given that the roots of quadratic equation x 2 – (a + 2)x + 3b = 1 are a and b. Find the values of a and b. 7. Given that the roots of quadratic equation 2x 2 + x = 6 are m and n. Form a quadratic equation with roots m + 2 and n + 2. 8. Given the equation 3x 2 + x = 4 has the roots of p and q. Form a quadratic equation with roots 2+p p and 2+q q 9. Given the equation 3bx 2 – (3b – 4)x = 6b + 1 where one of the root of the equation is negative of the other. Find, (a) the value of b (b) the roots of the equation. 10. Given the equation 3x 2 = – 11x – 6 has the roots of p and q Form a quadratic equation with roots 2p and 2q. 11. Given the equation (2x + 3)(x – 4) + b = 0 where one of the roots is four times of the other root. Find, (i) the roots of the equation
  • 18. MODULE 1 ADDITIONAL MATHEMATICS SPM (ii) the value of b. 12. Given the equation x 2 – a 5 x + 2a – 3 = 0 where one of the roots of the equation is reciprocal to the other. Find, (i) the value of a such that a > 0 (ii) the roots of the equation. SPM CLONE (QUADRATIC EQUATIONS) 1. Solve the quadratic equation x(3x + 5) = 4 Give your answer correct to three decimal places. 2. Solve the quadratic equation x(10 – 5x) = 4 Give your answer correct to three decimal places. 3. Solve the quadratic equation x x 1 2 4 3 =+ Give your answer correct to three decimal places. 4. Solve the quadratic equation 42 1 =+ x x Give your answer correct to four significant figures. 5. A quadratic equation x 2 – px + 2p – 3 = 0 has two equal roots. Find the possible values of p. QUADRATIC FUNCTIONS 1. Find the range of x in each of the following, (a) (3 + x)(2 – x) < 0 (b) ( x – 2)(x – 4) ≥ 0 (c) f (x) = 2x 2 – x – 10 and f (x) is always negative. (d) (x – 5) 2 ≤ x + 1 (e) (2x +1)(x – 2) ≤ 0 (f) (3 – 2x)(4 – x) < 0 (g) (5x + 1)(2 – x) ≥ 0 (h) 2x 2 + x ≤ 15 (i) x(x – 2) ≥ 0 (j) (3x + 1)(x – 2) ≥ 0 (k) (x – 2)(x – 4) ≥ 0
  • 19. MODULE 1 ADDITIONAL MATHEMATICS SPM 2. Express each of the following quadratic functions in the form of a(x + p) 2 + q. Hence, state the maximum or minimum value of the function and state the axis of symmetry. (a) f (x) = x 2 – 5x + 3 (b) f (x) = x 2 + 8x – 7 (c) f (x) = x 2 – 6x – 3 (d) f (x) = 4 + 8x – x 2 (e) f (x) = 9 – 6x – x 2 (f) f (x) = x 2 + 2x – 4 3. Express each of the following quadratic functions in the form of a(x + p) 2 + q. Hence, state the maximum or minimum value of the function and state the axis of symmetry. (a) f (x) = 3x 2 – 3x + 9 (b) f (x) = 2 – 4x – 2x 2 (c) f (x) = 5x 2 – 10x + 5 (d) f (x) = 2x 2 – 8x + 4 (e) f (x) = 5 + 20x – 5x 2 (f) f (x) = 9 – 6x – 3x 2 4 Sketch the graph for each of the following equations, (a) y = x 2 – 8x + 3 (b) y = 4 + 8x – x 2 (c) y = 2x 2 + 4x – 6 (d) y = 4 – 8x – 2x 2 (e) y = 3x 2 + 6x – 6 (f) y = 9 + 6x – 3x 2 5 Diagram shows the graph of y = 3(x – a) 2 + b. Find, (a) the values of a and b (b) the coordinates of A. 6 Diagram shows the graph of y = – 2x 2 + ax + b. Find the values of a and b. 7. Diagram shows the graph of (3, 4) x y A O (2, 10) x y O (1, – 8) x y O
