CHAPTER 1: FUNCTIONS SPM 1993 1. Given the function f : x → 3 – 4x and function g : x → x2 – 1, find (a) f -1 (b) f -1g(3) [5 marks] 2.Given the functions f, g and h as a f : x → 2x 3 g:x→ ,x≠2 x−2 h : x → 6x2 – 2 (i) (ii) determine function f h(x) find the value of g -1(-2) [7 marks] SPM 1995 1. Given the function f(x) = 3x + c and 4 inverse function f -1(x) = mx + . Find 3 (a) the value of m and c [3 marks] (b) (i) f(3) (ii) f -1f(3) [3 marks] 2. Given the function f : x → mx + n, g : x → (x + 1)2 – 4 and fg : x → 2(x + 1)2 – 5. Find (i) g2(1) (ii) the values of m and n (iii) gf -1 [5 marks] SPM 1996 hx + k , x≠2 x−2 2x − 5 and inverse function f -1 : x → , x≠3 x −3 Find (a) the values of h and k [3 marks] (b) the values of x where f(x) = 2x [3 marks] 1. Given the function f : x → 2. Given the function f : x → 2x + 5 and fg : x →13 – 2x, Find (i) function gf (ii) the values of c if gf(c2 + 1) = 5c - 6 [5 marks] SPM 1997 1. Given the functions g: x → px + q and g2 : x→ 25x + 48 (a) Find the value of p and q (b) Assume that p>0, find the value of x so that 2g(x) = g(3x + 1) b\ 3. Function m given that m : x → 5 – 3x2 . If p is a another function and mp given that mp : x → -1 – 3x2, find function p. [3 marks] SPM 1994 1. Given the functions f(x) = 2 – x and function g(x) = kx2 + n. If the composite function gf(x) = 3x2 – 12x + 8, find (a) the values of k and n [3 marks] 2 (b) the value of g (0) [2 marks] 2. The function f is defined as p+x f:x→ , for all value of x except 3 + 2x x = h and p is a constant. (i) determine the value of h (ii) the value of 2 maps by itself under function f. Find (a) the value of p (b) the value of another x which is mapped onto itself (c) f -1(-1) [7 marks] 1 SPM 1998 SPM 2001 1. Given the functions h(t) = 2t + 5t2 and v(t) = 2 + 9t Find (a) the value of h(t) when v(t) = 110 (b) the values of t so that h(t) = v-1(2) (c) function hv 1. Given the functions f(x) = 6x + 5 and g(x) = 2x + 3 , find (a) f g-1(x) (b) the value of x so that gf(-x) = 25 1. Given the function f : x → ax + b, a > 0 and f 2 : x → 9x – 8 Find (a) the values of a and b [3 marks] (b) (f -1)2(x) [3 marks] 2. Given the function f -1(x) = −1 ,x≠p p−x and g(x) = 3 + x. Find (a) f(x) [2 marks] -1 2 (b) the value of p if ff (p –1) = g[(2-p)2] SPM 1999 ( c) range of value of p so that fg-1(x) = x no real roots 1. Given the function f : x → k – mx. Find [5 (a) f -1(x) in terms of k and m [2 marks] marks] (b) the values of k and m, if f -1(14) = - 4 and f(5) = -13 [4 marks] SPM 2002 2. (a) The function g is defined as g : x → x + 3. Given the function fg : x → x2 +6x + 7. Find (i) function f(x) (ii) the value of k if f(2k) = 5k [7 marks] SPM 2000 1. Given the function g -1(x) = f(x) = 3x2 – 5. Find (a) g(x) 5 − kx and 3 [2 marks] 1. Given the function f(x) = 4x -2 and g(x) = 5x +3. Find (i) fg -1(x) x 2 (ii) the value of x so that fg-1( ) = 2 5 [5 marks] 2. (a) Given the function f : x →3x + 1, find f -1(5) [2 marks] (b) Given the function f(x) = 5-3x and g(x) = 2ax + b, where a and b is a constants. If fg(x) = 8 – 3x, find the values of a and b [3 marks] (b) the value of k when g(x2) = 2f(-x) [3 marks] 2. Given the function f : x → 4 – 3x. (a) Find 2 (i) f (x) (ii) (f2)-1(x) -1 2 (iii) (f ) [6 marks] 2 SPM 2003 P = {1, 2, 3} Q = {2, 4, 6, 8, 10} 1. Based on the above information, the relation between P and Q is defined by set of ordered pairs {(1,2), (1,4), (2,6), (2,8)}. State (a) the image of 1 (b) the object of 2 [2 marks] 2. Given that g : x → 5x + 1 and h : x → x2 – 2x +3, find (a) g-1(3) (b) hg(x) [4 marks] SPM 2004 1. Diagram 1 shows the relation between set P and set Q 3. Given the function h(x) = 6 , x ≠ 0 and x the composite function hg(x) = 3x, find (a) g(x) (b) the value of x so that gh(x) = 5 [4 marks] SPM 2005 1. In Diagram 1, the function h maps x to y and the function g maps y to z Determine (a) h-1(5) (b) gh(2) [2 marks] d∙ e∙ f∙ Set P Diagram 1 ∙ ∙ ∙ ∙ w x y z 2. The function w is defined as 5 w(x) = , x ≠ 2. Find 2−x (a) w-1(x) (b) w-1(4) [3 marks] 3. The following information refers to the functions h and g. h : x → 2x – 3 g : x → 4x - 1 Find gh-1 [ [3 marks] Set Q State (a) the range of the relation (b) the type of the relation [2 marks] 2. Given the function h : x → 4x + m and 5 h-1 : x → 2hk + , where m and k are 8 constants, find the value of m and of k. [3 marks] 3 write a relation between set A and set B [2 marks] 2. Diagram 1 shows the linear function h. find the value of x such that f ( x ) = 5 [ [2m] 4 . where m is a constant x (a) State the value of m (b) Using the function notation. DIAGRAM 1 (a) State the type of relation between set A and set B (b) Using the function notation. Given the function f : x → x − 3 . In diagram 1. set B shows the image of certain elements of set A Paper 2 1.SPM 2006 Paper 1 1. express h in terms of x [2 m] − DIAGRAM 2 Find the value of m [2 marks] 1 2 2. find 5 −1 (a) f ( x) [1 m] −1 (b) f g ( x) [2 m] ( c) h(x) such that hg ( x) = 2 x + 6 [3 m] SPM 2007 Paper 1 1. Diagram shows the function m−x h:x → . Given that f : x → 3 x − 2 and x g : x → + 1 . x ≠ 0 . find a) f(5) b) the value of k such that gf(5)=14 [ [3m] State 5 .3. Given the functions f ( x) = x − 1 and g ( x) = kx + 2 . for the domain 0 ≤ x ≤ 5 . find a) g −1 (6) b) hg (x) Find the value of a and b [3m] SPM 2008 Paper 1 1. where a and b are constants and (a) the value of t (b) the range of f(x) corresponding to the given domain [3 m] 2. Diagram 1 shows the graph of the function f ( x ) = 2 x − 1 . The following information is about the function h and the composite function h 2 . Given the function g : x → 5 x + 2 and h : x → x 2 − 4 x + 3 . [4m] 3. [5 marks] 2. Find the possible values of k and m. Write a quadratic equation in a form ax2 + bx + c = 0 [2 marks] SPM 1997 1. [4 marks] 2. Given that α and β are the roots of the equation x2 – 2x + k = 0. [2 marks] CHAPTER 2: QUADRATIC EQUATIONS SPM 1994 1. Find the values of h. The equation of px2 + px + 3q = 1 + 2x 1 have the roots and q p (a) Find the value of p and q (b) Next. Find the range of value of k if the equation x 2 + kx + 2k − 3 = 0 has no real roots [3 marks] 4. Given the equation x2 – 6x + 7 = h(2x – 3) have two equal real roots. Given that a and b are the roots of the 6 . One of the roots of the equation 2x2 + 6x = 2k . If α and β are the roots of the quadratic equation 2x2 – 3x – 6 = 0. while 2α and 2β are the roots of the equation x2 +mx +9=0. Find the possible values of p. Find the values of λ so that (3 – λ)x2 – 2(λ + 1)x + λ + 1 = 0 has two equal real roots. form another β α quadratic equation with roots and 3 3 [4 marks] SPM 1995 1. find the value of k [5 marks] 2. Prove that the roots of the equation (1 – p)x2 + x + p = 0 has a real and negative roots if 0 < p < 1 [5 marks] SPM 1996 1. by using the value of p and q in (a) form the quadratic equation with roots p and -2q SPM 1999 1.1 is double of the other root.equation x2 – (a + b)x + ab = 0. One of the roots of the equation x2 + px + 12 = 0 is one third of the other root. where k is a constant. SPM 1998 1.1 are the roots of the equation x2 + 5x = -4. Find the possible value of m and n. Given that m + 2 and n . [4 marks] 3. If m and n are the roots of the equation (2x – 3)(x + 4) + k = 0 and m = 4n. [6 marks] 3. Given that 1 and -5 are the roots of the 2 quadratic equation. Find the roots and the possible values of k. Find the possibles values of p. Find the values of m and k [4 marks] 2. [5 marks] SPM 2003 1. Form the quadratic equation which has 1 the roots -3 and . [5 marks] SPM 2002 1. Given that α β and are the roots of the 2 2 equation kx(x – 1) = 2m – x. [3 marks] SPM 2006 1. Find the range of values of p [3 marks] SPM 2004 1. Given that 2 and m are the roots of the equation (2x -1)(x + 3) = k(x – 1). has two equal roots Express h in terms of k [4 marks] SPM 2008 1. where k is a constant. Give your answer correct to four significant figures. Give your answer correct to three decimal places. where h and k are constants. where a. find the range of value of k if the equation has two different real roots. Give your answer in 2 the form ax2 + bx + c =0.4 has two distinct roots. Given the equation x2 + 3 = k(x + 1) has the roots p and q. A quadratic equation x 2 + px + 9 = 2 x has two equal roots. Find (a) the values of p and q [3 marks] (b) the range of values of k if the Equation 2x2 + px + q = k has no real roots [2 marks] 2. form another quadratic equation with roots 3α + 2 and 3β + 2. It is given that -1 is one of the roots of the quadratic equation x 2 − 4x − p = 0 Find the value of p [2 marks] SPM 2001 1. find the values of k and m. [3 marks] SPM 2007 1. If α and β are the roots of the quadratic equation 2 x 2 + 3 x − 1 = 0 .SPM 2000 1. The quadratic equation x(x + 1) = px . Solve the quadratic equation x(2x – 5) = 2x – 1. (a) Solve the following quadratic equation: 3x 2 + 5x − 2 = 0 (c) The quadratic equation hx 2 + kx + 3 = 0. If α + β = 6 and αβ = 3. [3 marks] 7 . [5 marks] 2. The equation 2x2 + px + q = 0 has the roots -6 and 3. Solve the quadratic equation 2x(x – 4) = (1 – x)(x + 2). b and c are constants [2 marks] SPM 2005 1. where k is a constant. [3 marks] CHAPTER 3: QUADRATIC FUNCTIONS SPM 1993 1. Find the range of values of n if 2n2 + n ≥ 1 [2 marks] SPM 1996 1. the minimum point is (2. Find (a) the values of p. where k. Given the quadratic equation f(x) = 6x – 1 – 3x2. (a) Express quadratic equation f(x) in the form k + m(x + n)2. Given that 3x + 2y – 1 = 0.2. Then sketch the graph for the function y. m and n are constants. In the diagram 1. Find the range of values of x if (a) x(x + 1) < 2 [2 marks] −3 ≥x (b) 1 − 2x 1. find the range of values of x if y < 5. 