Standard +0.3 This is a standard C3 calculus question involving differentiation of ln x and 1/x, finding stationary points, normals, and numerical methods. All techniques are routine for C3 level—finding where f'(x)=0, substituting to find coordinates, calculating gradient and normal equation, then solving an equation numerically. The algebraic manipulation is straightforward and each part follows logically from the previous one. Slightly easier than average due to clear structure and standard methods throughout.
The curve \(C\) has equation \(y = \text{f}(x)\), where
$$\text{f}(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$
The point \(P\) is a stationary point on \(C\).
Calculate the \(x\)-coordinate of \(P\). [4]
Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. [2]
The point \(Q\) on \(C\) has \(x\)-coordinate 1.
Find an equation for the normal to \(C\) at \(Q\). [4]
The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).
Show that the \(x\)-coordinate of \(R\)
satisfies the equation \(6 \ln x + x + \frac{2}{x} - 3 = 0\),
Figure 2 shows part of the curve C with equation y = f(x), where
f(x) = 0.5ex – x2.
The curve C cuts the y-axis at A and there is a minimum at the point B.
(a) Find an equation of the tangent to C at A. (4)
The x-coordinate of B is approximately 2.15. A more exact estimate is to be made of this
coordinate using iterations x = ln g(x ).
n + 1 n
(b) Show that a possible form for g(x) is g(x) = 4x. (3)
(c) Using x = ln 4x , with x = 2.15, calculate x , x and x . Give the value of x to 4
n + 1 n 0 1 2 3 3
decimal places. (2)
2
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3. (a) Sketch the graph of y = 2x + a, a > 0, showing the coordinates of the points where the
graph meets the coordinate axes. (2)
1
(b) On the same axes, sketch the graph of y = . (1)
x
(c) Explain how your graphs show that there is only one solution of the equation
x2x + a − 1 = 0. (1)
(d) Find, using algebra, the value of x for which x2x + 1 − 1 = 0. (3)
4. Figure 1
y
B(4 , 1)
3
−1 O A(2, 0) 3 x
Figure 1 shows a sketch of the curve with equation y = f(x), −1 ≤ x ≤ 3. The curve touches
the x-axis at the origin O, crosses the x-axis at the point A(2, 0) and has a maximum at the
point B(4 , 1).
3
In separate diagrams, show a sketch of the curve with equation
(a) y = f(x + 1), (3)
Answer
Marks
Guidance
(b) y =
f(x)
, (3)
(c) y = f(
x
), (4)
marking on each sketch the coordinates of points at which the curve
(i) has a turning point,
(ii) meets the x-axis.
3
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3
5. (i) Given that sin x = , use an appropriate double angle formula to find the exact value
5
of sec 2x.
(4)
(ii) Prove that
nπ
cot 2x + cosec 2x ≡ cot x, x≠ ,n∈Z.
2
(4)
4
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3x−1
6. The function f is defined by f: x → , x ∈ ¡ , x ≠ 3.
x−3
(a) Prove that f−1(x) = f(x) for all x ∈ ¡ , x ≠ 3. (3)
¡
(b) Hence find, in terms of k, ff(k), where x ≠ 3. (2)
Figure 3
y
6
2
−2 −1 O 2 x
−5
Figure 3 shows a sketch of the one-one function g, defined over the domain −2 ≤ x ≤ 2.
(c) Find the value of fg(−2). (3)
(d) Sketch the graph of the inverse function g−1 and state its domain. (3)
The function h is defined by h: x 2g(x – 1).
(e) Sketch the graph of the function h and state its range. (3)
5
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7. (i) (a) Express (12 cos θ – 5 sin θ) in the form R cos (θ + α), where R > 0 and
0 < α < 90°.
(4)
(b) Hence solve the equation
12 cos θ – 5 sin θ = 4,
for 0 < θ < 90°, giving your answer to 1 decimal place. (3)
(ii) Solve
8 cot θ – 3 tan θ = 2,
for 0 < θ < 90°, giving your answer to 1 decimal place. (5)
8. The curve C has equation y = f(x), where
1
f(x) = 3 ln x + , x > 0.
x
The point P is a stationary point on C.
(a) Calculate the x-coordinate of P. (4)
(b) Show that the y-coordinate of P may be expressed in the form k – k ln k, where k is a
constant to be found. (2)
The point Q on C has x-coordinate 1.
(c) Find an equation for the normal to C at Q. (4)
The normal to C at Q meets C again at the point R.
(d) Show that the x-coordinate of R
2
(i) satisfies the equation 6 ln x + x + – 3 = 0,
x
(ii) lies between 0.13 and 0.14. (4)
END
6
Question 8:
8
Tot
al
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1.
