OCR MEI C3 — Question 3

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeFind composite function expression
DifficultyModerate -0.8 This is a straightforward composition question requiring only basic understanding of function composition and logarithm laws. Students substitute g(x) into f to get ln(x³) = 3ln(x), then recognize this as a vertical stretch by factor 3. Both parts are routine C3 exercises with no problem-solving or novel insight required, making it easier than average.
Spec1.02u Functions: definition and vocabulary (domain, range, mapping)1.02w Graph transformations: simple transformations of f(x)
The functions \(f(x)\) and \(g(x)\) are defined for the domain \(x > 0\) as follows: $$f(x) = \ln x, \quad g(x) = x^3.$$ Express the composite function \(fg(x)\) in terms of \(\ln x\). State the transformation which maps the curve \(y = f(x)\) onto the curve \(y = fg(x)\).
Question 3:
3
6 Fig. 6 shows the triangle OAP, where O is the origin and Ais the point (0, 3). The point P(x, 0)
moves on the positive x-axis. The point Q(0, y) moves between O and A in such a way that
AQ(cid:2)AP (cid:1) 6.
y
A(0, 3)
Q(0, y)
O P(x, 0) x
Fig. 6
(i) Write down the length AQ in terms of y. Hence find APin terms of y, and show that
(y(cid:2)3)2 (cid:1) x2(cid:2)9. [3]
dy x
(ii) Use this result to show that (cid:1) . [2]
dx y(cid:2)3
dx dy
(iii) When x (cid:1) 4 andy (cid:1) 2, (cid:1) 2. Calculate at this time. [3]
dt dt
[Turn over
© OCR 2007 4753/01 Jan 07
y
A(0, 3)
Q(0, y)
O P(x, 0) x
O
3
6 Fig. 6 shows the curve y (cid:2) f(x), where f(x) (cid:2) 1 arctan x.
2
y
O
x
Fig. 6
(i) Find the range of the function f(x), giving your answer in terms of p. [2]
(ii) Find the inverse function f (cid:3)1(x). Find the gradient of the curve y (cid:2) f (cid:3)1(x) at the origin. [5]
(iii) Hence write down the gradient of y (cid:2) 1 arctan x at the origin. [1]
2
Section B (36 marks)
x2
7 Fig. 7 shows the curve y (cid:2) .It is undefined at x (cid:2) a;the line x (cid:2) ais a vertical asymptote.
1(cid:1)2x3
y
x = a
O x
Fig. 7
(i) Calculate the value of a, giving your answer correct to 3 significant figures. [3]
dy 2x (cid:3) 2x4
(ii) Show that (cid:2) . Hence determine the coordinates of the turning points of the
dx (1(cid:1)2x3)2
curve. [8]
(iii) Show that the area of the region between the curve and the x-axis from x (cid:2) 0 to x (cid:2) 1 is
1
ln 3. [5]
6
[Turn over
© OCR 2007 4753/01 June 07
y
O
x
O
Question 3:
3
6 Fig. 6 shows the triangle OAP, where O is the origin and Ais the point (0, 3). The point P(x, 0)
moves on the positive x-axis. The point Q(0, y) moves between O and A in such a way that
AQ(cid:2)AP (cid:1) 6.
y
A(0, 3)
Q(0, y)
O P(x, 0) x
Fig. 6
(i) Write down the length AQ in terms of y. Hence find APin terms of y, and show that
(y(cid:2)3)2 (cid:1) x2(cid:2)9. [3]
dy x
(ii) Use this result to show that (cid:1) . [2]
dx y(cid:2)3
dx dy
(iii) When x (cid:1) 4 andy (cid:1) 2, (cid:1) 2. Calculate at this time. [3]
dt dt
[Turn over
© OCR 2007 4753/01 Jan 07
y
A(0, 3)
Q(0, y)
O P(x, 0) x
O
3
6 Fig. 6 shows the curve y (cid:2) f(x), where f(x) (cid:2) 1 arctan x.
2
y
O
x
Fig. 6
(i) Find the range of the function f(x), giving your answer in terms of p. [2]
(ii) Find the inverse function f (cid:3)1(x). Find the gradient of the curve y (cid:2) f (cid:3)1(x) at the origin. [5]
(iii) Hence write down the gradient of y (cid:2) 1 arctan x at the origin. [1]
2
Section B (36 marks)
x2
7 Fig. 7 shows the curve y (cid:2) .It is undefined at x (cid:2) a;the line x (cid:2) ais a vertical asymptote.
1(cid:1)2x3
y
x = a
O x
Fig. 7
(i) Calculate the value of a, giving your answer correct to 3 significant figures. [3]
dy 2x (cid:3) 2x4
(ii) Show that (cid:2) . Hence determine the coordinates of the turning points of the
dx (1(cid:1)2x3)2
curve. [8]
(iii) Show that the area of the region between the curve and the x-axis from x (cid:2) 0 to x (cid:2) 1 is
1
ln 3. [5]
6
[Turn over
© OCR 2007 4753/01 June 07
y
O
x
O
The functions $f(x)$ and $g(x)$ are defined for the domain $x > 0$ as follows:
$$f(x) = \ln x, \quad g(x) = x^3.$$

Express the composite function $fg(x)$ in terms of $\ln x$.

State the transformation which maps the curve $y = f(x)$ onto the curve $y = fg(x)$.

\hfill \mbox{\textit{OCR MEI C3  Q3}}
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