3) of the function y = p(x + h)2 + k. (a) Find the range of value of x if 5x ≥ x2 [2 marks] (b) Find the range of value of p if x2 – (p + 1)x + 1 – p2 = 0 has no real roots. Find (i) the value of k (ii) the minimum point [4 marks] 2. [5 marks] 3. (b) Sketch the graph of function f(x) (c ) Find the range of value of p so that the equation 6x – 4 . Given that p = 5. (a) write f(x) in the form ax2 + bx + c [2 marks] (b) Curve y = kf(x) cut y-axis at the point (0. Without using differentiation method or drawing graph. find the minimum or maximum value of the function y = 2(3x – 1)(x + 1) – 12x – 1. f(x) = 0 is a quadratic equation which has the roots -3 and p.60). Determine whether the function f(x) has the minimum or maximum value and state the value of the minimum or maximum value. h and k (b) the equation of the curve when the graph is reflected on the x-axis [2 marks] 8 . [5 marks] 2.3x2 = p has two different real roots. [10 marks] SPM 1994 SPM 1995 1. have a minimum point p(6t. Given the quadratic equation 9 . Find the range of value of p so that f-1g(x) = x has no real roots SPM 2002 1. with m and n are constants. (a) state the value of m and n in terms of t (b) if t = 1. Show that the straight line and the curve does not intersect if k < 4 The graph show two curve y = 3(x-2)2 + 2p and y = x2 + 2x – qx + 3 that intersect in the two point at x-axis. Given that y = x + 2kx + 3k has a minimum value 2. (a) Given that f(x) = 4x2 – 1 Find the range of value of x so that f(x) is a positive (b) Find the range of value of x that satisfy inequality (x – 2)2 < (x – 2) SPM 1999 1. find the range of value of k so that the equation f(x) = k has a distinct roots 2.[3 marks] SPM 1997 1. (b) By using the value of k. determine the minimum or maximum point of the function y = 1 + 2x – 3x2. sketch the graph y = x2 + 2kx + 3k in the same axis (c) State the coordinate of minimum point for the graph y = x2 + 2kx + 3k SPM 1998 1. Find the range of values of k [5 marks] SPM 2001 1. Find the range of value of x if (x – 2)(2x + 3) > (x – 2)(x + 2) SPM 2000 1. Find (a) the value of p and q (b) the minimum value for the both curve 2. find two possible value of k. x ≠ p and p−x g(x) = 3 + x.(a) State the range of value of x for 5x > 2x2 – 3 (b) Given that the straight line 3y = 4 – 2x and curve 4x2 + 3y2 – k = 0. (a) Find the range of value of x so that 9 + 2x > 3 and 19 > 3x + 4 (b) Given that 2x + 3y = 6. Without using differentiation method or drawing graph.3t2). [4 marks] 2. 2 2. find the range of value of x when y < 5 2. Find the range of values of x if (a) 2(3x2 – x) ≤ 1 – x (b) 4y – 1 = 5x and 2y > 3 + x 3. Given that f-1 (x) = 1 . Quadratic function f(x) = 2[(x – m)2 + n]. Hence. (a) Without using differentiation method. The straight line y = 2x + k does not intersect the curve x2 + y2 – 6 =0 . state the equation of the axis of symmetry for the graph. where k is a constant. State curve (a) the value of p (b) the value of q (c ) the equation of the axis of symmetry SPM 2005 (paper 1) 1. where k is a constant. (b) the equation of the axis of symmetry (c) the coordinates of the maximum point [3 marks] SPM 2005 (paper 1) 1. The function f(x) = x2 – 4kx + 5k2 + 1 has a minimum value of r2 + 2k. Next. Diagram 2 shows the graph of a quadratic functions f(x) = 3(x + p)2 + 2. The y = f(x) has the minimum point (1. which has the roots p and q. Find the range of values of x for which x(x – 4) ≤ 12 [3 marks] 2. 2. find the values of k and r if the graph of the function is symmetrical about x = r2 . (a) By using the method of completing square. q). Diagram 2 shows the graph of a quadratic function f(x)=3(x + p)2 + 2. where p is a constant Find Diagram 2 (a) the value of k Diagram 2 The curve y = f(x) has the minimum point 10 . The straight line y = 5x – 1 does not intersect the curve y = 2x2 + x + p. Diagram 2 shows the graph of the function y = -(x – k)2 – 2. where p is a constant. show that r = k -1 [4 marks] (b) Hence. or otherwise. Find the range of values of p [3 marks] 2. Given that y = p + qx – x2 = k – (x + h)2 for all values of x (a) Find (i) h (ii) k in terms of p and/or q (b) the straight line y = 3 touches the curve y = p + qx – x2 (i) state p in terms of q (ii) if q = 2.x2 + 3 = k(x + 1). find the range of values of k if p and q has two distinct roots. where r and k are constants.1 [4 marks] SPM 2004 (paper 1) 1. sketch the graph for the curve SPM 2003 (paper 2) 1. where q is a constant. state the equation of the axis of symmetry for the curve. Find the range of the values of x for (2 x − 1)( x + 4) > 4 + x [2 marks] SPM 2007(paper 1) 1..p) and intersects the f(x)-axis at point A Dia gram 2 a) State the symmetry of the curve b) express f (x) in the form ( x + b) 2 + c . The quadratic function f ( x) = x 2 + 2 x − 4 can be expressed in the form f ( x) = ( x + m) 2 − n . [3 marks] 3.q). where q is a constant. Find the range of values of x for which 2 x 2 ≤ 1 + x [3 marks] 2. 11 coordinates of A [1m] b) By using the method of completing square. The curve has a maximum point at B(2. [4m] c) determine the range of values of x. Diagram 3 shows the graph of quadratic function y = f (x) . where p. where b and c are constants. n=…………. if f ( x ) ≥ −5 [2m] . Find the range of the value of x for ( x − 3) 2 < 5 − x . Find the value of m and of n [3 marks] Answer m=…………. Diagram 2 shows the curve of a quadratic function f ( x ) = − x 2 + kx − 5 . [3 m] SPM 2008 (paper 2) 1. The equation of the axis of symmetry is x = 3 State a) the range of values of p b) the value of q c) the value of r [3 m] 2. The straight line y = −4 is a tangent to the curve y = f (x) a) write the equation of the axis of SPM 2008 (paper 1) 1. find the value of k and of p. has a minimum value of -4. State the value of p the value of q c) the equation of the axis of symmetry [3 m] SPM 2006 1. The quadratic function f ( x) = p ( x + q ) 2 + r . q and r are constants. where m and n are constants..a) b) (1. 1m 1m 1m Diagram 2 SPM 1995 1. Given the total length of the aquarium is 440 cm and the total area of the glass used to make the aquarium is 6300 cm2. If perimeter of the net box is 48 cm and the total surface area is 135 cm3. If the area and the perimeter of the carpet are 3 8 m2 and 12 m. Find the value of x. find the measurements 4 of the room. Solve the simultaneous equation x2 – y + y2 = 2x + 2y = 10 SPM 1994 1. Calculate the possible values of v and w. Diagram 2 shows a rectangular room. Diagram 2 shows a rectangular pond JKMN and a quarter part of a circle KLM with centre M. x2 + 6xy + 6 = 0 2. where k and p are constants. Solve the following simultaneous equation and give your answer correct to two decimal places 2x + 3y + 1 = 0. A cuboids aquarium measured u cm × w cm × u cm has a rectangular base.the value of k and p CHAPTER 4: SIMULTENOUS EQUATIONS SPM 1993 1. Find the values of k and p x 2y 2. Solve the simultaneous equation 4x + y + 8 = x2 + x – y = 2 2. SPM 1997 1. Determine 1m SPM 1998 1. x + 6y = 3 3 y 2. 12 . Solve the simultaneous equation: x 2 + = 4 . Find the value of u and w SPM 1996 1. Given that (3k. If the area of the pond is 10 π m2 and the length JK is longer than the length of the curve KL by π m. -2p) is a solution for the simultaneous equation x – 2y = 4 and 2 3 + =1. shaded region is covered by perimeter of a rectangular carpet which is placed 1 m away from the walls of the room. The top part of it is uncovered whilst other parts are made of glass. 2k) is a solution for the equation x2 + py – 29 = 4 = px – xy . Given that (-1. Diagram 2 shows the net of an opened box with cuboids shape. He planted padi and yam on the areas as shown in the above diagram. SPM 2003 1. Without drawing the x graph. ABCD is a piece of paper in a rectangular shape. 2. Find the area of land planted with yam. Solve the simultaneous equation x 6y 2x + 3y = 9 and − = −1 y x SPM 2000 1. ABE is a semi-circle shape cut off from the paper. yam Pak Amin has a rectangular shapes of land. The yam is planted on a rectangular shape area. Its area is 28 cm2. Given the following equation: M = 2x − y N = 3x + 1 R = xy − 8 Find the values of x and y so that 2M = N = R 4. Solve the simultaneous equation y y 1 x 3 − + 3 = 0 and + − =0 3 x 2 2 2 SPM 2001 1. Solve the simultaneous equation 3x – 5 = 2y . y(x + y) = x(x + y) – 5 2. Given that x + y – 3 = 0 is a straight line cut the curve x2 + y2 – xy = 21 at two different point. Find the coordinates of the point 2. Given the curve y2 = 8(1 – x) and the y straight line = 4. calculate the coordinates of the intersection for the curve and the straight line.[use π = 22 ] 7 SPM 1999 1. the perimeter left is 26 cm. Diagram 2 shows. Given the area of the land planted with padi is 115 m2 and the perimeter of land planted with yam is 24 m. Solve the simultaneous equation 4x + y = −8 and x2 + x − y = 2 SPM 2004 13 . Find the integer values of x and y SPM 2002 1. 1. Solve the simultaneous equations 2 x 2 + y = 1 and 2 x 2 + y 2 + xy = 5 Give your answer correct to three decimal places [5 m] SPM 2007 1. Give your answers correct to three decimal places. Solve the simultaneous equations p − m = 2 and p2 + 2m = 8. Solve the following simultaneous equations: 2 x − y − 3 = 0 . Solve the following simultaneous equations : x − 3y + 4 = 0 x 2 + xy − 40 = 0 [5m] 14 . Solve the simultaneous equation 1 x + y = 1 and y2 − 10 = 2x 2 SPM 2006 1. 2 x 2 − 10 x + y + 9 = 0 [5 m] SPM 2008 1. SPM 2005 1. (a)Solve the following equations: (i) 4 log 2 x =5 (ii) 2 x . y = 4 when x = 2 and y = 8 when x = 5. state x in terms of y 2. 