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1. 1. ADDITIONAL MATHEMATICS FORM 4 2007 mozac 1 ADDITIONAL MATHEMATICS FORM 4 MODULE 1 FUNCTIONS SIMULTANEOUS EQUATIONS MODUL KECEMERLANGAN AKADEMIK
2. 2. ADDITIONAL MATHEMATICS FORM 4 2007mozac 2 1 FUNCTIONS  PAPER 1 1 A relation from set P = {6, 7, 8, 9} to set Q = {0, 1, 2, 3, 4} is defined by ‘subtract by 5 from’. State (a) the object of 1 and 4, (b) the range of the relation. Answer : (a)………………………… (b)………………………… 2 The arrow diagram below shows the relation between Set A and Set B. Set A Set B State (a) the range of the relation, (b) the type of the relation. Answer : (a)………………………… (b)………………………… 3 The function f is defined by f : x  2 – mx and f 1 (8) = 2, find the value of m. Answer : m = ………………………….          3 2 1 1 16 12 9 4 1
3. 3. ADDITIONAL MATHEMATICS FORM 4 2007mozac 3 4 Given the function : 3 4f x x  , find the value of m if 1 (2 1) .f m m   Answer : m = …………………………. 5 Given the functions : 2 4f x x  and 10 : , 2, 2 fg x x x    find (a) the function g, (b) the values of x when the function g mapped onto itself. Answer : (a)………………………… (b)………………………… 6 The function f is defined by : , 3 x a f x x h x     . Given that 1 (2) 8f   , Find (a) the value of h, (b) the value of a. Answer : (a) h = ……………………… (b) a = ………………………
4. 4. ADDITIONAL MATHEMATICS FORM 4 2007mozac 4 7 Given the functions : 2f x x  and 2 :g x mx n  . If the composite function fg is given by 2 : 3 12 8gf x x x  , find (a) the values of m and n, (b) 2 ( 1)g  . Answer : (a) m = ……………………… n = ……………………… (b)…………………………….. 8 Given the functions :f x px q  where p > 0 and 2 4 9:f x x  , find (a) the values of p and q, (b) 1 f  (5). Answer : (a) p = ……………………… q = ……………………… (b)…………………………….. 9 If 4 : , 3 3 f x x x    , : 3gf x x  and 4 3 : , 3 5 5 fh x x x    , find (a) the function g, (b) the function h. Answer : (a)………………………… (b)…………………………
5. 5. ADDITIONAL MATHEMATICS FORM 4 2007mozac 5 10 Given the function f : 7 2 .x x  Find (a) the range of f corresponds to the domain 1 3x  , (b) the value of x that maps onto itself. Answer : (a)………………………… (b) x = .…………………… 11 Given the function  xxf 3: p and 1 5 : 2 3 f x qx   , where p and q are constants. Find the values of p and q. Answer : (a) p = …………………… (b) q = …………………… 12 Given : 4 3f x x   , find (a) the image of –3, (b) the object which has the image of 5. Answer : (a)………………………… (b)…………………………
6. 6. ADDITIONAL MATHEMATICS FORM 4 2007mozac 6 13 The diagram below shows the mapping for the function 1 f and g. ` Given that f (x) = ax + b and g(x) = b x a , calculate the value of a and b. Answer : a = ………………………… b = ………………………… 14 Given that :h x  | 5x – 2 |, find (a) the object of 6, (b) the image which has the object –2. Answer : (a)………………………… (b)………………………… 15 Given that xxf 23:  and 1)( 2 xxg , find (a) f g(x), (b) g f(–1). Answer : (a)………………………… (b)………………………… ● ● ● 1 f g 2 6 4
7. 7. ADDITIONAL MATHEMATICS FORM 4 2007mozac 7  PAPER 2 16 The above diagram shows part of the function rqxpxxf  2 )( . Find (a) the values of p, q and r, (b) the values of x which map onto itself under the function f. 17 Given that functions f and g are defined as 2 : xxf  and :g x ax b  where a and b are constants. (a) Given that f(1) = g(1) and f(3) = g(5), find the values of a and b. (b) With the values a and b obtained from (a), find gg(x) and g1 . . 18 Given v(x) = 3x – 6 and w(x) = 6x – 1, find (a) vw1 (x), (b) values of x so that vw(2x) = x. 19 Given that the function 2 1 :   x xf , and the composite function 162: 21  xxxgf , find (a) the function of g (x), (b) g f (3), (c) f 2 (x). 20 Given that : 3 2f x x  and : 1 5 x g x   , find (a) f 1 (x), (b) f 1 g (x), (c) h(x) such that hg(x) = 2x + 6.       x f (x) 2 1 0 10 1 4