  • 20. MODULE 1 ADDITIONAL MATHEMATICS SPM y = 3x 2 + ax + b. Find the values of a and b. SPM CLONE (QUADRATIC FUNCTIONS) 1. Find the range of values of x for which x(x – 2) > 3 2. Find the range of values of x for which x(6x – 7) > 10 3. Find the range of values of x for which x(x – 13) ≥ ─ 42 4. Find the range of values of x for which 2x(x – 1) < 3 – x 5 Diagram shows the graph of the function y = ─ (x – h)2 – 2, where h is a constant. Find, (a) the value of h, (b) the equation of the axis of symmetry, (c) the coordinates of the maximum point. 6 Diagram shows the graph of the function y = ─ (x + p)2 + 10 with (3, q) as the maximum point of the curve. Given p and q are constants. State, (a) the value of p, (b) the value of q, (c) the equation of the axis of symmetry 7. Diagram shows the graph of the quadratic function f(x) = (x + b)2 – 8 with (─ 4, c) as its minimum point. Given b and c are constants. State, (a) the value of b, (b) the value of c, (c) the equation of the axis of (2, ─ 6) x y O ─ 6 (3, q) x y O (– 4, c) x y O
  • 21. MODULE 1 ADDITIONAL MATHEMATICS SPM symmetry. 8. The quadratic function f(x) = x2 – 4x + p has a minimum value of ─ 11. Find the value of p. 9 The function f(x) = a – bx – 3x2 has a maximum value of 6 when x = ─ 2. Find the value of a and b. 10. Given that f(x) = (x – 3)2 – 9 = hx2 ─ 6x + k – 2. Find, (a) the value of h and k. (b) the minimum value of f(x). 11. The function f(x) = x 2 – 4kx + 5k 2 + 1 has a minimum value of r 2 + 2k, where both r and k are constants. (a) By using the method of completing the square, show that r = k – 1. (b) Hence, or other wise, find the value of k and r if the graph of function is symmetrical at x = r 2 – 1. FUNCTIONS 1. Given the function f : x → 2x + 1 and g : x → 1 2 +x where x ≠ – 1. Find, (a) fg (b) gf (c) f 2 g 2. Given the function f (x) = 2−x x where x ≠ p and g(x) = 2x + 5 Find, (a) the value of p (b) (i) fg (ii) gf (iii) g 2 f 3. Given the function f (x) = 12 13 − + x x and g(x) = 2 1 + + x x Find, (a) fg (b) fg 2 4. Given the function f (x) = 52 13 − + x x and g(x) = 43 32 − + x x
  • 22. MODULE 1 ADDITIONAL MATHEMATICS SPM Find, (a) f – 1 g (b) gf – 1 (c) f – 1 g – 1 5. Given the function f (x) = 3 52 + + x x and g(x) = 2 13 − − x x . Find, (a) f – 1 g (b) g – 1 f – 1 6. Given the function f (x) = 2 13 − + x x and gf (x) = 1+x x . Find the function of g. 7. Given the function f (x) = 3x + 1 and gf (x) = 2x + 5. Find the function of g. 8. Given the function f (x) = 43 32 − + x x and fg(x) = 1 2 +x x . Find the function of g. 9. Given the function f (x) = 4x + 1 and fg(x) = 2 12 + + x x . Find the function of g. 10. Given the function f (x) = 13 12 − + x x and g(x) = 1 4 +x x Find, (a) fg(2) (b) gf (3) 11. Given the function f (x) = 1 4 +x x and g(x) = 2 3 +x x Find, (a) gf – 1 (5) (b) fg – 1 (4) 12. Given the function f (x) = 13 12 + + x x and g(x) = 2+x x If fg(x) = bx ax + + 4 2 , find the values of a and b. 13. Given the function f (x) = ax + b and f 2 (x) = 9x + 16 where a > 0. Find the values of a and b. 14. Given the function f (x) = 53 12 − + x x and g(x) = 2 4 +x x . If f – 1 g(x) = bx ax + + 10 2 , find the values of a and b.
