Math Repository
about
topic
al
year
ly
Yearly
2009
P1 Q4
Topical
Graphs
09 P1 Q4
2009 H2 Mathematics Paper 1 Question 4
Graphs and Transformations
Answers
(i)
f
(
27
)
+
f
(
45
)
=
11
{f(27)+f(45)=11}
f
(
27
)
+
f
(
45
)
=
11
(ii)
(iii)
110
3
{\frac{110}{3}}
3
110
Full solutions
(i)
f
(
27
)
+
f
(
45
)
=
f
(
3
)
+
f
(
1
)
=
(
2
(
3
)
−
1
)
+
(
7
−
1
2
)
=
5
+
6
=
11
■
\begin{align*} & f(27) + f(45) \\ & = f(3) + f(1) \\ & = \Big(2(3)-1\Big) + \left(7-1^2\right) \\ & = 5 + 6 \\ & = 11 \; \blacksquare \end{align*}
f
(
27
)
+
f
(
45
)
=
f
(
3
)
+
f
(
1
)
=
(
2
(
3
)
−
1
)
+
(
7
−
1
2
)
=
5
+
6
=
11
■
(ii)
(iii)
∫
−
4
3
f
(
x
)
d
x
=
2
∫
0
2
(
7
−
x
2
)
d
x
+
∫
2
4
(
2
x
−
1
)
d
x
+
∫
2
3
(
2
x
−
1
)
d
x
=
2
[
7
x
−
x
3
3
]
0
2
+
[
x
2
−
x
]
2
4
+
[
x
2
−
x
]
2
3
=
2
(
14
−
8
3
)
+
(
4
2
−
4
−
2
2
+
2
)
+
(
3
2
−
3
−
2
2
+
2
)
=
110
3
■
\begin{align*} & \int_{-4}^3 f(x) \; \mathrm{d}x \\ & = 2 \int_{0}^2 (7-x^2) \; \mathrm{d}x + \int_2^4 (2x-1) \; \mathrm{d}x + \int_2^3 (2x-1) \; \mathrm{d}x \\ & = 2 \left[ 7x-\frac{x^3}{3} \right]_{0}^2 + \left[ x^2 - x \right]_2^4 + \left[ x^2 - x \right]_2^3 \\ & = 2 \left( 14 - \frac{8}{3} \right) + \left( 4^2 - 4 - 2^2 + 2 \right) + \left( 3^2 - 3 - 2^2 + 2 \right) \\ & = \frac{110}{3} \; \blacksquare \end{align*}
∫
−
4
3
f
(
x
)
d
x
=
2
∫
0
2
(
7
−
x
2
)
d
x
+
∫
2
4
(
2
x
−
1
)
d
x
+
∫
2
3
(
2
x
−
1
)
d
x
=
2
[
7
x
−
3
x
3
]
0
2
+
[
x
2
−
x
]
2
4
+
[
x
2
−
x
]
2
3
=
2
(
14
−
3
8
)
+
(
4
2
−
4
−
2
2
+
2
)
+
(
3
2
−
3
−
2
2
+
2
)
=
3
110
■
Back to top ▲