8. 8. ADDITIONAL MATHEMATICS FORM 4 2007mozac 8 4 SIMULTANEOUS EQUATIONS  PAPER 2 1 Solve the equation 4x + y + 8 = x2 + x – y = 2. 2 Solve the simultaneous equations qp 1 3 2  = 2 and 3p + q = 3. 3 Solve the equation x2 – y + y2 = 2x + 2y = 10. 4 Solve the simultaneous equations and give your answers correct to three decimal places, 2m + 3n + 1 = 0, m2 + 6mn + 6 = 0. 5 Solve the simultaneous equations 1 3 x y = 3 and y2 – 1 = 2x. 6 Given (1, 2k) is the solution of the simultaneous equation x2 + py – 29 = 4 = px – xy, where k and p are constants. Find the values of k and p. 7 Solve the simultaneous equations 3 0 3 2 x y    and 3 2 1 0 2x y    8 Given (2k, 4p) is the solution of the simultaneous equations x – 3y = 4 and 9 7 4x y  = 1. Find the values of k and p. 9 Given the following equations : A = x + y B = 2x – 14 C = xy – 9 Find the values of x and y such that 3A = B = C
9. 9. ADDITIONAL MATHEMATICS FORM 4 2007mozac 9 10 Solve the simultaneous equations and give your answers correct to four significant figures, x + 2y = 2 2y2 – xy – 7 = 0 11 The straight line 3y = 1 – 2x intersects the curve y2 3x2 = 4xy – 6 at two points. Find the coordinates of the points. 12 If x = 2 and y = 1 are the solutions to the simultaneous equations ax + b2 y = 2 and 2 2 1 2 b x ay  , find the values of a and b. 13 The perimeter of a rectangle is 34 cm and the length of its diagonal is 13 cm. Find the length and width of the rectangle. 14 The difference between two numbers is 8. The sum of the squares and the product of the numbers is 19. Find the two numbers. 15 A piece of wire of length 24 cm is cut into two pieces, with one piece bent to form a square ABCD and the other bent to form a right-angled triangle PQR. The diagram below shows the dimensions of the two geometrical shapes formed. The total area of two shapes is 15 cm2 , (a) show that 6x + y = 21 and 2x2 + y(x + 1) = 30. (b) Find the value of x and y. x cm A (x + 1) cm y cm B x cm (x + 2) cm P D RC S
11. 11. ADDITIONAL MATHEMATICS FORM 4 2007mozac 2 2 QUADRATIC EQUATIONS  PAPER 1 1 One of the roots of the quadratic equation 2x2 + kx – 3 = 0 is 3, find the value of k. Answer : k = …………….……………. 2 Given that the roots of the quadratic equation x2 – hx + 8 = 0 are p and 2p, find the values of h. Answer : h = ………………………… 3 Given that the quadratic equation x 2 + (m – 3)x = 2m – 6 has two equal roots, find the values of m. Answer : m = ………………………… 4 Given that one of the roots of the quadratic equation 2x2 + 18x = 2 – k is twice the other root, find the value of k. Answer : k = ………………………… 5 Find the value of p for which 2y + x = p is a tangent to the curve y2 + 4x = 20.
12. 12. ADDITIONAL MATHEMATICS FORM 4 2007mozac 3 Answer : p = ………………………… 6 Solve the equation 2(3x – 1)2 = 18. Answer : …..………………………… 7 Solve the equation (x + 1)(x – 4) = 7. Give your answer correct to 3 significant figures. Answer : …..………………………… 8 Find the range of values of m such that the equation 2x 2 – x = m – 2 has real roots. Answer : …..………………………… 9 Find the range of values of x for which (2x + 1)(x + 3) > (x + 3)(x – 3).