  • 23. MODULE 1 ADDITIONAL MATHEMATICS SPM 15. Given the function f (x) = 2x + 3 and g(x) = 2+x x . Given that gf – 1 (x) = bx + + ax , find the values of a and b. 16. Given the functions f : x → 2x + 5 and g : x → 3 – 2x. Find the value of, (a) f 2 (b) gf 17. Given the functions g : x → 5x + 1 and h : x → x 2 – 2x + 3. Find the value of, (a) g2 (b) gh 18 Given that f : x → 2x + 5 and fg: x → 13 – 2x. Find g(x). 19. Given that the composite function gf(x) = 3x – 4, find f(x) if g(x) = 2 x . 20 Given that the composite function hg : x → 1 6 + x and h : x → 2x + 1. Find g(3). 21. Given that the function g : x → 3x + 2 and fg(x) = 27x + 17. Find f(x). 22. Given that f : x → ax + b, g : x → x 2 + 3 and fg : x → 3x 2 + 5. Find the value of a and b. 23. The functions f and g are defined by f : x → 5 – 3x and g : x → 2ax + b respectively, where a and b are constants. If the composite function fg(x) = 8 – 3x, find the value of a and b. 24 Given that f : x → 4 – 3x and g : x → 2x 2 + 5, find, (a) f – 1 (─ 8) (b) gf (x) 25. Given that f : x → 4x + 1 and g : x → x 2 – 2x – 6, find, (a) f – 1 (─ 3) (b) gf (x) 26 Given that g : x → 1 – 3x – x 2 and h – 1 : x → 2x – 1, find,
  • 24. MODULE 1 ADDITIONAL MATHEMATICS SPM (a) h(5) (b) gh – 1 (x) 27. The following information refers to the functions f and g. Find fg – 1 (x). 28. The following information refers to the functions f and g. Find f – 1 g(x) 29. Given the functions h : x → 2x + n and h – 1 : x → mx – 2 3 , where m and n are constants, find the value of m and of n. 30. Given the functions f : x → 4+ − x bax and f – 1 : x → 2 34 − −− x x , where a and b are constants, find the value of a and of b. 31 Given the functions g : x → p – 3x and g – 1 : x → 3 2 – qx , where p and q are constants, find the value of p and of q. f : x → 3x + 2 g : x → x – 3 f : x → 3x + 5 g : x → x x −3 To Learn from Mistakes
  • 25. MODULE 1 ADDITIONAL MATHEMATICS SPM DIFFERENTIATION Find dx dy for each of the following equations, (a) y = 2x 2 + 3x + 5 (b) y = x 2 (x 2 + 1) (c) y = 2 32 1 x xx ++ (d) y = (3x 2 + 1) 5 (e) y = (3x + 1)(2x + 1) 3 (f) y = 12 23 − − x x (g) y = 5 – x 2 – x 3 (h) y = (x 4 + 1)x 5 (i) y = 4 2 1 x xx ++ (j) y = ( )3 23 2 x− (k) y = (2x + 1) 3 (3x – 1) 4 (l) y = ( ) ( )2 3 32 12 x x − + (m) y = 4x 3 + 2x 2 + x 1 (n) y = 3x 2 (x 2 + x + 1)
  • 26. MODULE 1 ADDITIONAL MATHEMATICS SPM (o) y = x xx +2 4 (p) y = (5 – 2x 3 ) 4 (q) y = (1 – 3x) 2 (2x + 1) 3 (r) y = 2 43 13 x x − + RATES OF CHANGE FOR RELATED QUANTITIES, SMALL CHANGES AND APPROXIMATIONS 1. Given y = 2x 2 + x + 3, find the small change in y when x increases from 2 to 2.01. 2. Given A = 3B 2 + 2B + 3, find the small change in A when B decreases from 3 to 2.92. 3. Given p = 2q 2 + 5q – 3, find the small change in p when q increases from 4 to 4.02. 4. Given y = 5x 2 – x + 2, find the small change in y when x increases from 5 to 5.03. 5. Given A = 2B 2 – B – 3, find the small change in A when B decreases from 4 to 3.9. 6. Given y = 3x 2 + 2x + 3, find the small positive change in y when x decreases from 2 to 1.98. 7. Given y = 3x 2 + 2x + 2, find the small positive change in x when y increases from 10 to 10.5. 8. Given y = 2x 2 – x, find the small pysitive change in x when y increases
