| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.2 This is a straightforward C3 question on logarithms and inverse functions covering standard techniques: finding inverse functions, identifying domain/range, solving logarithmic equations, and sketching modulus functions. All parts are routine textbook exercises requiring direct application of learned methods with no novel problem-solving or insight needed. Slightly easier than the typical 0.0 benchmark question. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02v Inverse and composite functions: graphs and conditions for existence1.06d Natural logarithm: ln(x) function and properties1.06e Logarithm as inverse: ln(x) inverse of e^x |
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Question 8:
8
Total
PhysicsAndMathsTutor.com
EDEXCEL CORE MATHEMATICS PRACTICE PAPER 1
1. Express as a single fraction in its simplest form
x2 − 8x+15 2x2 +6x
×
x2 −9 (x−5)2
(4)
2. The root of the equation f(x) = 0, where
f(x) = x+ln2x−4
is to be estimated using the iterative formula x = 4−ln2x , with x = 2.4.
n+1 n 0
(a) Showing your values of x , x , x ,…, obtain the value, to 3 decimal places, of the root.
1 2 3
(4)
(b) By considering the change of sign of f(x) in a suitable interval, justify the accuracy of your
answer to part (a). (2)
3. The function f is defined by
f :x a 2x −a , x ∈°
where a is a positive constant.
(a) Sketch the graph of y = f(x), showing the coordinates of the points where the graph cuts the axes.
(2)
(b) On a separate diagram, sketch the graph of y = f(2x), showing the coordinates of the points where
the graph cuts the axes.
(2)
(c) Given that a solution of the equation f(x) = 1 x is x = 4, find the two possible values of a.
2
(4)
4. Prove that
1− tan2θ
≡cos2θ.
1+ tan2θ
(6)
2
PhysicsAndMathsTutor.com
EDEXCEL CORE MATHEMATICS PRACTICE PAPER 1 MARK SCHEME
3 x−4
5. Express + as a single fraction in its simplest form. (7)
x2 + 2x x2 −4
6. The function f, defined for x ∈° , x > 0, is such that
1
f′(x) = x2 – 2 + .
x2
(a) Find the value of f″(x) at x = 4. (3)
(b) Given that f(3) = 0, find f(x). (4)
(c) Prove that f is an increasing function. (3)
2 6
7. f(x) = − , x > 1
x−1 (x−1)(2x+1)
4
(a) Prove that f(x) = . (4)
2x+1
(b) Find the range of f. (2)
(c) Find f
−1(x).
(3)
(d) Find the range of f
−1(x).
(1)
8. The function f is given by
f :x a ln(3x −6), x ∈° , x > 2.
(a) Find f
−1(x).
(3)
(b) Write down the domain of f
−1
and the range of f
−1.
(2)
(c) Find, to 3 significant figures, the value of x for which f(x) = 3.
(2)
The function g is given by
g:x a ln3x −6 , x ∈° , x ≠ 2.
(d) Sketch the graph of y = g(x).
(3)
(e) Find the exact coordinates of all the points at which the graph of y = g(x) meets the coordinate
axes. (3)The function f is given by
$$f: x \mapsto \ln(3x - 6), \quad x \in \mathbb{R}, \quad x > 2$$
\begin{enumerate}[label=(\alph*)]
\item Find $f^{-1}(x)$. [3]
\item Write down the domain of $f^{-1}$ and the range of $f^{-1}$. [2]
\item Find, to 3 significant figures, the value of $x$ for which f(x) = 3. [2]
\end{enumerate}
The function g is given by
$$g: x \mapsto \ln|3x - 6|, \quad x \in \mathbb{R}, \quad x \neq 2$$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Sketch the graph of $y = g(x)$. [3]
\item Find the exact coordinates of all the points at which the graph of $y = g(x)$ meets the coordinate axes. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q8 [13]}}