13. 13. ADDITIONAL MATHEMATICS FORM 4 2007mozac 4 Answer : …..………………………… 10 Find the range of values of k such that the quadratic equation x2 + x + 8 = k(2x – k) has two real roots. Answer : …..…………………………  PAPER 2 11 The quadratic equation xqpxpx 10222  has roots 1 p and q. (a) Find the values of p and q. (b) Hence, form a quadratic equation which has the roots p and 3q. 12 (a) Given that and are the roots of the quadratic equation 2x2 + 7x – 6 = 0, form a quadratic equation with roots (+ 1) and (+ 1). (b) Find the value of p such that (p – 4)x2 + 2(2 – p)x + p + 1 = 0 has equal roots. Hence, find the root of the equation based on the value of p obtained. 13 (a) Given that 2 and m – 1 are the roots of the equation x2 + 3x = k, find the values of m and k. (b) Find the range of values of p if the straight line y = px – 5 does not intersect the curve y = x2 – 1. 14 (a) Given that 3 and m are the roots of the quadratic equation 2(x + 1)(x + 2) = k(x – 1). Find the values of m and k . (b) Prove that the roots of the equation x2 + (2a – 1)x + a2 = 0 is real when a  1 4 . 15 (a) Find the range of values of p where px2 + 2(p + 2)x + p + 7 = 0 has real roots. (b) Given that the roots of the equation x 2 + px + q = 0 are and 3, show that 3p 2 = 16q.
14. 14. ADDITIONAL MATHEMATICS FORM 4 2007mozac 5 3 QUADRATIC FUNCTIONS  PAPER 1 1 Solve the inequality 2(x – 3)2 > 8. Answer : …..………………………… 2 Find the range of values of p which satisfies the inequality 2p2 + 7p 4. Answer : …..………………………… 3 Find the range of values of m if the equation (2 – 3m)x2 + (4 – m)x + 2 = 0 has no real roots. Answer : …..………………………… 4 The quadratic function 4x2 + (12 – 4k)x + 15 – 5k = 0 has two different roots, find the range of values of k. Answer : …..…………………………
15. 15. ADDITIONAL MATHEMATICS FORM 4 2007mozac 6 5 Without using differentiation method find the minimum value of the function f(x) = 3x2 + x + 2. Answer : f (x)min = …………………… 6 Given that g(x) = 3x2 – 2x – 8, find the range of values of x so that g(x) is always positive. Answer : …..………………………… 7 The expression x2 – x + p, where p is a constant, has a minimum value 9 4 . Find the value of p. Answer : p = ………………………… 8 The quadratic functions 2 3 ( ) 3 ( 1) 2 k f x x         has a minimum value of 6. Find the value of k. Answer : k = …………………………
16. 16. ADDITIONAL MATHEMATICS FORM 4 2007mozac 7 9 (a) Express y = 1 + 20x – 2x2 in the form y = a(x + p)2 + q. (b) Hence, state (i) the minimum value of y, (ii) the corresponding value of x. Answer : (a) …………….…………….. (b) (i) ……….……………... (ii) ……………………… 10 Jawapan : p = …………………………… q = …………………………… r = …………………………… 0 33 (4, 1)   x y The diagram on the left shows the graph of the curve 2 ( )y p x q r   with the turning point at (4, 1). Find the values of p, q and r .
17. 17. ADDITIONAL MATHEMATICS FORM 4 2007mozac 8  PAPER 2 11 Given the function f (x) = 7 mx x2 = 16 (x + n)2 for all real values of x where m and n are positive, find (a) the values of m and n, (b) the maximum point of f(x), (c) the range of values of x so that f(x) is negative. Hence, sketch the graph of f(x) and state the axis of symmetry. 12 Given that the quadratic function f (x) = –2x2 – 12x – 23, (a) express f (x) in the form m(x + n)2 + p, where m, n and p are constants. (b) Determine whether the function f(x) has the minimum or maximum value and state its value. 13 Given that x2 – 3x + 5 = p(x – h)2 + k for all real values of x, vhere p, h and k are constants. (a) State the values of p, h and k, (b) Find the minimum or maximum value of x2 – 3x + 5 and the corresponding value of x. (c) Sketch a graph of f (x) = x2 – 3x + 5. (d) Find the range of values of m such that the equation x2 – 3x + 5 = 2m has two different roots.