3 x = 5 x +1 (b) Given that log 5 3 = 0.2) x when the metal is heated for x seconds.4 (ii) log 7 75 SPM 1997 1. If 3 − log 10 x = 2log 10 y. Solve the following equations: 15 . (a) Given that m = 2 r and n = 2 t . without using a calculator scientific or four-figure table . to increase the temperature of the metal from 30 0 C to 1500 0 C SPM 1996 1. find the value of x and y which satisfies the equation of n (b) Given that 2 r = 3 s = 6 t . Hence or otherwise.683 and log 5 7 = 1. in second. Calculate (i) the temperature of the metal when heated for 10. (a) If h = log m 2 and k = log m 3. state in terms of h and /or k (i) log m 9 (ii) log 6 24 (b)Solve the following equations: 1 (i) 4 2x = 32 (ii) log x 16 − log x 2 = 3 SPM 1994 1. (ii) log 8 m − log 4 n CHAPTER 5: INDICES AND LOGARITHMS SPM 1993 1.(a) 81(27 2 x ) = 1 (b) 5 t = 26. Express t in terms of r and s ( c) Given that y = kx m where k and m are constants. calculate (i) log 5 1. (a) Express 2 n + 2 − 2 n + 10(2 n −1 ) in a simplify terms (c) Solve the equation 3 x + 2 − 5 = 0 2.3 2. Find the values of k and m SPM 1995 1. Show that log 3 xy = 2 log 9 x + 2 log 9 y. find the value 3 b) The temperature of a metal increased from 30 0 C to T 0 C according to the equation T = 30(1. state in terms of r and/or t mn 3 (i) log 2 32 . Solve the following equations: (a) log 3 x + log 9 3x = −1 1 (b) 8 x + 4 = x x +3 4 2 2.209.4 seconds (ii) time. (a) Given that log 8 n = 1 . Find (a) log 2 45 9 (b) log 4 5 2. show that x 2 + y 2 = 7xy (b) Without using scientific calculator or four-figure mathematical tables. 3.3772 (ii) solve the equation 3 × a n −1 = 3 SPM 2000 1.(a) Find the value of 3 log3 7 without using a scientific calculator or four figure table.log 3 xy = 10 and log 9 xy 3 = log 9 y 2 2. (a)Simplify log16 7 Without using scientific calculator or four-figure mathematical tables (b) Given that 3 lg xy 2 = 4 + 2lgy .7924. SPM 2001 1. Given that log SPM 1999 1. (a) Solve 3 log 2 x = 81 (b) If 3 2 x = 8(2 3 x ).585 and log 2 5 = 2.lgx with the condition x and y is a positive integer. (a) Given that 2 log 3 (x + y) = 2 + log 3 x + log 3 y. 8 Calculate after how many years will the car cost less than RM 20 000 for the first time SPM 1998 1. Without using scientific calculator or four-figure mathematical tables. Show that xy = 10 (c) The total savings of a cooperation after n years is given as 2000(1 + 0. (a) Given that x = log 2 3. (b) Solve the equation 5 log x 3 + 2 log x 2 . Without using scientific calculator or fourfigure mathematical tables (i) prove that log a 27a = 3. find the value of 4 x . Hence find the value of 4 y if y=1+x (b) Given that log a 3 = 0. solve the equation log 9 [log 3 (4x – 5)] = log 4 2 (c ) After n year a car was bought the n 7 price of the car is RM 60 000 .log x 324 = 4 and give your answer correct to four significant figures. prove that 9 x log a = log a 8 8 log 12 49 × log 64 12 2. Calculate the minimum number of years required for the savings to exceed RM 4 000. Given that log 2 k = p and log 3 k = r x 4 = u and log y 5 = y State log 4 x 3 y in terms of u and/or w 2. (a) Given that log a 3 = x and log a 5 = y. 45 Express log a 3 in terms of x and y a (b) Find the value of log 4 8 + log r r (c ) Two experiments have been conducted to get relationship between two variables x and y. The equation x 3(9 ) = 27 y and log 2 y = 2 + log 2 (x – 2) were obtain from the first and second experiment respectively 16 .322. Given that log 2 3 = 1.07) n . Solve the equation 16 2 x −3 = 8 4 x [3 m] log 4 x = log 2 3 .Find log k 18 in terms of p and r 2. [3 m] 17 . (a) Given that 2 log 4 x − 4 log16 y = 3 State x in terms of y (b) Solve the simultaneous equation 2 m−1 × 32 k + 2 = 16 and 5 −3m × 125 3− k = 1 where m and k are constants SPM 2003. 27 m in terms of p and r express log m 4 SPM 2006 2 x −3 = 1. Solve the equation log 3 4 x − log 3 ( 2 x − 1) = 1 [3 marks] 3. Solve the equation 2 x + 4 − 2 x +3 = 1 [3 marks] 2. Given that log 5 2 = m and log 5 7 = p . Solve the equation 8 1 4 x+2 [3 marks] 2. Solve the equation 4 = 7x [4 marks] SPM 2004. Given that log 2 xy = 2 + 3 log 2 x − log 2 y .Given that log m 2 = p and log m 3 = r .P1 1. 8b express log 4 in terms of x and y c [4 marks] 2. show that xy – 100y 2 = 9 (b) Solve the equation (i) 3 x + 2 = 24 + 3 x (ii) log 3 x =log 9 ( 5 x + 6 ) SPM 2002 1. Given that log 2 = x and log 2 c = y . Given that log 2 T − log 4 v = 3 .P1 1. Find λ in terms of k (b) Solve the equation log 2 ( 7t − 2 ) − log 2 2t = −1 2. If 5 2 λ −1 = 15. Given that value of x. (a) Given that log 5 3 = k. express T in terms of V [4 marks] 2 x −1 2. Solve the equation 2 + log 3 ( x + 1) = log 3 x [3 marks] SPM 2007 1. Solve the equation 32 4 x = 4 8 x +6 [3 marks] 2.9 in terms of m and p SPM 2005. find the 2. express y in terms of x [3 marks] 3.P1 1. (a) Given that log 10 x = 2 and log 10 y = -1. express log 5 4. Given that 9(3 n −1) = 27 n [3 marks] SPM 2008(paper 1) 1. (5. find (a) the value of p and q CHAPTER 6: COORDINATE GEOMETRY (b) area of ABCD SPM 1993 1. 4). q) respectively. a parallelogram KLMN. Find the coordinates of P SPM 1994 1. point K(1. Find the equation of the straight line KM (c ) If the straight line KM intersects again 18 SPM 1993 . The above diagram show. (4. Point P moves such that PK:PL = 1:2 (a) Show that the equation of locus P is x 2 + y 2 − 4x = 0 (b) Show that the point M(2. Solutions to this question by scale drawing will not be accepted. 2) is on the locus P. C and D have a coordinates (2. Find the coordinates of N (d) Calculate the area of triangle OMN SPM 1994 1. Find (a) the gradient of PQ (b) the equation of straight line QR ( c) the coordinates of R SPM 1993 2. 0) and point L(-2. Solutions to this question by scale drawing will not be accepted Point P and point Q have a coordinate of (4. Given that ABCD is a parallelogram.1) and (2. -1) and (p. The straight line QR is perpendicular to PQ cutting x-axis at point R. Hence write down the equation of KL in the form of intercepts (b) ML is extended to point P so that L divides the line MP in the ratio 2 : 3. 3). From the above diagram.locus P at N. Points A. (a) Find the value of T. 0) are the two fixed points. 2). B. In the diagram. Solutions to this question by scale drawing will not be accepted. The straight line y = 4 x − 6 cutting the curve y = x 2 − x − 2 at point P and point Q (a) calculate (i) the coordinates of point P and point Q (ii) the coordinates of midpoint of PQ (iii) area of triangle OPQ where Q is a origin (b) Given that the point R(3. P. the straight line y = 2 x + 3 is the perpendicular bisector of straight line which relates point P(5. k) lies on straight line PQ (i) the ratio PR : RQ (ii) the value of k 2. 6) in the ratio 2SE = SF Find (i) the equation of the locus of S (ii) the coordinates of point when locus S intersect y-axis SPM 1995 1. SPM 1996 1.(b) the equation of the straight line passing through point L and perpendicular with straight line LMN 2. find the distance of PQ Graph on the above show that the straight line LMN Find (a) the value of r 19 . t) (a) Find the midpoint of PQ in terms of n and t (b) Write two equations which relates t and n ( c) Hence. 0) and F(2. Q and R are three points are on a line 2 y − x = 4 where PQ : QR = 1:4 Find (i) the coordinates of point P (ii) the equation of straight line passing through the point Q and perpendicular with PR (iii) the coordinates of point R (b) A point S moves such that its distance from two fixed points E(-1. 7) and point Q(n. (a)The above diagram. In the diagram. Find the coordinates of point C [3m] 20 . calculate the coordinates of point T (c) A point move such that its distance 1 from point S is of its distance from 2 point T. (i) Find the equation of the locus of the point (ii) Hence. The diagram shows the straight line graphs of PQS and QRT on the Cartesian plane. AB and BC are two straight lines that perpendicular to each other at point B. given that the area of rectangular TUVW is 58 units2 (d) Fine the equation of the straight line TU in the intercept form SPM 1997 2.2. Point P and point S lie on the x-axis and y-axis respectively. Q is the midpoint of PS (a) Find (i) the coordinates of point Q (ii) the area of quadrilateral OPQR [4m] (b)Given QR:RT = 1:3. Given the equation of the straight line AB is 3y + 2x − 9 = 0 (a) Find the equation of BC [3m] (b) If CB is produced. The diagram shows the vertices of a rectangle TUVW on the Cartesian plane (a) Find the equation that relates p and q by using the gradient of VW (b) show that the area of ∆TVW can 5 be expressed as p − q + 10 2 ( c) Hence. Point A and point B lie on x-axis and y-axis respectively. calculate the coordinates of point V. it will intersect the xaxis at point R where RB = BC. determine whether the locus intersects the x-axis or not SPM 1998 1. Given point A(−2. intersects the curve at point C. 9).1. 2 KL and LJ respectively. Q(5. 0). Point P move such that distance from point Q(0. (a) Find (i) the equation of the straight line JK (ii) the equation of the perpendicular bisector of straight line LJ [5m] 21 2. Point P divides the line segment AB in the ratio 2 : 3. where JPQR forms a parallelogram. Find (a) the coordinates of point P (b) the equation of straight line that is perpendicular to AB and passes through P. ACD and BCE are straight lines. Straight line BC. Find (a) the equation of the straight line AB [3m] (b) the equation of the straight line BC [3m] . Find the coordinates of point S [2m] (c ) Calculate the area of ∆PQR and hence.8) . Point S move so that its distance from point T(3. find the area of ∆JKL [3m] SPM 1999 1. In the diagram. Locus of the point P and S intersects at two points.−4) and point B (4. 1) is the same as its distance from point R(3.3 are midpoints of straight lines JK. produced=diperpanjangkan 3. Given C is the midpoint of AD. P(2. 