  • 27. MODULE 1 ADDITIONAL MATHEMATICS SPM from 6 to 6.1. 9. Given y = 5x 2 – x + 1, find the small negativechange in x when y I Increases from 7 to 7.3. 10. Given that y = 3x 2 + 3x + 1, when x decreases from 2 to 1.9. Find the small change in y. 11. Given that A = 3B 2 + 3B + 2, when B increases from 3 to 3.02. Find the small changed in A 12. Given p = 3q 2 – 2q + 1, when q decreases from 4 to 3.9. Find the small hange in p . 13. Given A = B 3 , when B increases from 3 to 3.02. Find the small changed in A.. 14. Given y = 2 4 x , when x decreases from 2 to 1.9. Find the small change in y 15 The volume of a sphere increases at the rate of 2π cm 3 s – 1 . Find the rate of change of the radius when the radius is 5 cm. 16. A cone has a height 6 cm. Its base radius changes at the rates of 0.5 cm s – 1 . Find the rate of change of the volume when the base radius is 3 cm. 17 A rectangle has borders of 3x cm and 4x cm. Given that x changes at the rate of 2 cm s – 1 . Find the rate of change of its area when the area is 48 cm 2 . 18 A cylinder has a changing height and base radius. Given that the cylinder's height is twice as that of the base radius and its volume increases at the rate of 200π cm 3 s – 1 . Find the rate of change of the radius when the radius is 5 cm. 19 A container has a volume of V = 2π x (3x + 5) where the volume increases at the rate of 200π cm 3 s – 1 . Find the rate of change of x when x = 5.
  • 28. MODULE 1 ADDITIONAL MATHEMATICS SPM 20. Given y = 2 50 +x when x changes at the rate of 3 unit s – 1 . Find the changing rate of y when x = 3. 21 The surface area of a cuboids is given as A = 3π x(x 2 + x 16 ) Its surface area increases at the rate of 24π cm 2 s – 1 . Find the increment of x when x is 6 cm. 22. A cylinder with the height of 4 cm has the base radius that is changing at the rate of 0.5 cm s – 1 . Find the rate of change of the cylinder volume when the radius is 5 cm. 23 A sphere is inflated with air at the rate of 400π cm 3 s – 1 . Find the rate of change of the sphere radius when the volume is 36π cm 3 . 24. A cube has a border of 2x cm where x changes at the rate of 0.5 cm s – 1 . Find the rate of change of the volume when x is 2 cm. 25. A cylinder has a changing base radius and height. Given that the cylinder's height is three times as that of the base radius and the radius changes at the rate of 0.5 cm s – 1 . Find the rate of change of the cylinder volume when the radius is 2 cm. 26. The cylinder radius with the height of 3 cm gets longer with the rate 0.5 cm s – 1 . Find the rate of change of the cylinder volume when the cylinder volume is 48π . 27. A sphere is inflated with air at the rate of 20π cm 3 s – 1 . Find the rate of change of the sphere radius when the radius is 5 cm. STATISTICS 1. Find a set of five positive integers which has a mode of 8, a mean of 6 and a median of 7.