18. 18. ADDITIONAL MATHEMATICS FORM 4 MODULE 3 INDICES AND LOGARITHMS COORDINATE GEOMETRY MODUL KECEMERLANGAN AKADEMIK
19. 19. 5 INDICES AND LOGARITHMS  PAPER 1 1 Simplify 3 32 27 93    x xx . Answer : ………………………………… 2 Express )5(1555 12212   xxx to its simplest form. Answer : ………………………………… 3 Show that 7 x + 7 x + 1 – 21(7 x – 1 ) is divisible by 5 for all positive integers of n. Answer : ………………………………… 4 Find the value of a if log a 8 = 3. Answer : a = ..………………………… 5 Evaluate 55log5 .
20. 20. Answer : ………………………………… 6 Given ma 10log and nb 10log . Express b a 100 log 3 10 in terms of m and n. Answer : ………………………………… 7 Given log 7 2 = p and q5log7 . Express 7 log 2 8 in terms of p and q. Answer : ………………………………… 8 Simplify 27log 243log13log 8 1364  . Answer : ………………………………… 9 Solve the equation xx 95 12  .
21. 21. Answer : ………………………………… 10 Solve the equation log 3 (2x + 1) = 2 + log3 (3x – 2). Answer : …………………………………  PAPER 2 11 The temperature of an object decreases from 80C to T C after t minutes. Given T = 80(08)t . Find (a) the temperature of the object after 3 minutes, (b) the time taken for the object to cool down from 80C to 25C. 12 (a) (i) Prove that 9log ab = 3 3 1 log log ) 2 ( a b . (ii) Find the values of a and b given that 3log 4 ab and 2 1 log log 4 4  b a . (b) Evaluate 1 1 5 5 3(5 ) n n n    . 13 The total amount of money deposited in a fixed deposit account in a finance company after a period of n years is given by RM20 000(104)n .Calculate the minimum number of years needed for the amount of money to exceed RM45 000. 14 (a) Solve the equation 5log 644 x  .
22. 22. (b) Find the value of x given that log 5 log 135x x  = 3. (c) Given 2 5 loglog 42  ba . Express a in terms of b. 15 (a) Solve the equation  9 3 16log log (2 1) log 4x   . (b) Given that 3log 5 a and 3log 7 b , find the value of p if 2 3 log3 ba p   .
23. 23. 6 COORDINATE GEOMETRY  PAPER 1 1 Given the distance between two points A(1, 3) and B(7, m) is 10 units. Find the value of m. Answer : m = …………………………… 2 Given points P(2, 12), Q(2, a) and R(4, 3) are collinear. Find the value of a. Answer : a = ………………………………… 3 Find the equation of a straight line that passes through B(3, 1) and parallel to 5x – 3y = 8. Answer : …………………………………
24. 24. 4 Find the equation of the perpendicular bisector of points A(1, 6) and B(3,0). Answer : ………………………………… 5 Given A(p, 3), B(3, 7), C(5, q) and D(3, 4) are vertices of a parallelogram. Find (a) the values of p and q, (b) the area of ABCD. Answer: (a) p = ………………………… q = ………………………… (b) ……………………………. 6 The points A(h, 2h), B(m, n) and C(3m, 2n) are collinear. B divides AC internally in the ratio of 3 : 2. Express m in terms of n. Answer : …………………………………
25. 25. 7 The equations of the straight lines AB and CD are as follows: AB : y = hx + k CD : 3 6        hx k y Given that the lines AB and CD are perpendicular to each other, express h in terms of k. Answer : ………………………………… 8 Given point A is the point of intersection between the straight lines 3 2 1  xy and x + y = 9. Find the coordinates of A. Answer : ………………………………… 9 Find the equation of the locus of a moving point P such that its distance from point R(3, 6) is 5 units. Answer : …………………………………
26. 26. 10 Given points K(2, 0) and point L(2, 3). Point P moves such that PK : PL = 3 : 2.Find the equation of the locus of P. Answer : …………………………………  PAPER 2 11 Given C(5, 2) and D(2, 1) are two fixed points. Point P moves such that the ratio of CP to PD is 2 : 1. (a) Show that the equation of the locus of point P is 034222  yxyx . (b) Show that point E(1, 0) lies on the locus of point P. (c) Find the equation of the straight line CE. (d) Given the straight line CE intersects the locus of point P again at point F, find the coordinates of point F. 12 Given points P(2, 3), Q(0, 3) and R(6, 1). (a) Prove that angle PQR is a right angle. (b) Find the area of triangle PQR. (c) Find the equation of the straight line that is parallel to PR and passing through point Q. 13 The diagram above shows a quadrilateral KLMN with vertices M(3, 4) and N(2, 4).Given the equation of KL is 5y = 9x – 20. Find (a) the equation of ML, (b) coordinates of L, (c) the coordinates of K, (d) the area of the quadrilateral KLMN. x M(3, 4) N(2,4) K L 0 y