2) is 3 units. The diagram shows the curve y 2 = 16 − 8 x that intersects the xaxis at point B and the y-axis at point A and D. In the diagram. and BC : CE = 1:4 Find (a) the coordinates of point C (b) the coordinates of point E (c ) the coordinates of the point of intersection between lines AB and ED produced [3m] 2. which is perpendicular to the straight line AB. (a) Find the equation of the locus of P (b) Show that the equation of the locus of point S is x 2 + y 2 − 6 x − 4 y + 4 = 0 ( c) Calculate the coordinates of the point of intersection of the two locus (d) Prove that the midpoint of the straight line QT is not lie at locus of point S (b) Straight line KJ is produced until it intersects with the perpendicular bisector of straight line LJ at point S. 7) and R 1 4 . (c) the coordinates of point C [4m] SPM 2000 2. The diagram shows a trapezium ABCD. Given the equation of AB is 3 y − 2 x − 1 = 0 Find (a) the value of k [3m] (b) the equation of AD and hence, find the coordinates of point A [5m] (c) the locus of point P such that triangle BPD is always perpendicular at P [2m] SPM 2001 1. Given the points P(8, 0) and Q(0, -6). The perpendicular bisector of PQ intersects the axes at A and B. Find (a) the equation of AB [3m] ∆AOB , where O is the (b) the area of origin. [2m] 2. Solutions to this question by scale drawing will not be accepted. Straight line x − 2 y = 6 intersects the x-axis and y-axis at point A and point B 1. The diagram shows a triangle ABC where A is on the y-axis. The equations of the straight line ADC and BD are y − 3 x + 1 = 0 and 3 y + x − 7 = 0 respectively. Find (a) the coordinates of point D (b) the ratio AD : DC respectively. Fixed point C is such that the gradient of line BC is 1 and straight line AC is perpendicular to the straight line AB. Find (a) the coordinates of points A and B [1m] (b) the equation of the straight lines AC and BC [5m] (c) the coordinates of point C [2m] (d) the area of triangle ABC [2m] 22 1. The diagram shows a triangle ABC with an area 18 units2 . the equation of the straight line CB is y − x + 1 = 0. Point D lies on the x-axis and divides the straight line CB in the ratio m : n. Find (a) the coordinates of point B (b) m : n 3. In the diagram, the equation of BDC is y = −6 . A point P moves such that its 1 distance from A is always the distance 2 of A from the straight line BC. Find (a) the equation of the locus of P (b) the x-coordinates of the point of intersection of the locus and the x-axis [5m] SPM 2002 2. A(1, 3), B and C are three points on the straight line y = 2 x + 1 . This straight line is tangent to curve x 2 + 5 y + 2 p = 0 at point B. Given B divides the straight lines AC in the ratio 1 : 2. Find (a) the value of p [3m] (b) the coordinates of points B and C [4m] (c) the equation of the straight line that passes through point B and is perpendicular to the straight lineAC [3m] 3. Given A(-1, -2) and B(2, 1) are two fixed points. Point P moves such that the ratio of AP and PB is 1 : 2. (a) Show that the equation of the locus of point P is x 2 + y 2 + 4 x + 6 y + 5 = 0 [2m] (b) Show that point C(0, -5) lies on the locus of point P [2m] (c) Find the equation of the straight line AC [3m] (d) Given the straight line AC intersects the locus of point P at point D. Find the coordinates of point D [3m] 23 SPM 2003(P1) 1. The points A(2h, h) , B ( p, t ) and C ( 2 p,3t ) are on a straight line. B divides AC internally in the ratio 2 : 3 Express p in terms of t [3m] 2. The equations of two straight lines are y x + = 1 and 5 y = 3x + 24 . 5 3 Determine whether the lines are perpendicular to each other [3m] 3. x and y are related by the equation y = px 2 + qx , where p and q are constants. A straight line is obtained y by plotting against x, as shown in x Diagram 1. y x (b) The tangent to the circle at point Q intersects the y-axis at point T. Find the area of triangle OQT [4m] SPM 2004(P1) 1. Diagram 3 shows a straight line graph of y against x x y x Diagram 1 Calculate the values of p and q [4m] P2(section B) 1. solutions to this question by scale drawing will not accepted. A point P moves along the arc of a circle with centre A(2, 3). The arc passes through Q(-2, 0) and R(5, k). (a) Find (i) the equation of the locus of the point P (ii) the values of k [6m] Given that y = 6 x − x 2 , calculate the value of k and of h [3m] 2. Diagram 4 shows a straight line PQ with x y the equation + = 1 . The point P lies 2 3 on the x-axis and the point Q lies on the yaxis 24 6). find the y-intercepts of CD [3m] SPM 2005(P1) 1. JK and RT. find the coordinates of D [2m] (c) Given that CD is perpendicular to AB. Diagram 5 shows the straight line AB which is perpendicular to the straight line CB at the point B (a) Find (i) 25 . The following information refers to the equations of two straight lines. Solutions to this question by scale drawing will not accepted. The point P moves such that PA : PB = 2 : 3.Find the equation of the straight line perpendicular to PQ and passing through the point Q [3m] 3. Digram 1 shows a straight line CD which meets a straight line AB at the point D . Find the equation of the locus of P [3m] P2(section A) 4. The point C lies on the y-axis Express p in terms of k [2m] P2(section B) 2. (a) write down the equation of AB in the form of intercepts [1m] (b) Given that 2AD = DB. JK RT : : y = px + k y = (k − 2) x + p where p and k are constant the equation of the straight line AB (ii) the coordinates of B [5m] (b) The straight line AB is extended to a point D such that AB : BD = 2 : 3 Find the coordinates of D [2m] (c) A point P moves such that its distance from point A is always 5 units. 3) and the point B is (4. which are perpendicular to each other. The point A is (-1. Find the equation of the locus of P [3m] SPM 2006(P1) 1. Point C lies on the straight line AB SPM 2007 Section A (paper 2) 1. the straight line AB has an equation y + 2 x + 8 = 0 . AB intersects the x-axis at point A and intersects the y-axis at point B (a) (b) (c) (i) of P (ii) Calculate the area. Solutions to this question by scale drawing will not be accepted Diagram 3 shows the triangle AOB where O is the origin. solutions by scale drawing will not be accepted In diagram 1. find the coordinates of C A point P moves such that its distance from point A is always twice its distance from point B Find the equation of the locus y + 2x + 8 = 0 Diagram 1 Point P lies on AB such that AP:PB = 1:3 Find (a) the coordinates of P [3 m] (b) the equations of the straight line that passes through P and perpendicular to AB [3 m] SPM 2007 (paper 1) 26 Hence. determine whether or not this locus intercepts the y-axis .The equation of the straight line CB is y = 2x − 1 Find the coordinates of B [3 marks] P2(section B) 1. of triangle AOB Given that AC:CB = 3:2. in unit2. intercept of 2 and is parallel to the straight line y + kx = 0 . (2.Determine the value of h and of k [3 marks] 2. The points (0. [3 marks] SPM 2008(paper 1) 1. Diagram 13 shows a straight line passing through S(3.y) moves such that PS = PT.6) and C(p.4) 1.t) and (-2. Find the equation of the locus of W [3m] (b) It is given that point P and point Q lie on the locus of W. Point S lies on the line PQ. B(4.3). (a) (b) Diagram 13 Write down the equation of the straight line ST in the form x y + =1 a b A point P(x. Given that the area 27 units. The vertices of a triangle are A(5. find the values of p. Find the equation of the locus of P [4 m] (a) A point W moves such that its distance from point S is always 2 1 2 2.0) and T(0. Diagram shows a triangle OPQ.-2). The straight line of the triangle is 4 unit2.x y + = 1 has a 6 h y. Calculate (i) the value of k. find the values of t.2). [3 m] SPM 2008 Section B (paper 2) 1.-1) are the vertices of a triangle. Given that the area of the triangle is 30 unit 2 . . 6. The mean for the numbers 6. x.(ii) the coordinates of Q [5m] (c) Hence. find the area. find the mode for the numbers when (i) x = y (ii) x ≠ y 1 37 . 2. 2. y is 5 (a) show that x + y = 12 (b) hence. draw a histogram 28 . in unit2. The below table shows the marks obtained by a group of students in a monthly test . Marks Number students 1-20 5 21-40 8 41-60 12 61-80 11 81-100 4 (a) On a graph paper. 10. of triangle OPQ [2m] CHAPTER 7: STATISTICS SPM 1993 1. find (c) if standard deviation is 2 the values of x 2. 2. x + 4. x + 7 and x − 3 has a mean of 7.and use it to estimate the modal mark (b) By calculating the cumulative frequency. The sum of these numbers is 150 whereas the sum of the squares of these numbers is 2890. find the median mark. Numbers of classes Numbers of pupils 6 35 5 36 4 30 . Find 29 The table shows the age distribution of 200 villagers. Find the standard deviation of these number (b) Find a possible set of five integers where its mode is 3. 2 x − 1. 2 x + 5. 2. without drawing an ogive (c) Calculate the mean mark SPM 1994 1. The list of numbers x − 2. of the number of pupils in each class (b) Age Numbers of villagers 1-20 50 21-40 79 41-60 47 61-80 14 81-100 10 Find (a) the maximum value of x if modal mark is 2 (b) the minimum value of x if mean mark more than 3 (c) the range of value of x if median mark is 2 2. The below table shows the marks obtained by a group of students in a monthly test . 3.Find (a) the value of x (b) the variance [6m] 2. (a) Find the mean and variance of the numbers in set A (b) If another number is added to the 10 numbers in set A. 6. Find the standard deviation of these numbers. calculate (i) the median (ii) the third quartile of their ages SPM 1996 1. Set A is a set that consist of 10 numbers. Marks Number of students 1 4 2 6 3 2 4 x 5 1 (i) the mean (ii) the standard deviation. (a) The table shows the results of a survey of the number of pupils in several classes in a school. the mean does not change. median is 4 and mean is 5. Without drawing a graph. (a) Given a list of numbers 3. 