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