  • 29. MODULE 1 ADDITIONAL MATHEMATICS SPM 2. If the mean of the set data 4, p, 5, 6, 5, 6 and 4 is 5, where p is a constant, find the mode. 3. A set of numbers, p + 3, 2p – 5, p + 7, p and 3p + 7 has a mean of 12. Find the value of p and the median. 4. The mean of the set of data 47, 60, m, n, 23, 50, 33 is 32. Find the value of m + n. 5. The mean of the set data 3, p + 2, p 2 – 2, 3p is 6. Find the possible values of p. 6. Given the set of data 4, 5, 6, 8, 9, k, determine the value of k if the median is 7. 7. The set of data m – 4, m – 3, 3m – 5, m 2 – 1, m + 3, m + 6 is arranged in ascending order. If the median is 2, find the positive value of m. 8. The set of data k, 6, 8, 12, 15 is arranged in ascending order. If the mean is equal to the median, find the value of k. 9. The mean of the sed data 2, k, 2k, 2k + 1, 6, 7 and 9 is p.. If 2 is subtracted from each value of the data, the new mean becomes . 2 1p + . Find the values of k and p 10. The mean of a set of four positive integers is 5. When a number x is taken out from the set, the mean becomes 4. Find the value of x. Age (years) 1 2 3 4 5 Number of children 3 5 2 m 4 11. The table above shows the ages of children in a nursery. Find,
  • 30. MODULE 1 ADDITIONAL MATHEMATICS SPM (a) the maximum value of m if the mode age is 2 years., (b) the minimum value of m if the mean age is more than 3 years. Score 0 1 2 3 Frequency 3 4 5 p 12. The table above shows the scores obtained by a student in a game. (a) If the mean score is 2, find the value of p. (b) If the median score is 2, find the range of values of p. Number of absentees 0 1 2 3 4 Number of classes 4 8 6 5 2 13. The table above shows the number of absentees on a certain day for all the 25 classes in a school. Find the median of the number of absentees. 14. Given a set of number 3, 5, 6, 8, 9, 10, 11 and 12. Find the variance and standard deviation of the set of numbers. 15. The mean of a set of numbers, 6, 8, x, 12 and 13 is 10. Find, (a) the value of x, (b) the variance and standard deviation 16. The mean of a set of number, 7, 14, 15, m, 2m, 47 and 52, is 27. Find, (a) the value of m, (b) the variance and standard deviation of the set of numbers. 17. A set of six numbers has a mean of 8 and a variance of 4. When a number p is added to the set, the mean remains unchanged. Find, (a) the value of p, (b) the new variance.
  • 31. MODULE 1 ADDITIONAL MATHEMATICS SPM Height (cm) Number of plants 20 – 29 5 30 – 39 8 40 – 49 15 50 – 59 12 60 – 69 7 70 – 79 3 18. The table above shows the distribution of the heights of plants in a garden. Without drawing an ogive, calculate the median height of the plants. Score Frequency 10 – 19 4 20 – 29 8 30 – 39 16 40 – 49 28 50 – 59 14 60 – 69 10 19. The table above shows the frequency of a certain distribution. Without drawing an ogive, find the median, the first quartile and the third quartile. Score Frequency 1.0 – 1.4 4 1.5 – 1.9 5 2.0 – 2.4 7 2.5 – 2.9 6 3.0 – 3.4 3 3.5 – 3.9 5 20. The table above shows the frequency of a certain distribution. Without drawing an ogive, find the median, the first quartile and the third quartile.
  • 32. MODULE 1 ADDITIONAL MATHEMATICS SPM Score Cumulative Frequency 20 – 29 4 30 – 39 12 40 – 49 28 50 – 59 56 60 – 69 70 70 – 79 80 21. The table above shows the cumulative frequency of a certain distribution. Without drawing an ogive, find the median and the interquartile range 22. For the set numbers, 13, 15, 19, 17, 20, 11, and 12, determine, (a) the median (b) the interquartile range. 23. For the set numbers, 10, 12, 15, 14, 22, 19, and 13, determine, (a) the median (b) the interquartile range. SPM CLONE 1. A set of examination marks x 1, x 2, x 3, x 4, x 5, x 6 has a mean of 12 and standard deviation of 5. (a) Find, (i) the sum of the marks, ∑ x, (ii) the sum of the squares of the marks, ∑ x 2 . (b) Each mark is multiplied by 3 and then 2 is added to it. Find for the new set of marks, (i) the mean, (ii) the variance. 2. A set of numbers x 1, x 2, x 3, x 4, x 5, x 6 has a mean of 6 and standard deviation of 2.5. (a) Find, (i) the sum of the numbers, ∑ x, (ii) the sum of the squares of the numbers, ∑ x 2 . (b) Each number in the set is multiplied by 4 and then 5 is added to each.
