New Senior Secondary Mathematics Advanced Exercise Ch. 03: Functions and Graphs ADVANCED EXERCISE CH. 03: FUNCTIONS AND GRAPHS [Finish the following questions if you aim at DSE Math Level 4] Q1 [CE Math 92 9] (Modified) 2 Figure 3 shows the graph of y = 2 x − 4 x + 3 , where x ≥ 0 . P(a, b) is a variable point on the graph. A rectangle OAPB is drawn with A and B lying on the x- and y-axes respectively. (a) (i) (ii) Find the area of rectangle OAPB in terms of a. Find the two values of a for which OAPB is a square. (b) Suppose the area of OAPB is 3 3 2 . Show that 4 a − 8a + 6a − 3 = 0 . 2 Q2 [CE Math 91 6] 2 The curve y = x − 6 x + 5 meets the y-axis at A and the x-axis at B and C as shown in Figure 2. (a) (b) Find the coordinates of A, B and C. The line y = x + 5 passes through A and meets the curve again at D. Find the coordinates of D. Q3 [Misc] 2 Given that the vertex of a quadratic curve y = ax + bx + c is (3, 2). If the curve passes through (-1, 4), find the values of a, b and c. Q4 [Misc] 11 2 By finding the minimum value of 3 x − 4 x + 5 , determine the maximum value of the expression 2 3x − 4 x + 5 and the corresponding value of x. [Finish the following questions if you aim at DSE Math Level 5] Q5 [CE Math 90 6] In the figure, the curve y = x + px + q cuts the x-axis at the two points A (α , 0) and B (β , 0) . M (−2, 0) is the mid-point of AB. 2 (a) Express α + β in terms of p. Hence find the value of p. (b) If α + β = 26 , find the value of q. Page 1 2 2 New Senior Secondary Mathematics Advanced Exercise Ch. 03: Functions and Graphs Q6 [CE AMath 86I 3] The maximum value of the function f ( x) = 4k + 18 x − kx (k is a positive constant) is 45. Find k. 2 Q7 [Misc] Given the maximum value of the function f ( x) = 2 + 9 k 2 x − x is 18, where k ≠ 0 . Find the values of k. Q8 [Misc] 2 The function y = ax + 12 x + c attains its minimum value -10 when x = − 13 . Find the values of a and c. 2 [Finish the following questions if you aim at DSE Math Level 5*] Q9 [CE Math 82 11] 2 In Figure 5, O is the origin. The curve C1 : y = x − 10 x + k (where k is a fixed constant) intersects the x-axis at the points A and B. (a) (b) (c) 2 By considering the sum and the product of the roots of x − 10 x + k = 0 , or otherwise, (i) find OA + OB , (ii) find OA × OB in terms of k. M and N are the mid-points of OA and OB respectively (see Figure 5). (i) Find OM + ON . (ii) Find OM × ON in terms of k. 2 Another curve C2 : y = x + px + r (where p and r are fixed constants) passes through the points M and N. (i) Using the results in (b) or otherwise, find the value of p and express r in terms of k. (ii) If OM = 2, find k. Q10 [CE AMath 91I 9] Let f ( x) = x + 2 x − 2 and g ( x) = −2 x − 12 x − 23 . 2 (a) 2 2 Express g ( x) in the form a ( x + b) + c , where a, b and c are real constants. Hence show that g ( x) < 0 for all real values of x. (b) (c) Let k1 and k2 ( k1 > k 2 ) be the two values of k such that the equation f ( x) + kg ( x) = 0 has equal roots. (i) Find k1 and k2 . (ii) Show that f ( x) + k1 g ( x) ≤ 0 and f ( x) + k 2 g ( x) ≥ 0 for all real values of x. Using (a) and (b), or otherwise, find the greatest and least values of Page 2 f ( x) g ( x) . New Senior Secondary Mathematics Advanced Exercise Ch. 03: Functions and