27. 27. 14 In the above diagram, PQRS is a trapezium.QR is parallel to PS and QRS = PSR = 90. (a) Find (i) the equation of the straight line RS, (ii) the coordinates of S. (b) The line PQ produced meets the line SR produced at T.Find (i) the coordinates of T, (ii) the ratio of PQ : QT. 15 The above diagram shows a rectangle ABCD with vertices B(3, 3), A and C are points On the x-axis and y-axis respectively. Given that the equation of the straight line AB is 2y = x + 3, find (a) the coordinates of A, (b) the equation of BC, (c) the coordinates of C, (d) the area of triangle ABC, (e) the area of rectangle ABCD. C B(3,3) A D 0 x y Q(2, 7) P(0, 1) R(10, 11) S 0 x y
28. 28. ADDITIONAL MATHEMATICS FORM 4 2007mozac 1 ADDITIONAL MATHEMATICS FORM 4 MODULE 4 STATISTICS CIRCULAR MEASURE
29. 29. ADDITIONAL MATHEMATICS FORM 4 2007mozac 2 7 STATISTICS  PAPER 1 1 The mean of a list of numbers x – 1, x + 3, 2x + 4, 2x – 3, x + 1 and x – 2 is 7. Find (a) the value of x, (b) the variance of the numbers. Answer: (a) x = .……………………… (b) …………………………… 2 The mean of a list of numbers 3k , 5k + 4, 3k + 4 , 7k – 2 and 6k + 6 is 12. Find (a) the value of k, (b) the median of the numbers. Answer: (a) k = .……………………… (b) …………………………… 3 Given a list of numbers 8, 9, 7, 10 and 6. Find the standard deviation of the numbers. Answer: ………………………….
30. 30. ADDITIONAL MATHEMATICS FORM 4 2007mozac 3 4 The set of positive numbers 3, 4, 7, 8,12, x, y has a mean 6 and median 7. Find the possible values of x and y. Answer: x = …………………………….. y = …………………………….. 5 The test marks of a group of students are 15, 43, 47, 53, 65, and 59. Determine (a) the range, (b) the interquartile range of the marks. Answer: (a) …………………………… (b) …………………………… 6 The mean of five numbers is q . The sum of the squares of the numbers is 120 and the standard deviation of the numbers is 4m. Express q in terms of m. Answer : ……………………………
31. 31. ADDITIONAL MATHEMATICS FORM 4 2007mozac 4 7 The sum of the 10 numbers is 170 and the sum of the squares of the numbers is 2930. Find the variance of the 10 numbers. Answer: ……………………………… 8 Score 0 1 2 3 4 Frequency 7 10 p 15 8 The table shows the scores obtained by a group of contestants in a quiz. If the median is 2, find the minimum value of p. Answer: ……………………………… 9 The numbers 3, 9, y , 15, 17 and 21 are arranged in ascending order. If the mean is equal to the median, determine the value of y. Answer : y = …………………………… 10 Number 41 – 45 46 – 50 51 – 55 56 – 60 61 – 65 Frequency 6 10 12 8 4 The table above shows the Additional Mathematics test marks of 40 candidates. Find the median of the distribution. Answer:............................................. Number of goals 1 2 3 4 5
32. 32. ADDITIONAL MATHEMATICS FORM 4 2007mozac 5 11 The table above shows the number of goals score in each match in a football tournament. Calculate the mean and the standard deviation of the data. Answer : mean = …………………………… standard deviation = ……………... 12 Given the set of positive numbers n, 5, 11. (a) Find the mean of the set of numbers in terms of n. (b) If the variance is 14, find the values of n. Answer: (a) …………………………… (b) n = ..……………………… 13 The mean and standard deviation for the numbers x1, x2, …, xn are 74 and 26 respectively. Find the (a) mean for the numbers 3x1 + 5 , 3x2 + 5, …, 3xn + 5, (b) variance for the numbers 4x1 + 2 , 4x2 + 2, …, 4xn + 2. Answer: (a) …………………………… (b) …………………………… Frequency 7 6 4 2 1
