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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 &#x2013; 4 3 (2 + x) = 1 &#x2013; 2x (b) 2p3 32p &#x2212; &#x2212; = 5 (c) 2m = 3 &#x2013; 2 m (d) 4p &#x2013; 3(1 + p) = 2 5 (e) p 3 + 2 = 7 (f) p 4 + 2p 3 = 2 (g) 7x &#x2013; 2 3 = 1 &#x2013; x (h) 2 3m + = 3 &#x2013; 2m (i) 5m &#x2013; 3 = 2 &#x2013; 3(1 + m) (j) 2 5x = 3 &#x2013; 2x 2. Expand the following (a) (2x &#x2013; 3)(3x + 5) (b) (4x + 7)(2x + 3) (c) (5p &#x2013; 3) 2 (d) (4p + 3)(4p &#x2013; 3) (e) (3x + 4)(2x &#x2013; 4) (f) (4p &#x2013; 3)(2p &#x2013; 3) (g) (5k + 3)(2k &#x2013; 3) (h) (3x &#x2013; 7) 2 (i) (4m &#x2013; 3)(2m + 5) (j) (7h &#x2013; 3)(2h + 1) 3. Factorise the following completely (a) 2x 2 + x &#x2013; 10 (b) 3x 2 + 10x &#x2013; 8 (c) 3p 2 &#x2013; 12 (d) 4p 2 &#x2013; 2p (e) 5m 2 + 12m (f) 4x 2 &#x2013; 12x + 5 (g) 5p 2 + p &#x2013; 6 (h) x 2 &#x2013; 4x (i) 5x 2 &#x2013; 45xy (j) 4x 2 &#x2013;15x + 9 (k) 7p 2 &#x2013; 20p (l) 8p 2 &#x2013; 6p + 1 (m) 3p 2 &#x2013; 12p (n) 2x 2 &#x2013; 18 (o) 2x 2 &#x2013; 3x &#x2013; 9 (p) 3x 2 &#x2013; 7x &#x2013; 6 (q) 4p 2 &#x2013; 11p + 6 (r) p&#x2013; 9p 2 (s) 2m 2 &#x2013; 17m + 8 (t) 7p 2 &#x2013; 16p + 4 (u) 3m 2 &#x2013; 28m + 9 (v) 2m 2 &#x2013; 8m (w) 3p 2 + 11p +6 (x) 1 &#x2013; 16y 2 (y) 3m 2 &#x2013; 13m + 10 (z) 9p 2 &#x2013; 19p + 2 4. Solve the equations below,
• 2. MODULE 1 ADDITIONAL MATHEMATICS SPM (a) (3x + 1)(2x &#x2013; 1) = 4 (b) p 3 + 2p = 7 (c) 3x(x + 1) = x + 5 (d) 2x 2 = 3(x + 1) &#x2013; 1 (e) p 4 + 3p = 8 (f) 5x(x &#x2013; 1) = x + 8 (g) (4m + 1)(2m + 1) = 6 (h) 3x(x + 1) = 6 (i) p 3 + 3 2p = 3 (j) (2m &#x2013; 1)(m &#x2013; 1) = 6 (k) 2x 2 + 3 = 3(x + 1) + 2 (l) 3x 2 = 2x + 8 (m) 2x 2 = 3(x &#x2013; 1) + 12 (n) 5 62 +x = x 5. Find the value of x and y which satisfy the equations below, 3x + 2y = 0 2x &#x2013; 3y = 26 6. Find the value of x and y which satisfy the equations below, 4x + 5y = &#x2013; 1 3x + 2y = 8 7. Find the value of x and y which satisfy the equations below, 3x + 2y = &#x2013; 4 x &#x2013; 3y = 17 8. Find the value of m and n which satisfy the equations below, 3m + 4n = 5 2m &#x2013; 3n = 9 9. Find the value of m and n which satisfy the equations below, 4m &#x2013; 7n = 23 6m + 2n = &#x2013; 3 10. Find the value of p and q which satisfy the equations below, 5p + 3q = 2 5 3p &#x2013; 2q = 11 SIMULTANEOUS EQUATIONS
• 3. MODULE 1 ADDITIONAL MATHEMATICS SPM 1. Solve the following pairs of simultaneous equations, (a) 5p &#x2013; 3q = 2 (b) 4p &#x2013; 3q = 2p 2 &#x2013; 3q 2 = 5 5p 2 &#x2013; p&#x2013; q 2 = 3 (c) y + 2x = 4 (d) 2p &#x2013; 3q = 4 y 2 + 2x &#x2013; x 2 = 5 4p 2 + q &#x2013; 2q 2 + 2 = 0 (e) 3p + 2q = 1 (f) 3p + 4q = 7 9p 2 - 3p + 2q 2 = 8 9p 2 + 3p &#x2013; 4q 2 = 8 (g) x y + x = 5 &#x2013; y (h) 2m &#x2013; 3n = 5 3y &#x2013; 2x = 2 4m 2 + 4mn + n 2 = 1 (i) 2x &#x2013; 3y = 5 (j) 2p &#x2013; q = 2p 2 &#x2013; 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 &#x2013; 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 &#x2013; y) = 3x 2 + 1 (b) 2x + y = x &#x2013; 2y + 1 = x 2 + 3(x &#x2013; y) (c) y + 2x = 1 (d) 2p &#x2013; 3q = 4 y 2 + 2x &#x2013; x 2 = 3 4p 2 + q &#x2013; 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 &#x2013; y (h) 2m &#x2013; 3n = 2 3y &#x2013; 2x = 2 m 2 + 4mn &#x2013; 7n 2 = 5 (i) 2x &#x2013; 3y = 5 (j) 2p &#x2013;3q = 2p 2 &#x2013; q 2 = 3 2x 2 -3xy + y 2 = 7 INDICES Solve each of the following equations, 1. 