Questions - Edexcel C3 - A-Level Maths

Show that $\mathrm { f } ( x ) = 0$ has a root $\alpha$ between $x = 0.75$ and $x = 0.85$ The equation $\mathrm { f } ( x ) = 0$ can be written as $x = [ \arcsin ( 1 - 0.5 x ) ] ^ { \frac { 1 } { 2 } }$.
Use the iterative formula $$x _ { n + 1 } = \left[ \arcsin \left( 1 - 0.5 x _ { n } \right) \right] ^ { \frac { 1 } { 2 } } , \quad x _ { 0 } = 0.8$$ to find the values of $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$, giving your answers to 5 decimal places.
Show that $\alpha = 0.80157$ is correct to 5 decimal places.

Edexcel C3 2012 June Q8

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ Given that $\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$,

find the value of $R$ and the value of $\alpha$.
Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for $0 \leqslant x < 180 ^ { \circ }$, giving your answers to 1 decimal place.
Express $14 \cos ^ { 2 } x - 48 \sin x \cos x$ in the form $a \cos 2 x + b \sin 2 x + c$, where $a , b$, and $c$ are constants to be found.
Hence, using your answers to parts (a) and (c), deduce the maximum value of $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$

Edexcel C3 Q1

Express as a single fraction in its simplest form

$$\frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 9 } \times \frac { 2 x ^ { 2 } + 6 x } { ( x - 5 ) ^ { 2 } }$$

Edexcel C3 Q2

The root of the equation $\mathrm { f } ( x ) = 0$, where

$$f ( x ) = x + \ln 2 x - 4$$ is to be estimated using the iterative formula $x _ { n + 1 } = 4 - \ln 2 x _ { n }$, with $x _ { 0 } = 2.4$.

Showing your values of $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$, obtain the value, to 3 decimal places, of the root.
By considering the change of sign of $\mathrm { f } ( x )$ in a suitable interval, justify the accuracy of your answer to part (a).

Edexcel C3 Q3

3. The function $f$ is defined by $$f : x \text { a } | 2 x - a | , \quad x \in ^ { \circ }$$ where $a$ is a positive constant.

Sketch the graph of $y = \mathrm { f } ( x )$, showing the coordinates of the points where the graph cuts the axes.
On a separate diagram, sketch the graph of $y = \mathrm { f } ( 2 x )$, showing the coordinates of the points where the graph cuts the axes.
Given that a solution of the equation $\mathrm { f } ( x ) = \frac { 1 } { 2 } x$ is $x = 4$, find the two possible values of $a$.

Edexcel C3 Q4

4. Prove that $$\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } \equiv \cos 2 \theta$$ \section*{EDEXCEL CORE MATHEMATICS PRACTICE PAPER 1}

Edexcel C3 Q5

Express $\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }$ as a single fraction in its simplest form.
The function f , defined for $x \in ^ { \circ } , x > 0$, is such that

$$\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 2 + \frac { 1 } { x ^ { 2 } }$$

Find the value of $\mathrm { f } ^ { \prime \prime } ( x )$ at $x = 4$.
Given that $\mathrm { f } ( 3 ) = 0$, find $\mathrm { f } ( x )$.
Prove that f is an increasing function.

Edexcel C3 Q7

7. $\quad \mathrm { f } ( x ) = \frac { 2 } { x - 1 } - \frac { 6 } { ( x - 1 ) ( 2 x + 1 ) } , x > 1$

Prove that $\mathrm { f } ( x ) = \frac { 4 } { 2 x + 1 }$.
Find the range of f.
Find $\mathrm { f } ^ { - 1 } ( x )$.
Find the range of $\mathrm { f } ^ { - 1 } ( x )$.