8. [6m] SPM 1995 1. If each element of the data is reduced by 2. is m. 8 Find (a) the values of m and k [4m] (b) the variance of the new data [2m] 2. 9.Length (mm) 20-29 30-39 40-49 50-59 60-69 70-79 80-89 Numbers of fish 2 3 7 12 14 9 3 (b) use a graph paper to answer this question The data in the table shows the monthly salary of 100 workers in a company. The table shows a set of numbers which has been arranged in an ascending order where m is a positive integer Set numbers Frequency 1 1 m-1 3 5 1 m+3 2 8 2 10 1 (ii) Based on the data. 8. (a) The following data shows the number of pins knocked down by two players in a preliminary round of bowling competition. 9. find the possible values of mode SPM 1998 1. find the percentage of the numbers of fish which has a length more than 55 mm SPM 1997 1. Player A: 8. draw an ogive to show distribution of the workers’ monthly salary From your graph. 7. 8. 8. 8. k. the new median 5m is . 9 Using the mean and the standard deviation. (i) The table shows the length of numbers of 50 fish (in mm) (a) calculate the mean length (in mm) of the fish (b) draw an ogive to show the distribution of the length of the fish (c) from your graph. 9. estimate the number of workers who earn more than RM 3 200 Numbers of workers 10 12 16 22 20 12 6 2 Monthly Salary (RM) 500-1 000 1 001-1 500 1 501-2 000 2 001-2 500 2 501-3 000 3 001-3 500 3 501-4 000 4 001-4 500 (a) express median for the set number in terms of m (b) Find the possible values f m (c) By using the values of m from (b). The mean of the data 2. 8. 3k. determine the better player to represent the state based on their consistency [3m] 30 . 12 and 18 which has been arranged in an ascending order. 6 Player B: 7. for a certain game with a mean of 8 and standard deviation of 3 (a) calculate Σx and Σx 2 (b) A number of scores totaling 180 with a mean of 6 and the sum of the squares of these scores of 1 200. the mean becomes 10. x. y has a mean of 5 and a standard deviation of 6. [4m] Marks Number of students <10 2 <20 8 <30 21 <40 42 <50 68 <60 87 <70 98 <80 100 2. Calculate the mean [4m] the median [3m] the standard deviation [3m] of the distribution SPM 2001 1. 6. 3. 6. The table shows the results 100 students in a test (a) Based on the table above. (a) Given that four positive integers have a mean of 9. The table shows the frequency distribution of the marks obtained by 100 pupils Marks 6-10 11-15 16-20 21-25 Marks26-30 0-9 31-35 Frequency 36-40 31 Number of pupils 12 20 27 16 13 10 2 . x. 9. The table shows the distribution of marks in a physics test taken by 120 pupils. 7 [3m] 2. 7. 4. Find the possible values of x and y 2. The frequency distribution of marks for 30 pupils who took a additional mathematics test is shown in the table Marks 20-39 49-59 60-79 80-99 Frequency 6 5 14 5 (b) Without drawing an ogive. copy complete the table below [2m] [2m] (b) Find the standard deviation of the set of numbers below: 5. Calculate the mean and variance of the remaining scores in set X. calculate the median mark [3m] (c) Find the mean mark [3m] SPM 2000 1. estimate the interquartile range of this distribution.2. is taken out from set X. [7m] SPM 1999 1. Set X consist of 50 scores. Find the value of y Marks Number of pupils 20-29 2 30-39 40-49 50-59 60-69 70-79 14 35 50 17 2 (a) By using a graph paper. The set of numbers integer positive 2. draw a histogram and estimate the modal mark [4m] (b) Without drawing an ogive.When a number y is added to these four integers. 6. Scores Number of pupils Each mark is multiplied by 2 and then is added to it.p2 section A 1. find (a) The median (b) The percentage of excellent pupils if the score for the excellent category is 31. A set of data consist of 10 numbers. (i) the mean (ii) the variance [4m] SPM 2004. paper 1 1.5 Find (i) the sum of the marks. The sum of the squares of the numbers is 100 and the standard deviation is 3k Express m in terms of k [3] paper 2.5 SPM 2003. x5 . (a) Find the mean and variance of the 10 numbers [3] (b) Another number is added to the set of data and the mean is increased by 1 Find the value of this number (ii) the standard deviation of the set 11 numbers [4 marks] 1 1 3 1 6 2 x 3 y 1 14 1 [5m] Find the values of x and y 2. Diagram 2 is a histogram which represents the distribution of the marks 32 . for the new set of marks. [7m] SPM 2002 1. The mean of four numbers is m . The scores are arranged in an ascending order. The table shows the distribution of scores obtained by 9 pupils in a competition. find Scores Number of pupils 5 ≤ 0 ≤1 5 ≤1 0 ≤2 5 ≤2 0 ≤3 ≤ 35 By drawing an ogive. ∑x (ii) the sum of the squares of the marks. The number of pupils is 40. The table shows the scores obtained by a number of pupils in a quiz. the sum of the number is 150 and the sum of the squares of the data is 2 472. x 2 . x 4 . Given the mean score is 8 and the third quartile is 11. Find. By drawing an ogive.(i) Calculate the variance [3m] (b) (ii) Construct a cumulative frequency table and draw an ogive to show the distribution of their marks. From the ogive. find the percentage of pupils who scored between 6 to 24.section A 1. A set of examination marks x1 .p2 section A 1. ∑ x 2 [3m] (a) SPM 2005. x6 has a mean of 5 and a standard deviation of 1. x3 . A set of positive integers consists of 2. Calculate the value of k [3 marks] (b) Use the graph paper to answer this question 33 SPM 2007 Paper 2 1. Table 1 shows the cumulative frequency distribution for the scores of 32 students in a competition . Find the mode score [4 marks] (c) What is the mode score if the score of each pupil is increased by 5? [1 mark] SPM 2006 paper 1 1. calculate the median mark [3m] (b) Calculate the standard deviation of the distribution [4m] Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis. (a) Without using an ogive. The variance for this set of integers is 14. (a) It is given that the median score of the distribution is 42. Table 1 shows the frequency distribution of the scores of a group of pupils in a game.obtained by 40 pupils in a test.section A 1. 5 and m. Find the value of m [3 marks] paper 2. draw a histogram to represent the frequency distribution of the Score Number of pupils 10-19 1 20-29 2 30-39 8 40-49 12 50-59 k 60-69 1 scores. the new mean is 8.set. The sum of the numbers is 60 and the sum of the squares of the numbers is 800 Find for the five numbers (a) the mean (b) the standard deviation [3 m] SPM 2008(Paper 1) 1.5 [3m] SPM 2008(Paper 2) 1. <40 42 <50 68 Table 1 Score Number of students <10 2 <20 8 <30 21 (a) Based on table 1. state the modal class Marks 10-19 20-29 30-39 40-49 50-59 Number of candidates 4 x y 10 8 [6m] Table 2 [1 m] (b) Without drawing an ogive.5. A set of seven numbers has a mean of 9 (a) Find ∑ x (b) When a number k is added to this SPM 1993 34 . Hence. find the interquatile range of the distribution [5 m] SPM 2007 Paper 1 1. copy and complete Table 2 Marks Freque ncy 0-9 10-19 20-29 30-39 40-49 Given that the median mark is 35. Table 5 shows the marks obtained by 40 candidates in a test. find the value of x and of y. A set of data consists of five numbers. Point P move such that PB = BC = BA. The diagram shows two arcs. 2. Given the ratio OS:SR = 3:1. Show that ∠PMQ = 120 0 b. show that the area of the shaded region is 25(π + 1) unit 2 35 . PS and QR. the area of the shaded region [6m] SPM 1995 The diagram shows a semicircle with centre O and diameter AOC. in terms of π and r.CHAPTER 8: CIRCULAR MEASURE 1. Find the value of the angle θ (in degrees and minutes) so that the length of arc of the circle AB same with the total of diameter AOC and length of arc of the circle BC The diagram shows a semicircle ABCD with centre D. Find. In the diagram. Find (a) the angle θ in radian (b) the area of the shaded region PQRS [6m] SPM 1994 1. of two circles with centre O and with radii OS and OR respectively. Locus for the point P is a circle with centre B. (a) Find the distance of BC (b) Show that the equation of locus P is x 2 + y 2 = 4 x + 6 y + 37 (c) (i) Find the area of major sector BAPC in terms of π (ii) Hence. M and N are the centers of two congruent circles with radius r cm respectively. 36 radians into degrees [2m] (a) The diagram shows a piece of cake with a uniform cross-section in the shape of a sector OPQ of a circle with centre O and radius 20 cm. Given the length of arc PQ is 20 cm. OR = 4 cm. Given ∠OPQ = θ rad.(c) the area of the shaded region SPM 1997 (a) Convert 2. The diagram shows a piece of wire in the shape of a sector OPQ of a circle with centre O . and the length of arc QT = 4. The diagram shows. AOB is a semicircle with centre D and AEB is a length of arc of the sector with centre C. The equation of AB x y + =1 is 12 6 Calculate (a) the area of ∆ABC (b) ∠ACB in radians The diagram shows two sectors OPQ and ORS of two concentric circle with centre O. The length of arc PQ is 15 cm and the thickness of the cake is 8 cm. find The angle θ in radians (b) the area of sector OPQ [5m] SPM 1996 1. Find ∠QST in rad (a) 36 . The diagram show semicircle PQR with centre O and sector QST of a circle with centre S. and the length of radius OS =6 Find (i) the value of θ (ii) the perimeter of the shaded region [4m] 2. The length of the wire is 100 cm. Find (a) the angle of this sector in radians (b) the total surface area of the cake [5m] 2. Given ST = 5. the length of arc PQ is twice the length of radius OQ.5 cm. (i) (ii) (b) 64020' into radians 4. Given that the length of arc DCF is 18. The diagram shows a sector MJKL with centre M and two sectors PJM and QML. The diagram shows two sectors OPQR and OST of two concentric circle with centre O having the same area. Find (a) the radius of sector MJKL [2m] The diagram shows a traditional Malay kite. TQS is an arc of circle with centre R and radius 10 cm. (b) the perimeter of the shaded region [2m] (c) the area of sector PJM [2m] (d) the area of the shaded region [4m] SPM 1999 1. find (a) the length of OQ [3m] (b) the area of region swept by the pendulum [2m] 2. The diagram shows the position of a simple pendulum that swings from P to Q.4 cm. Given that APB is an arc of a circle with centre O and radius 25 cm. of two circles with centre P and Q respectively. wau bulan. ANBQ is a semicircle with centre N and diameter 30 cm. OR = 8 cm. Calculate (a) ∠AOB 37 . and the length of arc PQ same as that arc QR Find (a) the length of PS (b) the length of arc ST 2.