Graphs Q11 [CE AMath 86II 6] 2 A straight line through C(3, 2) with slope m cuts the curve y = ( x − 2) at the points A and B. If C is the mid-point of AB, find the value of m. Q12 [CE AMath 81I 5] a 2 for all real values of x. Let f ( x) = x + ax + b , where a and b are real. Show that f ( x) ≥ f − 2 2 Hence, or otherwise, find the minimum value of x − 13 x + 5 . Q13 [Misc] 2 x + 1 2 = x + x − 1 , find f ( x) Given f 3 Q14 [Misc] Let f ( x) = − x + 10 x + 5 . p and q are two unequal real numbers such that f ( p) = f (q ) . Find the value of 2 p+q 2 Q15 [Misc] 2 4 4 2 2 (a) Prove that the quadratic function y = 4 x + 4hkx + h + k − h k can never be negative if h and k are real constants. (b) If the minimum value of y is 0, express h in terms of k. Q16 [Misc] (a) Prove that the expression ( x − 20)( x − 6) + k is positive for all real values of x if k > 49. (b) Hence, or otherwise, show that the expression [( y − 5)( y + 4)][( y − 3)( y + 2)] + 50 Q17 [Misc] Show that 0 < 6 2 x + 6 x + 11 ≤ 3 for all real values of x. Q18 [Misc] Given that f ( x) = x + 4 x + a + 3 , where a is a constant. 2 (a) Find the range of values of a such that f ( x) is never negative. (b) Determine the nature of the roots of the equation af ( x) = ( x + 2)( a −1) . 2 [Finish the following questions if you aim at DSE Math Level 5**] Q19 [CE AMath 84I 8] 1 < 0. 2 Let f ( x) = 5 x + bx + c , where b and c are real, c > 0 and f 2 (a) Show that the equation f ( x) = 0 has two distinct real roots. (b) Let α and β (i) (ii) By expressing f ( x) in factor form, show that 0 < α < If 1 2 Page 3 (α < β ) be the roots of f ( x) = 0 . −α = β − 1 2 1 <β. 2 , find the value of b and hence the range of values of c. is positive for all real values of y. New Senior Secondary Mathematics Advanced Exercise Ch. 03: Functions and Graphs Q20 [Misc] 2 2 In the figure, the curve C1 : y = x + 4 x − 1 cuts the x-axis at A and B and C2 : y = x + x − 1 cuts the x-axis at P and Q. (a) (b) Without finding the coordinates of A, B, P and Q, find the length of (i) AB (ii) PQ. Find AP + BQ . (c) Using the results of (a) and (b), find PB. (d) α is the smaller roots of x + 4 x − 1 = 0 and β is the smaller roots of x + x − 1 = 0 . Find the value of β − α 2 2 [Finish the following questions if you want extra knowledge / aim at difficult questions] Q21 [IMO Prelim 87-88] If f ( x) f ( y) − f ( xy ) = x + y for all real x and y, find f ( x) . Q22 [Misc] Let f be a function such that for all integers m and n, f ( m) is an integer and f (mn) = f ( m) f (n) . It is given that f (m) > f ( n) when 9 > m > n , f (2) = 3 and f (6) > 22 , find the value of f (3) . (Hint: Consider the rage of f (3) and the value of f ( 4) .) Q23 [Misc] −2 x + 1, when x < 1 Given that f ( x) = . If d is the maximum integral solution of f ( x) = 3 , find the value of d. 2 x − 2 x, when x ≥ 1 Answers Q1(a)(i) 2 −4 Q4 Max value = 3, + 3 (ii) 1.5 or 1 = Q5(a) % + Q9(a)(i) 10 (ii) k (b)(i) 5 (ii) (c)(i) (c) −1 ≤ + , - , Q19(b)(ii) Page 4 ≤ = −5, c < Q11 2 Q2(a) =− , 0,5 1,0 = 4 (b) -5 Q6 9 or % = −5, & = (ii) 24 Q12 . Q13 / = Q20(a)(i) 2√5 (ii) √5 (b) 3 (c) 5,0 (b) Q10(a) ' 9 √ 5 −5 (d) √ 6 7,12 # Q7 ± = −2 Q14 5 # +3 Q3 = , Q8 = =− , = , = 29 − 5 (b)(i) ( = 1, ( = − Q15(b) ℎ = ±( Q18(a) Q21 / = +1 ) ≥1 Q22 8 Q23 3