  • 33. MODULE 1 ADDITIONAL MATHEMATICS SPM For the new set of numbers, find, (i) the mean, (ii) the variance. 3. A set of data consists of 8 numbers. The sum of the numbers is 180 and the sum of the squares number is 4200. (a) Find the mean and variance of the 8 numbers. (b) Another number is added to the set of data and the mean is increased by 3. Find, (i) the value of this number, (ii) the standard deviation of the set of 9 numbers. 4. A set of data consists of six numbers. The sum of the numbers is 72 and the sum of the squares number is 958. (a) Find the mean and variance of the six numbers. (b) Another number is added to the set of data and the mean is increased by 2. Find, (i) the value of this number, (ii) the standard deviation of the set of 9 numbers. 5. The mean of eight numbers is m . The sum of of the squares of the numbers is 120 and the standard deviation is 3h. Express m in terms of h. INDEX NUMBER 1. A school had 2000 students in the year 2003. The enrolment in 2004 decreased to 1800 students. Calculate the index number that shows the changing in the number of students in 2004 based on 2003. 2. The total number of cars sold by the EM Company increased from 3000 units in 1998 to 3500 in 1999. Calculate the index number that shows the increase base on 1998. 3. The price of a bed decreased from RM 1500 in 2004 to RM 1000 in 2005. Calculate the price index of a bed in 2005 if the base year is 2004. 4. The frequency of giving speeches had increased from 120 speeches
  • 34. MODULE 1 ADDITIONAL MATHEMATICS SPM in 1998 to 250 speeches in 2000. Calculate the index number that shows the increase based on 1998. 5. A kilogram of sugar in 1985 was RM 1.20 and the price in 2005 was RM 1.50. Calculate the price index per a kilogram of sugar in 2005 based on 1985. 6. The price of a 21” TV in 2005 was RM 250. The price index of the television in 2005 based on the year 2000 is 80. Find the price of the television in 2000. 7. In the year 2000, a group of language teachers had collected 3000 essay. The index number the essay collection in 2005 based on 2000 is 120. Find the total number of essays collected in 2005. 8. In 1995, the price of a refrigerator was RM 1200. The price index of the refrigerator in 1998 is 150 based on 1995. Find the price of the refrigerator in 1998. 9. The price index of a commodity in 2001 based on 2000 is 120. If the price of commodity in 2000 was RM 2000, find its price in 2001. 10. The price indices of a commodity in 2004 and 2005 based on 2000 are 125 and 120 respectively. If the price of the commodity in 2004 is RM 100, calculate its price in 2005. 11. The price indices of a digital camera in 2002 and 2000 based on 1999 are 110 and 125 respectively. If the price of the camera in 2000 was RM 1200, find its price in 2002. 12. The price indices of a hand phone in 2000 and 2002 based on 1999 are 120 and 80 respectively. If the hand phone price in 2000 was RM 1080, find the price in 2002. Product Cost in 2000 (RM) Cost in 2005 (RM) Price index A P 200 125 B 120 Q 120
  • 35. MODULE 1 ADDITIONAL MATHEMATICS SPM C 40 60 R 13. The table above shows the prices of three products A, B and C in 2000 and 2005. Using the year 2000 as the base year, calculate the values of P, Q and R. Furniture Price index Weightage Chair 110 1 Table 112 4 Bed 125 y Mattress 108 3 14. The table above shows the price indices and weightages of a few furniture. If the composite price index is 113.2, calculate the value of y. Item Price in 2004 (RM) Price in 2005 (RM) Index number P a 2.00 120 Q 4.00 5.00 b R 3.00 c 115 15. The index number for 2004 with the year 2005 as the base year are given in the table above. (a) Calculate the values of a, b and c. (b) If the weightages of P, Q and R are 4, 8, and 3, respectively, Find the composite index in 2004 based on 2005 Product Price in 2000 (RM) Price in 2002 (RM) Weightage A 20 16 3