(b) the area of the shaded region [4m] SPM 1998 1.6 radians. ∠POQ = 0. Given OPS and OQT are straight lines.6 rad.75 cm. Given the angle of major JML is 3. If the angle POQ is 80 and the length of arc PQ is 14. that has an axis of symmetry OR. Given the length of arc PQR is 7. KOL is an arc of a circle with centre T.68 cm. with centre O and radius 6 cm. The diagram shows two sectors OAP and OBQ. OPQR of a circle with centre O and radius 5 cm. Given AB is parallel to KL. The diagram shows a sector.(b) the area of segment AGBH ( c) the area of the shaded region (Use π = 3. of two concentric circle with centre O. and the ratio of the length of arc AP to the length of arc BQ is 2:3 Calculate ∠BOQ in degrees [5m] 2.142 ) SPM 2000 1.5 rad. OB = 3 AO . AB = 6 cm and ∠KOL = 1200 (a) Find ∠AOB [1m] (b) Calculate the area of segment ABT [4m] (c) Show that the perimeter of the shaded region is 6 3 + 6 + 2π [5m] 38 . Calculate the radius of semicircle DAECF [1m] the angle θ in radians [3m] the area of sector QAEC [2m] the area of the shaded region [4m] The diagram shows a circle. KATBL. Given ∠AOP = 0. find (a) ∠POR in radians [2m] (b) the area of the shaded region [4m] 2. (a) (b) (c) (d) The diagram shows semicircle DAECF with centre Q and rhombus QAPC. SPM 2001 1. Diagram 1 shows a sector ROS with centre O The diagram shows two sectors OAB and OCD of two concentric circles with centre O . Find the value of θ in rad [3m] paper 2(section A) 1. Find (a) the value of k [3m] (b) the difference between the areas of sector OAB and OCD [2m] 2. OD = k cm and perimeter of the figure is 35 cm.24 cm and the perimeter of the sector ROS is 25 cm. Diagram 1 The length of the arc RS is 7. [4m] 39 . Diagram 1 shows the sector POQ. Given OB = (k + 2) cm. centre O with radius 10 cm The point R on OP is such that OR : OP = 3 : 5 In the diagram.SPM 2002 1. [3m] (b) the area of the shaded region . Given O is the midpoint of AC. ABCD is a rectangle and OAED is a sector of a circle with centre O and radius 6 cm. SPM 2003 paper 1 1. where AOD and BOC are straight lines. in cm2. in rad.Calculate (a) ∠AOD in radians [2m] (b) the perimeter of the shaded region [4m] ( c) the area of the shaded region [4m] Diagram 1 Calculate (a) the value of θ . of the arc JKL [4m] the area. Using π = 3.142) [3m] paper 2(section B) 1. The point A lies on OP. the point B lies on OQ and AB is perpendicular to OQ. find the length. in cm2. in terms of π [2m] (a) ∠POQ = π radian 6 40 . JQK is a tangent to the circle at Q. centre O Calculate the angle α .142. in cm. in cm. The straight lines. in cm. intersect the circle at P and R respectively. (Give your answer correct to four significant figures) (b) the length. Diagram 1 shows a sector POQ of a circle. of the radius of the circle [3m] paper2 (section B) 1. centre O and radius 5 cm.51 cm. JLK is an arc of a circle. The length of OA = 8 cm and 45. (use π = 3. Diagram 1 shows a circle with centre O SPM 2005 paper 1 1. OPQR is a rhombus. in radians. of the radius. centre O. of the shaded region [4m] SPM 2004 paper 1 1. find (a) the value of θ . Diagram 1 shows a circle with centre O Given that the length of the major arc AB is The length of the minor arc AB is 16 cm and the angle of the major sector AOB is 2900 . Diagram 4 shows a circle PQRT.the length. JO and KO. 142 ) Calculate (a) the length. in cm. PCQ is a semicircle with centre O and has a radius of 8 m. of the shaded region SPM 2006 paper 1 1. in cm. ∠XAY = 1.956 radians [use π = 3. (c) the area. of the flower bed 41 . in cm2.1 radians and the lengths of arc AB = 7 cm. in m. Diagram 7 Given that OB = 10 cm.It is given that OA : OP = 4 : 7 (Use π = 3. The shaded region is a flower bed and has to be fenced. in m . in m2 of the lawn [2m] (b) the length. RAQ is sector of a circle with centre A and has a radius of 14 m. of the shaded region Sector COQ is a lawn. Diagram 4 shows the plan of a garden. calculate (c) the value of θ in radian (d) the area in cm2. AY = 4 cm. of the fence required for fencing the flower bed [4m] 2 (c) the area. of the shaded region.142] Calculate (a) the area. It is given that AC = 8 cm and ∠COQ = 1. of AP (b) the perimeter. Diagram 7 shows sector OAB with centre O and sector AXY with centre a paper 2 1. of the arc AB [5 m] 2 (b) the area in cm .142] Find (a) ∠ QOR. of the arc BC the area.142] Calculate (a) the length. centre O and radius 10 cm inscribed in a sector APB of a circle. in cm. of shaded region [5 m] SPM 2008 paper 1 1. Given that P. Diagram 4 shows a circle. are tangents to the circle at point Q and point R. [Use π = 3. Diagram 18 shows a circle with centre O and radius 10 cm. in radians (b) the area. in cm 2 . respectively. Q and R are points such that OP = PQ and ∠ OPR = 900. in cm.[4m] AP and PB. Diagram 4 shows a sector BOC of a circle with centre O (a) (b) It is given that AD = 8 cm and BA =AO = OD = DC = 5 cm Find the length. The straight lines. 1. of the shaded region [4 m] SPM 2007 paper 2 1. in cm2 of the coloured Region 42 . centre P. SPM 2007 Paper 1 [use π = 3. (a) show that θ = 1. in cm of the minor arc QR [3m] (c) calculate the area. The smaller circle has centre Y and radius 8 cm. in cm2. [5m] SPM 2008 paper 2 1. Diagram shows two circles. The straight line PQ is a common tangent to the circle at point P and point Q. [use π = 3.37 (to two decimal places) [2m] 43 . The larger circle has centre X and radius 12 cm. of the colored region.142] Given that ∠PXR = θ radian.[4m] (b) calculate the length. The circle touch at point R. Given p = 2 x − 3 and y = − 3 .98) 4 dy in terms of x dx the small change in y.− is 2. find f '(x) 4x − 3 dy using dx 2. Find the coordinates at the curve y = (2 x − 5) 2 where the gradient of the normal for the curve 1 is 4 SPM 1996 CHAPTER 9: DIFFERENTATION 1. Hence. The gradient of the curve y = hx + k at x2 SPM 1993 1. Given y = x(3 − x) . Given f ( x) = find f ' (x) x −1 2. express d2y dy y 2 + x + 12 in terms of x. Differentiate y = 4 − 3 using the x SPM 1995 1 − 2x3 1. find the first principle (c) Find d 1 dx 2 x + 1 7 the point − 1.98 SPM 1997 lim n 2 − 4 (a) Find the value of n → 2 n − 2 (b) Given f ( x) = (2 x − 3) 5 find f ′′ (x) 2. find the value of x that satisfy the first principle 44 . Differentiate x 4 (1 + 3 x) 7 with respect to x. Find the values of h 2 and k 3.equation d2y dy y 2 + x + 12 dx dx 3. Given that f ( x) = SPM 1994 1. when x decreases from 2 to 1. dx dx Hence. (a) Given that y = 3 x 2 + 5 . 1 − 2x 2 . find dx x 16 estimate the value of (1. Given y = 16 dy if x = 2 . Find p2 (a) the approximate change in x if thE rate of change in p is 3 units per second 2. 4 . The diagram shows a wooden block consisting of a cone on top of a cylinder with radius of x cm. Given the slant height of the cone is 2x cm.2 cm s-1. and the volume of the cylinder is 24 π cm 3 a) Prove that the total surface area of the block. find f ' ( x) The diagram shows a container in the shape of a pyramid. Find the (a) The diagram shows a rectangle JKLM inscribed in a circle. Water is poured into the container so that its surface area is 4p2 cm2 and its height from the vertex of the pyramid is h cm. A cm2. A cm 2 . Given that f ( x) = 4 x(2 x − 1) 5 . The square base of the pyramid has an area of 36 cm2 and the height of the pyramid is 4 cm. calculate the rate of change in the volume of the space that is not filled with water if h = 2 cm.minimum 3. [5m] (b) 2. Given JK = x cm and KL = 6 cm show that the area of the shaded region. (a) [5m] SPM 1998 1. is given by πx 2 A= − 6 x + 9π 4 (b) Calculate the value of x so that the area of the shaded region is a 45 . (i) Show that the volume of the container that is filled with water is 3 V = (64 − h 3 ) 4 (ii) If the rate of change in the height of water is 0. is given by 2 16 A = 3π x + [3m] x b)Calculate the minimum surface area of the block [3m] c) Given the surface area of the block changes at a rate of 42 π cm 2 s −1 . Given y = t − 2t 2 and x = 4t + 1 dy (a)Find .142 ) (i) Calculate the rate of change in the radius of the circle (ii) Hence. find r in terms of k 4.0). its length increases at a rate of 0. calculate the radius of the circle after 4 second SPM 2000 1. find 1 − 3x 2 ) 5 [4m] [2m] [2m] 2. find the corresponding small increase in t. Find (i) the coordinates of point A and the value of k (ii) the equation of the tangent at The diagram shows a box with a uniform cross section ABCDE . find the approximate increase in the surface area of the block [2m] SPM 1999 1. Differentiate the following expressions with respect to x (a) 1 + 3 x 4 (b) 2x + 5 x4 + 3 (x f ( x) = −2 . [2m] 3 (a) Given y = 3 x 2 − 4 x + 6 . Find the equation of the tangent to the curve y = 2 x 2 + r at the point x = k . [2m] d) Given the radius of the cylinder increases from 4 cm to 4.01.003 cm. x increases by 2%. in terms of x dx (b) If x increases from 3 to 3. Given AB = ED = (30-6x) cm. When x = 5 . Given f ' (0) 3 (b) A piece of wire 60 cm long is bent to form a circle. If the tangent passes through the point (1.of change of its radius when its radius is 4 cm. BC = 3x cm.