9 x = 27 1 &#x2013; x 2. 3 x = 3 x + 1 &#x2013; 6 3. 14k 2 5&#x2212; = 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 &#x2013; y 10. 3 x + 2 + 3 x + 4 = 90 11. 3 x + 1 = 2 x 7 x + 1 12. 9 3 &#x2013; 2x = 243 13. 5 x + 1 = 5 x &#x2013; 1 + 24 14. 2x 2 3&#x2212; = 54 15. 2 x &#x2013; 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 &#x2013; 2 + 24 20. 2p 2 5&#x2212; = 16 1 21. 7 p + 2 + 7 p = 350 22. 5 x &#x2013; 1 &#x2013; 5 x &#x2013; 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 &#x2013; 2 + 124 27. 3p 2 3 = 24 28. 3 x + 1 = 108 &#x2013; 3 x SPM CLONE (INDICES) 1. Find the value of x for each of the following, (a) 81 3 4 = &#x2212; x (b) 5 2 40 3 2 = &#x2212; x (c) 5 4 5 1 9 3 &#x2212; = x x (d) 5 2 5 4 12 3 x x = &#x2212; (e) 8 13 =&#x2212; x (f) 8 7 4 1 32 xx = 2. Solve the following equations, (a) ( ) 1832 3 =x (b) 0 32 1 8 1 =&#x2212;&#x2212; x x (c) 1 2x 24 1 16 + + &#xD7; = xx (d) mm 323 6482 =&#xD7; &#x2212; (e) 153 0 102 327 7 9 &#x2212;+ &#x2212; &#xD7; = yy y (f) 21972 =&#xD7; xx (g) 100853 =&#xD7; xx (h) 153 5 1 =&#xD7;&#xF8F7; &#xF8F8; &#xF8F6; &#xF8EC; &#xF8ED; &#xF8EB; &#x2212;x x (i) ( ) 4 32 1 8 y y =&#xD7; (j) p8p4 29216 &#xD7;= LOGARITHMS A. Solve each of the following equations, (a) log 2 x &#x2013; log 2 (1 &#x2013; x) = 4 (b) log 3 3y = 2 + log 3 (1 &#x2013; 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 &#x2013; 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 &#x2013; p) (j) 4)(2log 5 &#x2212;y = 5 1 (k) log x 8 + log x 4 = 5 (l) log 3 (m &#x2013; 2) = log 3 m &#x2013; 2 (m) y2log 3 = 9 (n) log 3 y = 2 + log 3 (1 &#x2013; y) (o) 1 log +y 4 + 1 log +y 8= 3 5 (p) 2)(3log 5 &#x2212;p = 25 (q) log 5 p = 1 &#x2013; log 5 (3p &#x2013; 2) (r) log 3 (x &#x2013; 6) = 4 - log 3 3x (s) log 5 5x + log 5x = -1 (t) 2)(3log 2 &#x2212;y = 2 1 (u) log 2 x + log 4 x = 2 3 (v) log 5 (4m &#x2013; 1) = 1 &#x2013; 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 &#x2013; log 3 (x &#x2013; y) = 2 (f) log 4 y = 2 &#x2013; log 2 x (g) log 3 y = 2 + log 9 x (h) log 5 (y &#x2013; 1) + 2 = log 5 x (i) log 2 (y &#x2013; x) &#x2013; 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 &#x2013; log 5 500 + log 5 10 (c) log 3 8 &#x2013; log 3 108 &#x2013; log 3 6 (d) log 25 4 . log 8 36 . log 216 125 (e) log 5 10 &#x2013; log 5 4 + log 5 250 (f) log 2 3 + log 2 12 &#x2013; log 2 24 &#x2013; log 2 48 (g) log 27 8 . log 125 9 . log 4 25 (h) log 3 6 + log 3 162 &#x2013; log 3 108 (i) log 4 49 . log 125 8 . log 7 25 (j) log 3 8 &#x2013; log 3 6 &#x2013; 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 &#x2013; 1) 3 )-(4 log m = 3 (f) 2 31)-( 2 log k = 8 (g) (m &#x2013; 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 &#x2212; (b) log 4 3 2 8 = m (c) 0.4771 = log 10 (4 &#x2013; 2x) (d) 64 1 log n = 3 (e) log 10 15 + log 10 5 3 &#x2500; x = 2 log 10 0.3 2. Solve log 2 5x &#x2013; log 2 (x + 3) = 1
