| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Sketch exponential graphs |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 transformations and function composition. Parts (a)-(b) test routine graph transformations, (c) requires simple substitution, (d) involves solving a logarithmic equation, and (e) tests function composition with exponentials and logarithms. All techniques are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties |
Total
Question 9:
9
Total
1. Use the derivatives of sin x and cos x to prove that the derivative of tan x is sec2 x. (4)
3
2. The function f is given by f:x α 2 + , x ∈
x+2
2
(cid:0)
PhysicsAndMathsTutor.com
, x ≠ –2.
3
(a) Express 2 + as a single fraction. (1)
x+2
(b) Find an expression for f–1(x). (3)
(c) Write down the domain of f–1. (1)
3. (a) Express as a fraction in its simplest form
2 13
+ .
x−3 x2 + 4x−21
(3)
(b) Hence solve
2 13
+ =1.
x−3 x2 + 4x−21
(3)
x2 + 4x+3
4. (a) Simplify .
x2 + x
(2)
(b) Find the value of x for which log (x2 + 4x + 3) – log (x2 + x) = 4.
2 2
(4)
5. (i) Prove, by counter-example, that the statement
“sec(A+ B)≡secA+secB, for all A and B”
is false
(2)
(ii) Prove that
nπ
tanθ+cotθ≡2cosec2θ, θ≠ , n∈
2
3
(cid:0)
.
(5)
6. (a) Prove that
1−cos2θ nπ
≡ tan θ , θ ≠ , n ∈
sin2θ 2 (cid:0)
PhysicsAndMathsTutor.com
.
(3)
(b) Solve, giving exact answers in terms of π,
2(1 – cos 2θ ) = tan θ , 0 < θ < π .
(6)
7. Given that y = log x, x > 0, where a is a positive constant,
a
(a) (i) express x in terms of a and y,
(1)
(ii) deduce that ln x = y ln a.
(1)
dy 1
(b) Show that = .
dx xlna
(2)
The curve C has equation y = log x, x > 0. The point A on C has x-coordinate 10. Using the
10
result in part (b),
(c) find an equation for the tangent to C at A.
(4)
The tangent to C at A crosses the x-axis at the point B.
(d) Find the exact x-coordinate of B.
(2)
PhysicsAndMathsTutor.com
8. The curve with equation y = ln 3x crosses the x-axis at the point P (p, 0).
(a) Sketch the graph of y = ln 3x, showing the exact value of p.
(2)
The normal to the curve at the point Q, with x-coordinate q, passes through the origin.
(b) Show that x = q is a solution of the equation x2 + ln 3x = 0.
(4)
(c) Show that the equation in part (b) can be rearranged in the form x =
1e−x2
.
3
(2)
(d) Use the iteration formula x = 1e −x n 2 , with x = 1, to find x , x , x and x . Hence write
n + 1 3 0 3 1 2 3 4
down, to 3 decimal places, an approximation for q.
(3)
4
9. Figure 3
y
(0, c)
O (d, 0) x
Figure 3 shows a sketch of the curve with equation y = f(x), x ≥ 0. The curve meets the
coordinate axes at the points (0, c) and (d, 0).
In separate diagrams sketch the curve with equation
(a) y = f
−1(x),
(2)
(b) y = 3f(2x).
(3)
Indicate clearly on each sketch the coordinates, in terms of c or d, of any point where the curve
meets the coordinate axes.
Given that f is defined by
f : x → 3(2 −x ) − 1, x ∈
5
(cid:0) , x ≥ 0,
(c) state
(i) the value of c,
(ii) the range of f.
(3)
(d) Find the value of d, giving your answer to 3 decimal places.
(3)
The function g is defined by
g : x → log x, x ∈
2
(cid:0)
PhysicsAndMathsTutor.com
, x ≥ 1.
(e) Find fg(x), giving your answer in its simplest form.
(3)
END\includegraphics{figure_3}
Figure 3 shows a sketch of the curve with equation $y = \text{f}(x)$, $x \geq 0$. The curve meets the coordinate axes at the points $(0, c)$ and $(d, 0)$.
In separate diagrams sketch the curve with equation
\begin{enumerate}[label=(\alph*)]
\item $y = \text{f}^{-1}(x)$, [2]
\item $y = 3\text{f}(2x)$. [3]
\end{enumerate}
Indicate clearly on each sketch the coordinates, in terms of $c$ or $d$, of any point where the curve meets the coordinate axes.
Given that f is defined by
$$\text{f}: x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item state
\begin{enumerate}[label=(\roman*)]
\item the value of $c$,
\item the range of f.
\end{enumerate} [3]
\item Find the value of $d$, giving your answer to 3 decimal places. [3]
\end{enumerate}
The function g is defined by
$$\text{g}: x \mapsto \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Find fg(x), giving your answer in its simplest form. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q9 [14]}}