(a) The straight line 4 y + x = k is the normal to the curve y = ( 2 x − 1) 2 − 3 at point A. CD = 4x and AF = 2 cm (i) Show that the volume of the box V cm 3 . is given by equation V = 300 x − 48 x 2 (ii) Calculate (b) the value of x that makes V a maximum (c) the maximum value of V point A 46 .1 cm s −1 (use π = 3. Find the corresponding rate of change of y. when the wire is heated. Show that the area of the shaded region A = 4+π 2 20 y − y 4 b. find the length and width of the rectangle that make the area of the shaded region a maximum Given y = 2 x 3 − 5 x 2 + 7 . 2. 3). find an expressions for in dx terms of x and hence.4. when y decreases from 3 to 2. the values of h and k b. Hence.(b) The diagram shows a toy in the shape of a semicircle with centre O. dx find (i) the small change in x. Given that AC = x cm and the area of triangle ABC is A dL cm. find dy the value of at the point (2.142 . x-coordinate of the turning point of the graph of the function 3.98 (ii) the rate of change in y. Given p = (1 + t ) + t 2 dp Find and hence find the values of t dt dp =7 where dt 2. Find a. Given the perimeter of ABCD is 40 cm a. Given that graph of function k f ( x) = hx 3 + 2 has gradient function x 96 f ' ( x) = 3 x 2 − 3 where h and k are x constants. at the instant when x = 2 and the rate of change in x is 0.6 unit per second [5m] SPM 2002 5 2 3 1. (a) y = 2x – x2 47 . Using π = 3. Diameter AB can be adjusted so that point C which lies on the circumference can move such that AC + CB = 40 cm. find the maximum area of triangle ABC SPM 2001 4 + 3r 1. Given f ( r ) = find limited value 5 − 2r of f (r ) when r → ∞ The diagram shows a circle inside rectangle ABCD such that the circle is constantly touching the two sides of the rectangle. -12) is 3 x 2 − 6 x .4 m (use π = 3. x Given that y increases at a constant rate of 4 units per second. find the rate of change of x when x = 2 [3m] paper 2(section B) 3. Volume of a 1 2 cone = πr h ) [4m] 3 SPM 2003 paper2(section A) 1. The gradient function of a curve which passes through A(1. (i) If the radius of the ink blot increases at a constant rate of 18 mm for every 6 second.2 m3 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 0. (a) Diagram 2 shows a conical container of diameter 0.6 m and height 0.Water is poured into the container at a constant rate of 0.01 [3m] dy = 2 x + 2 and 3. Given that y = 14 x (5 − x) . Differentiate 3 x 2 (2 x − 5) 4 with respect to x [3m] 2. (a) Given that dx y = 6 when x = −1 . Given straight lines AB and PQ touch the curve at point O and point R respectively. find y in terms of x [3m] (b) Hence.02 mm [4m] paper2(sectionB) 4. Find (a) the equation of the curve [3m] (b) the coordinates of the turning points of 48 . Find the coordinates of point R [4m] x2 d2y dy + ( x − 1) + y = 8 2 dx dx (b) A drop of ink falls on a piece of paper and forms an expanding ink blot in the shape of a circle. find the value of x if SPM 2004 1. find the approximate value of the area ink blot at the instant when its radius is 5. calculate (a) the value of x when y is a maximum (c) the maximum value of y [3m] 2. Two variables x and y are related by the 2 equation y = 3 x + . find the rate of change in the area of ink blot at the instant when its radius is 5 mm (ii) Using differentiation.(a) The diagram shows the curve y = 3 x − x 2 that passes through the origin.142.5m. where AB and PQ are perpendicular to each other. Given that y = x 2 + 5 x . use differentiation to find the small change in y when x increases from 3 to 3. where h is a small value [3m] 2 x2 ( 3 x − 5) 2 [4m] .3) is parallel to the straight line y + x − 5 = 0 . The volume of water. It is given that the 1 gradient of the normal at P is − 4 Find the coordinates of P [3m] 2. Find (a) the value of p (b) the equation of the curve SPM 2006 Paper 1 1.the curve and determine whether each of the turning points is a maximum or a minimum [5m] SPM 2005 1. where k is a constant. Find the value of p [3 m] SPM 2008 Paper 1 1. V cm3. Water is poured into the container at the rate of 10 cm3 s-1. Given that h( x) = h" (1) 1 dy when x =1 dx (b) express the approximate change in y. where p is small value (a) find the value of SPM 2007 Paper 2 1. the approximate change in y when x changes from 4 to 4 + h. The point P lies on the curve y = ( x − 5) 2 . in terms of h. It is given that y = u = 3x − 5 . evaluate 2. Given that y = 3 x 2 + x − 4 49 . in terms of p. where 3 dy in terms of x dx [4m] 3. Two variables x and y are related by the 16 equation y = 2 . The tangent to the curve at the point (1. Find the rate of change of the height of water. at the instant when its height is 2 cm [3m] paper2(sectionA) 3. in cm s-1. A curve with gradient function 2 x − has a turning point at (k. where p is a constant. The curve y = x − 32 x + 64 has a minimum point at x = p . where p is a constant. A curve has a gradient function px 2 − 4 x . Find 2 7 u . 3 where h cm is the height of the water in the container. The curve y = f (x) is such that dy = 3kx + 5 . 8) (a) Find the value of k [3 m] (b) determine whether the turning point is a maximum or minimum point [2 m] ( c) find the equation of the curve [3 m] SPM 2007 Paper 1 1. when x changes from 1 to 1 + p. x Express. in a 1 3 container is given by V = h + 8h . dx The gradient of the curve at x = 2 is 9 Find the value of k [2 m] 2 2. BFC. 50 . divide by three parts. ABC. calculate the total length is needed (b) Calculate ∠BCA (c ) Given that the area of ∆FCG same with the area ∆ADE . The diagram shows a land form triangle. [2m] The diagram shows a ∆ PQR (a) Calculate obtuse angle PQR (b) Sketch and label another triangle which is different from triangle PQR in the diagram.2. ADB. where the lengths of PQ and QR as well as angle PQR are maintained. The normal to the curve y = x 2 − 5 x at point P is parallel to the straight line y = − x + 12 . Find the equation of the normal to the curve at point P. SPM 1993 1. [4m] [1m] (c ) If the length of PR is reduced while the length of PQ and angle PQR are maintained. and AEGC is a straight line 12 CHAPTER 10: SOLUTION OF that sin ∠BAC = Given TRIANGLE 13 (a) if the fence want to build along the boundary BC. Calculate the length of GC SPM 1994 1. calculate the length of PR so that only one ∆ PQR can be form [2m] 2. The diagram shows a cuboid. sin ∠ADC = 4 where 5 ∠ADC is an obtuse angle. points A. [5m] 2. (a) the length of AC correct to two decimal places [3m] (b) ∠ABC [2m] SPM 1996 1. Calculate 51 . D and E lie on a flat horizontal surface. Given that AB = 3 cm. SPM 1995 1. calculate (a) the length of AD (b) ∠DAE SPM 1997 In the diagram. In the diagram. Given BCD is a straight line. calculate the length of CD 2. B. calculate the area of the slanting face. BCD is a straight line.In the diagram. C. BC = 4 cm and ∠ABC = 90 0 and vertex D is 4 cm vertically above B. Calculate (a) ∠JQL [4m] (b) the area of ∆JQL [2m] The diagram shows a pyramid with ∆ABC as the horizontal base. ∠ACB is an obtuse angle and the area of ∆ADE = 20 cm2. BD = 5 cm. Find (a) ∠BDC (b) the length of AD The diagram shows a trapezium ABCD Calculate (a) ∠CBD (b) the length of straight line AC 2. BDE and ADC are a straight lines. In the diagram. Calculate (a) ∠VTU (b) the area of plane VTU SPM 1999 1. 1. JKL is a triangle with side JK = 10 cm. Calculate (a) ∠JLK (b) the area of ∆JKL SPM 2000 52 .456 and sin ∠JKL = 0. VD is vertical and base ABCD is horizontal. CD = 8 cm and AE = 12 cm. The diagram shows a triangle ABC Calculate (a) the length of AB (b) the new area of triangle ABC if AC is lengthened while the lengths of AB.1.36 . BC and ∠BAC are maintained [3m] SPM 1998 The diagram shows a pyramid VABCD with a square base ABCD. Given that sin ∠KJL = 0. BC = 7cm. 2. VR = 15 cm and ∠VQR = 80 0 Calculate (a) the length of QR (b) the area of the slanting face The diagram shows a prism with a uniform triangular cross-section PTS. PQR is a straight line. The diagram shows a cyclic quadrilateral ABCD. PV = 10 cm. The lengths of straight lines DC and CB are 3 cm and 6 cm respectively. Calculate the length of PS The diagram shows a quadrilateral ABCD. Given PQ = 4 cm. show that cos α = 29 2. Find the total surface area of the rectangular faces [5m] SPM 2003 53 . The diagram shows a pyramid with a triangular base PQR whish is on a horizontal plane. 