• 10. MODULE 1 ADDITIONAL MATHEMATICS SPM 3. Solve log 3 (2x + 7) &#x2013; log 3 x = 2 4. Solve log 2 (3x &#x2013; 2) = 1 &#x2013; log 2 (x + 1) 5. Solve log 4 (x + 4) &#x2013; log 4 (3x + 1) = 2 1 6. Solve log 2 5x &#x2013; log 2 (1 + 8x) = &#x2013; 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 &#x2013; 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 &#x2013; 2x &#x2013; 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 = &#x2013; 3x &#x2013; 7 and passes through the point (3, 2) (i) which passes through the point (&#x2013;1, 3) and is parallel to the straight line 3y = &#x2013; 2x &#x2013; 5. (j) which is perpendicular to the straight line 4y &#x2013; 3x &#x2013; 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 (&#x2500; 4, 2) C B (6, 0) x y O A (1, 5) 2y = 3x &#x2013; 18 C B x y O P (5, 7) R Q (&#x2500; 4, &#x2500; 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 = &#x2013; 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 (&#x2013; 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 = &#x2013; 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 = &#x2013; 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 &#x2013; x = 0. Given that PR is perpendicular to the straight line 2y &#x2013; 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 = &#x2013; 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(&#x2013; 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(&#x2013; 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, &#x2013; 12 ) Q A(1, 2) CB(7, &#x2013; 1) A(&#x2013; 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 &#x2013; 2 and n &#x2013; 2. 2. Given the equation x 2 &#x2013; 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 &#x2013; 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 &#x2013; (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 &#x2013; (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 &#x2013; (3b &#x2013; 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 = &#x2013; 11x &#x2013; 6 has the roots of p and q Form a quadratic equation with roots 2p and 2q. 11. Given the equation (2x + 3)(x &#x2013; 4) + b = 0 where one of the roots is four times of the other root. Find, (i) the roots of the equation
• 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 &#x2013; 5x + 3 (b) f (x) = x 2 + 8x &#x2013; 7 (c) f (x) = x 2 &#x2013; 6x &#x2013; 3 (d) f (x) = 4 + 8x &#x2013; x 2 (e) f (x) = 9 &#x2013; 6x &#x2013; x 2 (f) f (x) = x 2 + 2x &#x2013; 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 &#x2013; 3x + 9 (b) f (x) = 2 &#x2013; 4x &#x2013; 2x 2 (c) f (x) = 5x 2 &#x2013; 10x + 5 (d) f (x) = 2x 2 &#x2013; 8x + 4 (e) f (x) = 5 + 20x &#x2013; 5x 2 (f) f (x) = 9 &#x2013; 6x &#x2013; 3x 2 4 Sketch the graph for each of the following equations, (a) y = x 2 &#x2013; 8x + 3 (b) y = 4 + 8x &#x2013; x 2 (c) y = 2x 2 + 4x &#x2013; 6 (d) y = 4 &#x2013; 8x &#x2013; 2x 2 (e) y = 3x 2 + 6x &#x2013; 6 (f) y = 9 + 6x &#x2013; 3x 2 5 