  • 36. MODULE 1 ADDITIONAL MATHEMATICS SPM B 30 33 4 C 40 48 5 D 15 21 8 16. The table above shows the prices of four products A, B, C, dan D in 2000 and 2002. (a) Calculate the index numbers of each products A, B, C, dan D in 2002 with 2000 as the base year. Then calculate the composite index in 2002 with 2000 as the base year. (b) If all the products are sold at RM 150 per unit in 2000, calculate the selling price in 2002. SOLUTION OF TRIANGLES 1. The diagram shows a cuboid. Calculate, (a) the length of BH (b) ∠DBH (c) ∠AHF 2. The diagram shows a pyramid with a triangular base ABC. Given AB = AC = 15 cm, BC = 20 cm vertex V is 12 cm vertically above A. Calculate, (a) the length of BV (b) ∠VBC (c) the area of triangle VBC 3. The diagram shows a pyramid with a horizontal square base ABCD of side 12 cm. Vertex V is 8 cm vertically above the centre of the base. Calculate, (a) the angle between side VA and the base ABCD, (b) the angle between the planes 8 cm A B CD E F GH 5 cm 6 cm B A C V A B CD V
  • 37. MODULE 1 ADDITIONAL MATHEMATICS SPM VAB and VCD. 4. The diagram shows a quadrilateral PQRS. Calculate, (a) ∠PSR in degrees, (b) ∠PQR in degrees, (c) the area of the quadrilateral 5. The diagram shows the points P, Q, R, S, and T on a horizontal plane. Given QRS is a straight line, ∠PRQ is obtuse and the area of triangle PST is 25 cm 2 . Calculate, (a) the length of PS, (b) ∠SPT 6. The diagram shows a triangle ABC. (a) Find the length of AC (b) A quadrilateral ABCD is now formed with AC as the diagonal, ∠ACD =20° and AD = 7 cm. Find the two possible values of ∠ADC. (c) Using the acute ∠ADC from part (b), find, (i) the length of CD, (ii) the area of quadrilateral ABCD. P Q R V P Q R S 10.2 cm 6.5 cm 8.5 cm 14.1 cm 35.8° Q P R S T 9.5 cm 7.5 cm 11.2 cm 5.8 cm 30° A B C 15 cm 11 cm 70°
  • 38. MODULE 1 ADDITIONAL MATHEMATICS SPM 7. The above diagram shows a pyramid with ∆PQR as its horizontal base and V as its vertex. Given the angle between the inclined plane VQR and the base is 45°, VQ = VR = 3.4 cm and PQ = PR = 3.8 cm. Calculate, (a) the length of QR if the area of the base is 4.5 cm 2 , (b) the length of PV if the angle between PV and the base is 40°, (c) the area of triangle VPQ. 8. The diagram shows a quadrilateral PQRS where ∠PQR is acute. Calculate, (a) ∠PQR (b) ∠PSR (c) the area, in cm 2 , of quadrilateral PQRS. 9. The diagram shows a triangle ABC. (a) Calculate the length, in cm, of BC, (b) A quadrilateral ABDC is now formed so that BC is a diagonal, ∠BCD = 40° and BD = 8 cm. Calculate the two possible values of ∠BDC. (c) Using the acute ∠BDC from (b), calculate, (i) the length, in cm, of CD, (ii) the area, in cm 2 , of quadrilateral ABCD. P Q R S 11.9 cm 4.8 cm 9.3 cm 9.5 cm41° A B C 140° 7 cm 4 cm A B C D 36°
  • 39. MODULE 1 ADDITIONAL MATHEMATICS SPM 10. The diagram above shows two triangles, ABC and ACD with BC = 10 cm, AC = 8.5 cm and CD = 15.6 cm. (a) Find the length of AB, (b) If the area of triangle ACD is 53.64 cm 2 , find, (i) the value of the obtuse angle ACD, (ii) the length of AD. 11. The diagram shows, ABC and CDE are straight lines. Find, (a) the length of BD, (b) the value of ∠CAE, (c) the area of quadrilateral ABDE. 12. The diagram shows a triangle ACD and B is a point on AC. Calculate, (a) the length of CD, (b) ∠ ADB, (c) the length of AC. 99% of all FAILURES come from people who have the habit of MAKING EXCUSES You Never Fail E A B D C 4.3 m2.1 m 2.1 m 2 m 47° A B C D 11.5 cm 5 cm 80° 45°
  • 40. MODULE 1 ADDITIONAL MATHEMATICS SPM GOOD LUCK

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