33. 33. ADDITIONAL MATHEMATICS FORM 4 2007mozac 6 14 The mean of the data 2, h, 3h, 11, 12 and 17 which has been arranged in an ascending order, is p. If each of the element of the data is reduced by 2, the new median is 8 9 p. Find the values of h and p. Answer: h = …………………………… p = …………………………… 15 The table above shows a set of numbers arranged in ascending order where p is a positive integer. (a) Express the median of the set of the of numbers in terms of p. (b) Find the possible values of p. Answer: (a) ………………………….. (b) p = …………………...….  PAPER 2 16 A set of examination marks x1, x2, x3, x4, x5, x6 has a mean of 7 and a standard deviation of 14. (a) Find (i) the sum of the marks, x. (ii) the sum of the squares of the marks, x2 . (b) Each mark is multiplied by 3 and then 4 is added to it. Find, for the new set of marks, (i) the mean, (ii) the variance. Number 2 p – 1 7 p + 4 10 12 Frequency 2 4 2 3 3 2
34. 34. ADDITIONAL MATHEMATICS FORM 4 2007mozac 7 17 Length (mm) 16 – 19 20 – 23 24 – 27 28 – 31 32 – 35 36 – 39 Frequency 2 8 18 15 6 1 The table above shows the lengths of 50 leaves collected from a tree. (a) Calculate (i) the mean, (ii) the variance length of the leaves. (b) Without drawing an ogive, find the interquartile range length of the leaves. 18 Set R consists of 40 scores, y, for a certain game with the mean of 9 and standard deviation of 5. (a) Calculate y and y2 . (b) A number of scores totaling 200 with a mean of 10 and the sum of the squares of these scores of 2700, is taken out from set R. Calculate the mean and variance of the remaining scores in set R. 19 A set of data consists of 10 number. The sum of the numbers is 150 and the sum of the squares of the numbers is 2 472. (a) Find the mean and variance of the 10 numbers. (b) Another number is added to the set of data and the mean is increased by 1. Find (i) the value of this number, (ii) standard deviation of the set of 11 numbers. 20 The table shows the frequency distribution of the scores of the scores of a group of pupils in a game. Score Number of pupils 10 – 19 1 20 – 29 2 30 – 39 8 40 – 49 12 50 – 59 m 60 – 69 1 (a) It is given that the median score of the distribution is 42. Calculate the value of m. (b) Use the graph paper provided by the invigilator to answer this question. Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis, draw a histogram to represent the frequency distribution of the scores. Find the mode score. (c) What is the mode score if the score of each pupil is increased by 5?
35. 35. ADDITIONAL MATHEMATICS FORM 4 2007mozac 8 8 CIRCULAR MEASURE  PAPER 1 1 Convert (a) 5420to radians. (b) 406 radians to degrees and minutes. Answer : (a) .......................................... (b) ......................................... 2 Answer : ...................................... 3 The area of a sector of a circle with radius 14 cm is 147 cm2 . Find the perimeter of the sector. Answer :....................................... The diagram on the left shows a sector OAB with centre O and radius 9 cm. Given that the perimeter of the sector OAB is 30 cm. Find the angle of AOB in radian. O A B 9 cm9 cm
36. 36. ADDITIONAL MATHEMATICS FORM 4 2007mozac 9 4 Answer :....................................... 5 Answer : ..................................... 6 Answer : ..................................... 2 rad 6 cm O A B The diagram on the left shows a circle with a sector OAB and centre O . Find the area of the major sector OAB in cm2 and state your answer in terms of π. O R Q P 2 cm 10 cm The diagram on the left shows a sector of a circle OPQ with centre O and OPR is a right angle triangle. Find the area of the shaded region. O A B The diagram on the left shows an arc of a circle AB with centre O and radius 4 cm. Given that the area of the sector AOB is 6 cm2 . Find the length of the arc AB.