2. Figure 2
y
C
A
O • B x
Figure 2 shows part of the curve C with equation y = f(x), where
f(x) = 0.5ex – x2.
The curve C cuts the y-axis at A and there is a minimum at the point B.
(a) Find an equation of the tangent to C at A. (4)
The x-coordinate of B is approximately 2.15. A more exact estimate is to be made of this
coordinate using iterations x = ln g(x ).
n + 1 n
(b) Show that a possible form for g(x) is g(x) = 4x. (3)
(c) Using x = ln 4x , with x = 2.15, calculate x , x and x . Give the value of x to 4
n + 1 n 0 1 2 3 3
decimal places. (2)
2
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3. (a) Sketch the graph of y = 2x + a, a > 0, showing the coordinates of the points where the
graph meets the coordinate axes. (2)
1
(b) On the same axes, sketch the graph of y = . (1)
x
(c) Explain how your graphs show that there is only one solution of the equation
x2x + a − 1 = 0. (1)
(d) Find, using algebra, the value of x for which x2x + 1 − 1 = 0. (3)
4. Figure 1
y
B(4 , 1)
3
−1 O A(2, 0) 3 x
Figure 1 shows a sketch of the curve with equation y = f(x), −1 ≤ x ≤ 3. The curve touches
the x-axis at the origin O, crosses the x-axis at the point A(2, 0) and has a maximum at the
point B(4 , 1).
3
In separate diagrams, show a sketch of the curve with equation
(a) y = f(x + 1), (3)
(b) y = |f(x)|, (3)
(c) y = f(|x|), (4)
marking on each sketch the coordinates of points at which the curve
(i) has a turning point,
(ii) meets the x-axis.
3
PhysicsAndMathsTutor.com
3
5. (i) Given that sin x = , use an appropriate double angle formula to find the exact value
5
of sec 2x.
(4)
(ii) Prove that
nπ
cot 2x + cosec 2x ≡ cot x, x≠ ,n∈Z.
2
(4)
4
PhysicsAndMathsTutor.com
3x−1
6. The function f is defined by f: x → , x ∈ ¡ , x ≠ 3.
x−3
(a) Prove that f−1(x) = f(x) for all x ∈ ¡ , x ≠ 3. (3)
¡
(b) Hence find, in terms of k, ff(k), where x ≠ 3. (2)
Figure 3
y
6
2
−2 −1 O 2 x
−5
Figure 3 shows a sketch of the one-one function g, defined over the domain −2 ≤ x ≤ 2.
(c) Find the value of fg(−2). (3)
(d) Sketch the graph of the inverse function g−1 and state its domain. (3)
The function h is defined by h: x 2g(x – 1).
(e) Sketch the graph of the function h and state its range. (3)
5
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7. (i) (a) Express (12 cos θ – 5 sin θ) in the form R cos (θ + α), where R > 0 and
0 < α < 90°.
(4)
(b) Hence solve the equation
12 cos θ – 5 sin θ = 4,
for 0 < θ < 90°, giving your answer to 1 decimal place. (3)
(ii) Solve
8 cot θ – 3 tan θ = 2,
for 0 < θ < 90°, giving your answer to 1 decimal place. (5)
8. The curve C has equation y = f(x), where
1
f(x) = 3 ln x + , x > 0.
x
The point P is a stationary point on C.
(a) Calculate the x-coordinate of P. (4)
(b) Show that the y-coordinate of P may be expressed in the form k – k ln k, where k is a
constant to be found. (2)
The point Q on C has x-coordinate 1.
(c) Find an equation for the normal to C at Q. (4)
The normal to C at Q meets C again at the point R.
(d) Show that the x-coordinate of R
2
(i) satisfies the equation 6 ln x + x + – 3 = 0,
x
(ii) lies between 0.13 and 0.14. (4)
END
6
The curve $C$ has equation $y = \text{f}(x)$, where
$$\text{f}(x) = 3 \ln x + \frac{1}{x}, \quad x > 0.$$
The point $P$ is a stationary point on $C$.
\begin{enumerate}[label=(\alph*)]
\item Calculate the $x$-coordinate of $P$. [4]
\item Show that the $y$-coordinate of $P$ may be expressed in the form $k - k \ln k$, where $k$ is a constant to be found. [2]
\end{enumerate}
The point $Q$ on $C$ has $x$-coordinate 1.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find an equation for the normal to $C$ at $Q$. [4]
\end{enumerate}
The normal to $C$ at $Q$ meets $C$ again at the point $R$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Show that the $x$-coordinate of $R$
\begin{enumerate}[label=(\roman*)]
\item satisfies the equation $6 \ln x + x + \frac{2}{x} - 3 = 0$,
\item lies between 0.13 and 0.14. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q8 [14]}}