Diagram shows the graph of y = 3(x &#x2013; a) 2 + b. Find, (a) the values of a and b (b) the coordinates of A. 6 Diagram shows the graph of y = &#x2013; 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, &#x2013; 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 &#x2013; 2) &gt; 3 2. Find the range of values of x for which x(6x &#x2013; 7) &gt; 10 3. Find the range of values of x for which x(x &#x2013; 13) &#x2265; &#x2500; 42 4. Find the range of values of x for which 2x(x &#x2013; 1) &lt; 3 &#x2013; x 5 Diagram shows the graph of the function y = &#x2500; (x &#x2013; h)2 &#x2013; 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 = &#x2500; (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 &#x2013; 8 with (&#x2500; 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, &#x2500; 6) x y O &#x2500; 6 (3, q) x y O (&#x2013; 4, c) x y O
• 21. MODULE 1 ADDITIONAL MATHEMATICS SPM symmetry. 8. The quadratic function f(x) = x2 &#x2013; 4x + p has a minimum value of &#x2500; 11. Find the value of p. 9 The function f(x) = a &#x2013; bx &#x2013; 3x2 has a maximum value of 6 when x = &#x2500; 2. Find the value of a and b. 10. Given that f(x) = (x &#x2013; 3)2 &#x2013; 9 = hx2 &#x2500; 6x + k &#x2013; 2. Find, (a) the value of h and k. (b) the minimum value of f(x). 11. The function f(x) = x 2 &#x2013; 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 &#x2013; 1. (b) Hence, or other wise, find the value of k and r if the graph of function is symmetrical at x = r 2 &#x2013; 1. FUNCTIONS 1. Given the function f : x &#x2192; 2x + 1 and g : x &#x2192; 1 2 +x where x &#x2260; &#x2013; 1. Find, (a) fg (b) gf (c) f 2 g 2. Given the function f (x) = 2&#x2212;x x where x &#x2260; 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 &#x2212; + x x and g(x) = 2 1 + + x x Find, (a) fg (b) fg 2 4. Given the function f (x) = 52 13 &#x2212; + x x and g(x) = 43 32 &#x2212; + x x
• 22. MODULE 1 ADDITIONAL MATHEMATICS SPM Find, (a) f &#x2013; 1 g (b) gf &#x2013; 1 (c) f &#x2013; 1 g &#x2013; 1 5. Given the function f (x) = 3 52 + + x x and g(x) = 2 13 &#x2212; &#x2212; x x . Find, (a) f &#x2013; 1 g (b) g &#x2013; 1 f &#x2013; 1 6. Given the function f (x) = 2 13 &#x2212; + 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 &#x2212; + 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 &#x2212; + 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 &#x2013; 1 (5) (b) fg &#x2013; 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 &gt; 0. Find the values of a and b. 14. Given the function f (x) = 53 12 &#x2212; + x x and g(x) = 2 4 +x x . If f &#x2013; 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 &#x2013; 1 (x) = bx + + ax , find the values of a and b. 16. Given the functions f : x &#x2192; 2x + 5 and g : x &#x2192; 3 &#x2013; 2x. Find the value of, (a) f 2 (b) gf 17. Given the functions g : x &#x2192; 5x + 1 and h : x &#x2192; x 2 &#x2013; 2x + 3. Find the value of, (a) g2 (b) gh 18 Given that f : x &#x2192; 2x + 5 and fg: x &#x2192; 13 &#x2013; 2x. Find g(x). 19. Given that the composite function gf(x) = 3x &#x2013; 4, find f(x) if g(x) = 2 x . 