2. Express the length of BD in terms of (a) α (b) β 11 Hence. Vertex V is vertically above P. Given AD is the longest side of triangle ABD and the area of triangle ABD is 10 cm2 Calculate (a) ∠BAD (b) the length of BD ( c) the length of BC SPM 2001 1. Given the volume of the prism is 315 cm3.1. In the diagram. SPM 2002 1. V is the vertex of the tent and the angle between the inclined plane VBC and the base is 500 Given that VB = VC = 2.1.2 m and AB = AC = 2. of quadrilateral ABCD [8m] (b) A triangle A'B'C' has the same measurements as those given for triangle ABC.6 m.50. in cm2. A'C' = 12. The diagram shows a quadrilateral ABCD such that ∠ABC is acute (i) Sketch the triangle A'B'C' (ii) State the size of ∠ A'B'C' [2m] 54 .5 cm and ∠B 'A'C' = 40. but which is different in shape to triangle ABC SPM 2004 1.3 cm. that is. C'B' = 9. The diagram shows a tent VABC in the shape of a pyramid with triangle ABC as the horizontal base. calculate (a) the length of BC if the area of the base is 3 m2 (b) the length of AV and the base is 250 (c ) the area of triangle VAB (a) Calculate (i) ∠ABC (ii) ∠ADC (iii) the area. in cm. in cm. in cm2. of CD (ii) the area.kitaran vertical = mencancang horizontal = mengufuk obtuse angle = sudut cakah slanting face = permukaan condong acute = tirus formed = dibentuk diagonal = pepenjuru SPM 2006 SPM 2005 1. of BD [2 m] (c) ∠ABD [3 m] 2 (d) the area. in cm. ∠ACD = 40 0 and AD = 16 cm Calculate the two possible values of ∠ADC [2m] (c ) By using the acute ∠ADC from (b). of the quadrilateral ABCD cyclic quadrilateral=sisi empat 55 . Diagram 5 shows a quadrilateral ABCD 1. The diagram shows triangle ABC Diagram 5 The area of triangle BCD is 13 cm2 and ∠BCD is acute Calculate (a) ∠BCD [2 m] (b) the length. of AC [2m] (b) A quadrilateral ABCD is now formed so that AC is a diagonal. quadrilateral ABCD [3 m] (a) Calculate the length. in cm . calculate (i) the length. in cm 2 . The table below shows the monthly expenses of Ali’s family Year Expenses Food Transportation Rental Electricity & water 1998 RM 320 RM 80 RM 280 RM 40 1992 RM 384 RM 38 RM 322 RM 40 Calculate the length.CHAPTER 11: INDEX NUMBER SPM 2006 1. The table that follows shows the price indices in the year 1993 based on the year 1990 ii. The pie chart below shows the distribution of the monthly expenses in the Yusnis’ household in the year 1990. of ∆A ’BC [6 M] 56 . Point A’ lies on AC such that A’ B = AB (i) sketch ∆A ’BC (ii) calculate the area. in cm. Diagram 7 shows quadrilateral ABCD SPM 1993 1. if Ali’s monthly income in the year 1998 is RM 800. of AC (b) ∠ACB Find the composite index in the year 1992 by using the year 1998 as the base year. Hence. find the monthly income required in the year 1992 so that the increases in his income is in line with the increases in his expenses [5m] [4 M] SPM 1994 1. of the monthly expenses in the Yusnis’ household (b) the total monthly expenses in the year 1993. changes to price indices from the year 1994 to 1996 and their weightages respectively. and the composite price index in the year 1995 is 130. The below table shows the price indices and weightages of three items in the year 1995 based on the year 1990. [2m] Wood Cement Iron Steel 5 4 2 1 Calculate the composite price index in the Item Price Index Weightage Shirt 100 N Trousers 110 6 Bag 140 2 Shoes 100 4 year 1996 [3m] Food item Fish Prawn Chicken Beef Cuttlefish SPM 1997 1.Given the price of item R in the year 1990 and 1995 are RM 30 and RM 33 respectively. correct to the nearest integer. Given the composite price index in the year 1994 is RM 114 Calculate (a) the value of n (b) the price of a shirt in 1994 if its price in 1990 is RM 40 SPM 1996 1. if the total monthly expenses of the Yusris’ household in the year 1990 is RM 850 SPM 1995 1. correct to the nearest ringgit. Price index 140 120 125 115 130 Weightages 4 2 4 3 X 57 .40 and 160. calculate the price of a kilogram of rice in the year 1990. Using the year 1990 as the base year. The table below shows the price indices and weightages of four items in the year 1994 based on the year 1990. Item Price Index 1994 180 116 140 124 Changes to Price index from 1994 to 1996 Increases 10% Decreases 5% No change No change Weightages Calculate (a) the composite price index. (a) In the year 1995. the price and price index of a kilogram of a certain grade of rice are RM 2.Monthly expenses Food House rental Entertainment Clothing Others Price Index 130 115 110 115 130 (b) The above table shows the price indices in the year 1994 using 1992 as the base year. calculate its price in the year 1993 SPM 1999 1. Wi SPM 2001 1. find the value of y 58 . and C with their respective weightages. find the value of y if the composite price index is 113 Item A B C SPM 2002 1. B. is 127. By using 1998 as the base year.Item P Q R Price index 120 150 m Weightage 2 n 3 [2m] [2m] Index number. using the year 1996 as the base year. The composite index number of the data in the below table is 108 Find the value of x SPM 2000 1. The table below shows the prices indices of items A. price indices and the number of three items Pric e Year 1999 55 40 80 Price Index (Base year 1999) 120 150 125 Calculate (a) the value of m (b) the value n 105 5-x 94 x 120 4 SPM 1998 1. The price index of a certain item in the year 1997 is 120 when 1995 is used as the base year and 150 when 1993 is used as the base year. Given the price of P in the year 1996 is RM 12. The table below shows the prices.5.y Item (RM) Year 2000 66 x 100 Number of items 200 500 y A B C (a) Find the value x (b) If the composite price index of the three items in the year 2000 using year 2000 as the base year is 136. Given the price of the item in the year 1995 is RM 360. Hence. Ii Weightages.20 [6m] [4m] Price index x 98 123 Weightage 5 y 14 . The table below shows the price indices and weightages of 5 types of food items in the year 1998 using the year 1996 as the base year.00 and increases to RM 13.80 in the year 1999. Given the composite price index in the 1998. Calculate (a) the value of x [4m] (b) the price of a kilogram of chicken in the year 1998 if the price of a kilogram of chicken in the year 1996 is RM 4. calculate the value of x. Q.SPM 2003 Type of item A B C Price (RM) in 1996 70 80 60 Price (RM) in 1998 105 100 67. R. S and T for the year 1990.50 Price in 1995 RM 0. as well as their weightages (a) Using the year 1996 as the base year.70 RM 2. B and C in the year 1996 and 1998.00 z P Q R S T Price Index in 1995 based on 1990 175 125 y 150 120 (a) Find the value of (i) x (ii) y (iii) z (b) Calculate the composite index for the items in the year 1995 based on the year 1990 59 .00 RM 6.00 RM 4. 35 30 25 20 15 10 5 0 P Q R ITEMS S T WEEKLY COST (RM) Items Price in 1990 x RM 2. The table below shows the prices of three items A. calculate the price indices of items A.50 RM 9. Table 1 shows the prices and the price indices for the items.00 RM 2. The diagram below show is a bar chart indicating the weekly cost of the items P. find the values of x and y [5m] 1.50 RM 5.50 Weightage % Y X 2x 2. B and C (b) Given the composite price index of these items in the year 1998 based on the year 1996 is 140. calculate the corresponding cost of making these biscuits in the year 2001 if the cost in the year 2004 was RM 2985 [5m] 60 . Q.( c) The total monthly cost of the items in the year 1990 is RM 456 (d) The cost of the items increases by 20% from the year 1995 to the year 2000. Q. Q. R and S which are the main ingredients in the production of a type of biscuits (a) Calculate (i) the price of S in the year 1993 if its price in the year 1995 is RM 37. y and z [3m] (b) (i) calculate the composite index for the cost of making these biscuits in the year 2004 based on the year 2001 (ii) Hence. Find the composite index for the year 2000 based on the year 1990 SPM 2005 1. P. Diagram below shows a pie chart which represents the relative amount of the ingredients P. The table below shows the prices and the price indices for the four ingredients P. The table below shows the price indices and percentage of usage of four items.00 y 0. Calculate the value of x (ii) the price of a box of biscuits in the year 1993 if the corresponding price in the year 1995 is RM 32 [5m] P Q R S Item P Q R S (i) Price index for the year 1995 based on the year 1993 135 x 105 130 Percentage of usage (%) 40 30 10 20 (a) Find the value of x.60 z 0.00 2. R and S used in making biscuits of a particular kind. and S used in making these biscuits Ingredients Price per kg Year Year 200 2004 1 0. R.80 1.40 0.40 Price index for the year 2004 based on the year 2001 x 140 150 80 SPM 2004 1.70 (ii) the price index of P in the year 1995 based on the year 1991 if its price index in the year 1993 based on the year 1991 is 120 [5m] (b) The composite index number of the lost of biscuits production for the year 1995 based on the year 1993 is 128. 20 110 R 4. used to produce a kind of toy Diagram 6 shows a pie chart which represents the relative quantity of components used Diagram 6 (a) Find the value of x and of y [3m] (b) Calculate the composite index for the production cost of the toys in the year 2006 based on the year 2004 61 .20 1.127. S and T. Table shows the prices of the ingredients (a) The index number of ingredient P in the year 2005 based on the year 2004 is 120. The price per kilogram of ingredient R in the year 2005 is RM 2.00 6. Q. P.00 x Y 4.50 125 Q x 2. R.70 y T 2. Q. A particular kind of cake is made by using four ingredients P. Calculate the value of x and of y [3m] (c) The composite index for the cost of making the cake in the year 2005 based on the year 2004 is 1.00 4.00 more than its corresponding price in the year 2004. R and S.40 SPM 2006 1.5 (c) The cost of making these biscuits is Calculate expected to increase by 50% from the (i) The price of a cake in the year year 2004 to the year 2007 2004 if its corresponding price Find the expected composite index for in the year 2005 is RM30.60 the year 2007 based on the year 2001 (ii) the value of m if the [2m] quantities of ingredients P.00 2.00 2. Table 4 shows the prices and the price indices of five components. Calculate the value of w [2m] (b) The index number of ingredient R in the year 2005 based on the year 2004 is 125.80 140 SPM 2007 Ingredient P Q R S Price per kilogram (RM) Year 2004 Year 2005 5.00 150 S 3. R and S used are in the ratio of 7 : 3 : m : Component Price (RM) for the Price index for the 2 year year 2006 based on the year 2004 [3m] P 1.50 4.00 W 2. Q. calculate the corresponding cost in the year 2008 [4m] 62 .[3m] (c) The price of each component increases by 20% from the year 2006 to the year 2008 Given that the production cost of one toy in the year 2004 is RM 55.