37. 37. ADDITIONAL MATHEMATICS FORM 4 2007mozac 10 7 Answer : ...................................... 8 Answer :....................................... 9 Answer : ...................................... O P Q R S2 cm 0.8 rad The diagram shows two sectors OPQ and ORS of concentric circles with centre O. Given that POQ = 08 radian and OP = 3PR, find the perimeter of the shaded region. The diagram shows a semicircle of OPQR with centre O. Given that OP = 10 cm and QOR = 30. Calculate the area of the shaded region. P O R Q 30 10 cm The diagram shows a circle with centre O. Given that the major arc AB is 16cm and the minor arc AB is 4cm. Find the radius of the circle. O A B
38. 38. ADDITIONAL MATHEMATICS FORM 4 2007mozac 11 10 Answer : ...................................... 11 Answer : (a) r = ................................... (b) θ= ................................... 12 Answer : ……………………………… O R S  The diagram on the left shows a sector ROS with centre O. Given the length of the arc RS is 724 cm and the perimeter of the sector ROS is 25 cm. Find the value in radians. O A B  r cm The diagram on the left shows a sector with centre O. Given that the perimeter and the area of the sector is 14 cm and 10 cm2 respectively. Find (a) the value of r, (b) the value of θin radians. O A B 60 8 cm The diagram on the left shows a sector OAB of a circle with centre O. Find the perimeter of the shaded segment.
39. 39. ADDITIONAL MATHEMATICS FORM 4 2007mozac 12 13 Answer : (a) .......................................... (b) .........................................  PAPER 2 14 The above diagram shows two arcs AB and DE, of two circles with centre O. OBD and OCE are straight lines. Given OB = BD, find (a) the length of arc AB, (b) the area of segment DE, (c) the area of the shaded region. 15 The diagram on the right shows the position of a simple pendulum which swings from P and Q. Given that POQ = 25° and the length of arc PQ is 12.5 cm, calculate (a) the length of OQ, (b) the area swept out by the pendulum. O P Q O P Q R S T The diagram on the left shows a circle PRTSQ with centre O and radius 3 cm.Given RS = 4 cm and POQ = 130. Calculate (a) ROS , in degrees and minutes, (b) the area of segment RST, (c) the perimeter of the shaded region. 70 OA B C D E6 cm
40. 40. ADDITIONAL MATHEMATICS FORM 4 2007mozac 13 16 The diagram above shows a semicircle ACBE with centre C and a sector of a circle OADB with O. Given BAO = 35and OA = OB = 7 cm. Calculate (a) the diameter AB, (b) the area of the triangle AOB, (c) the area of the shaded region, (d) the perimeter of the shaded region. 17 The diagram above shows two circles PAQB with centres O and A respectively. Given that the diameter of the circle PAQB = 12 cm and both of the circles have the same radius. (a) Find POA in radians. (b) Find the area of the minor sector BOP. (c) Show that the area of the shaded region is (12 – 9 3 ) cm 2 the perimeter of the shaded region is (4+ 6 3 ) cm. O A B P O A B C D E   35
41. 41. ADDITIONAL MATHEMATICS FORM 4 2007mozac 14 18 The diagram below shows the plan of a garden. PCQ is a semicircle with centre O and has radius of 8 cm. RAQ is a sector of a circle with centre A and has a radius of 14 m. Sector COQ is a lawn. The shaded region is a flower bed and has to be fenced. It is given that AC = 8 m and COQ = 1956 radians. Using π= 3142, calculate (a) the area, in m2 , of the lawn, (b) the length, in m, of the fence required for fencing the flower bed, (c) the area, in m2 , of the flower bed. R Q R C P A O
42. 42. MODUL BIMBINGAN EMaS 2007 ADDITIONAL MATHEMATICS FORM 4 2007mozac 1 ADDITIONAL MATHEMATICS FORM 4 MODULE 5 DIFFERENTIATIONS
43. 43. ADDITIONAL MATHEMATICS FORM 4 2007mozac 2 9 DIFFERENTIATIONS  PAPER 1 1 Given y = 4(1 – 2x)3 , find dy dx . Answer : ………………………………… 2 Differentiate 3x2 (2x – 5)4 with respect to x. Answer : ………………………………… 3 Given that 2 1 (3 5) ( ) x h x   , evaluate h’’(1). Answer : …………………………………