20 Given that the composite function hg : x &#x2192; 1 6 + x and h : x &#x2192; 2x + 1. Find g(3). 21. Given that the function g : x &#x2192; 3x + 2 and fg(x) = 27x + 17. Find f(x). 22. Given that f : x &#x2192; ax + b, g : x &#x2192; x 2 + 3 and fg : x &#x2192; 3x 2 + 5. Find the value of a and b. 23. The functions f and g are defined by f : x &#x2192; 5 &#x2013; 3x and g : x &#x2192; 2ax + b respectively, where a and b are constants. If the composite function fg(x) = 8 &#x2013; 3x, find the value of a and b. 24 Given that f : x &#x2192; 4 &#x2013; 3x and g : x &#x2192; 2x 2 + 5, find, (a) f &#x2013; 1 (&#x2500; 8) (b) gf (x) 25. Given that f : x &#x2192; 4x + 1 and g : x &#x2192; x 2 &#x2013; 2x &#x2013; 6, find, (a) f &#x2013; 1 (&#x2500; 3) (b) gf (x) 26 Given that g : x &#x2192; 1 &#x2013; 3x &#x2013; x 2 and h &#x2013; 1 : x &#x2192; 2x &#x2013; 1, find,
• 24. MODULE 1 ADDITIONAL MATHEMATICS SPM (a) h(5) (b) gh &#x2013; 1 (x) 27. The following information refers to the functions f and g. Find fg &#x2013; 1 (x). 28. The following information refers to the functions f and g. Find f &#x2013; 1 g(x) 29. Given the functions h : x &#x2192; 2x + n and h &#x2013; 1 : x &#x2192; mx &#x2013; 2 3 , where m and n are constants, find the value of m and of n. 30. Given the functions f : x &#x2192; 4+ &#x2212; x bax and f &#x2013; 1 : x &#x2192; 2 34 &#x2212; &#x2212;&#x2212; x x , where a and b are constants, find the value of a and of b. 31 Given the functions g : x &#x2192; p &#x2013; 3x and g &#x2013; 1 : x &#x2192; 3 2 &#x2013; qx , where p and q are constants, find the value of p and of q. f : x &#x2192; 3x + 2 g : x &#x2192; x &#x2013; 3 f : x &#x2192; 3x + 5 g : x &#x2192; x x &#x2212;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 &#x2212; &#x2212; x x (g) y = 5 &#x2013; x 2 &#x2013; x 3 (h) y = (x 4 + 1)x 5 (i) y = 4 2 1 x xx ++ (j) y = ( )3 23 2 x&#x2212; (k) y = (2x + 1) 3 (3x &#x2013; 1) 4 (l) y = ( ) ( )2 3 32 12 x x &#x2212; + (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 &#x2013; 2x 3 ) 4 (q) y = (1 &#x2013; 3x) 2 (2x + 1) 3 (r) y = 2 43 13 x x &#x2212; + 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 &#x2013; 3, find the small change in p when q increases from 4 to 4.02. 4. Given y = 5x 2 &#x2013; x + 2, find the small change in y when x increases from 5 to 5.03. 5. Given A = 2B 2 &#x2013; B &#x2013; 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 &#x2013; 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 &#x2013; 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 &#x2013; 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 2&#x3C0; cm 3 s &#x2013; 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 &#x2013; 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 &#x2013; 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 200&#x3C0; cm 3 s &#x2013; 1 . Find the rate of change of the radius when the radius is 5 cm. 19 A container has a volume of V = 2&#x3C0; x (3x + 5) where the volume increases at the rate of 200&#x3C0; cm 3 s &#x2013; 1 . Find the rate of change of x when x = 5.