44. 44. ADDITIONAL MATHEMATICS FORM 4 2007mozac 3 4 Differentiate the following expressions with respect to x. (a) (1 + 5x2 )3 (b) 2 43 4   x x Answer : (a) ………………………………… (b) ………………………………… 5 Given a curve with an equation y = (2x + 1)5 , find the gradient of the curve at the point x = 1. Answer : ………………………………… 6 Given y = (3x – 1)5 , solve the equation 2 2 12 0 d y dy dx dx  Answer : …………………………………
45. 45. ADDITIONAL MATHEMATICS FORM 4 2007mozac 4 7 Find the equation of the normal to the curve 53 2  xy at the point (1, 2). Answer : ………………………………… 8 Given that the curve qxpxy  2 has the gradient of 5 at the point (1, 2), find the values of p and q. Answer : p = ……………………………… q = ……………………………… 9 Given (2, t) is the turning point of the curve 142  xkxy . Find the values of k and t. Answer : k = ……………………………… t = ……………………………… 10 Given 22 yxz  and xy 21 , find the minimum value of z. Answer : …………………………………
46. 46. ADDITIONAL MATHEMATICS FORM 4 2007mozac 5 11 Given 12 tx and 54  ty . Find (a) dx dy in terms of t , where t is a variable, (b) dx dy in terms of y. Answer : (a) …………………………… (b) …………………………… 12 Given that y = 14x(5 – x), calculate (a) the value of x when y is a maximum, (b) the maximum value of y. Answer : (a) ………………………………… (b) ………………………………… 13 Given that y = x2 + 5x, use differentiation to find the small change in y when x increases from 3 to 301. Answer : …………………………………
47. 47. ADDITIONAL MATHEMATICS FORM 4 2007mozac 6 14 Two variables, x and y, are related by the equation y = 3x + 2 x . Given that y increases at a constant rate of 4 units per second, find the rate of change of x when x = 2. Answer : ………………………………… 15 The volume of water, V cm3 , in a container is given by 31 8 3 V h h  , where h cm is the height of the water in the container. Water is poured into the container at the rate of 10 cm3 s1 . Find the rate of change of the height of water, in cm s1 , at the instant when its height is 2 cm. Answer : ……………………………
48. 48. ADDITIONAL MATHEMATICS FORM 4 2007mozac 7  PAPER 2 16 (a) Given that graph of function 2 3 )( x q pxxf  , has gradient function 2 3 192 ( ) 6f x x x    where p and q are constants, find (i) the values of p and q , (ii) x-coordinate of the turning point of the graph of the function. (b) Given 3 29 ( 1) 2 p t t   . Find dt dp , and hence find the values of t where 9. dp dt  17 The gradient of the curve 4 k y x x   at the point (2, 7) is 1 2 4 , find (a) value of k, (b) the equation of the normal at the point (2, 7), (c) small change in y when x decreases from 2 to 197. 18 The diagram above shows a piece of square zinc with 8 m sides. Four squares with 2x m sides are cut out from its four vertices.The zinc sheet is then folded to form an open square box. (a) Show that the volume, V m3 , is V = 128x – 128x2 + 32x3 . (b) Calculate the value of x when V is maximum. (c) Hence, find the maximum value of V. 8 m 8 m 2x m 2x m2x m 2x m 2x m 2x m 2x m 2x m
49. 49. ADDITIONAL MATHEMATICS FORM 4 2007mozac 8 19 (a) Given that 12p q  , where 0p and 0.q  Find the maximum value of .2 qp (b) The above diagram shows a conical container of diameter 8 cm and height 6 cm. Water is poured into the container at a constant rate of 3 cm3 s1 . Calculate the rate of change of the height of the water level at the instant when the height of the water level is 2 cm. [Use = 3142 ; Volume of a cone = hr2 3 1  ] 20 (a) The above diagram shows a closed rectangular box of width x cm and height h cm. The length is two times its width and the volume of the box is 72 cm3 . (i) Show that the total surface area of the box, A cm2 is x xA 216 4 2  , (ii) Hence, find the minimum value of A. (b) The straight line 4y + x = k is the normal to the curve y = (2x – 3)2 – 5 at point E. Find (i) the coordinates of point E and the value of k, (ii) the equation of tangent at point E. 6 cm 8 cm h cm x cm 2x cm