Edexcel C3 Q8

8. The function f is given by $$f : x \text { a } \ln ( 3 x - 6 ) , \quad x \in ^ { \circ } , \quad x > 2$$

Find $\mathrm { f } ^ { - 1 } ( x )$.
Write down the domain of $\mathrm { f } ^ { - 1 }$ and the range of $\mathrm { f } ^ { - 1 }$.
Find, to 3 significant figures, the value of $x$ for which $\mathrm { f } ( x ) = 3$. The function g is given by $$g : x \text { a } \ln | 3 x - 6 | , \quad x \in ^ { \circ } , \quad x \neq 2$$
Sketch the graph of $y = \mathrm { g } ( x )$.
Find the exact coordinates of all the points at which the graph of $y = \mathrm { g } ( x )$ meets the coordinate axes.

Edexcel C3 Q1

The function $f$ is given by

$$\mathrm { f } : x \propto \frac { x } { x ^ { 2 } - 1 } - \frac { 1 } { x + 1 } , x > 1$$

Show that $\mathrm { f } ( x ) = \frac { 1 } { ( x - 1 ) ( x + 1 ) }$.
Find the range of f. The function $g$ is given by $$\mathrm { g } : x \propto \frac { 2 } { x } , x > 0$$
Solve $\operatorname { gf } ( x ) = 70$.

Edexcel C3 Q3

3. The function f is even and has domain $\mathbb { R }$. For $x \geq 0 , \mathrm { f } ( x ) = x ^ { 2 } - 4 a x$, where $a$ is a positive constant.

In the space below, sketch the curve with equation $y = \mathrm { f } ( x )$, showing the coordinates of all the points at which the curve meets the axes.
Find, in terms of $a$, the value of $\mathrm { f } ( 2 a )$ and the value of $\mathrm { f } ( - 2 a )$. Given that $a = 3$,
use algebra to find the values of $x$ for which $\mathrm { f } ( x ) = 45$.

Edexcel C3 Q4

4. $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 1$$ The equation $\mathrm { f } ( x ) = 0$ has only one positive root, $\alpha$.

Show that $\mathrm { f } ( x ) = 0$ can be rearranged as $$x = \sqrt { \left( \frac { 4 x + 1 } { x + 1 } \right) } , x \neq - 1$$ The iterative formula $x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }$ is used to find an approximation to $\alpha$.
Taking $x _ { 1 } = 1$, find, to 2 decimal places, the values of $x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$.
By choosing values of $x$ in a suitable interval, prove that $\alpha = 1.70$, correct to 2 decimal places.
Write down a value of $x _ { 1 }$ for which the iteration formula $x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }$ does not produce a valid value for $x _ { 2 }$. Justify your answer.

Edexcel C3 Q5

5. The functions $f$ and $g$ are defined by $$\begin{aligned} & \mathrm { f } : x \alpha \quad | x - a | + a , x \in \mathbb { R }
& \mathrm {~g} : x \alpha \quad 4 x + a , \quad x \in \mathbb { R } \end{aligned}$$ where $a$ is a positive constant.

On the same diagram, sketch the graphs of f and g , showing clearly the coordinates of any points at which your graphs meet the axes.
Use algebra to find, in terms of $a$, the coordinates of the point at which the graphs of f and g intersect.
Find an expression for $\mathrm { fg } ( x )$.
Solve, for $x$ in terms of $a$, the equation $$\mathrm { fg } ( x ) = 3 a$$ \section*{6.}

Edexcel C3 Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{79c1c4df-3812-4295-b01d-4724eda3457d-4_656_791_315_386}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = 10 + \ln ( 3 x ) - \frac { 1 } { 2 } \mathrm { e } ^ { x } , 0.1 \leq x \leq 3.3 .$$ Given that $\mathrm { f } ( k ) = 0$,

show, by calculation, that $3.1 < k < 3.2$.
Find $\mathrm { f } ^ { \prime } ( x )$. The tangent to the graph at $x = 1$ intersects the $y$-axis at the point $P$.
1. Find an equation of this tangent.
2. Find the exact $y$-coordinate of $P$, giving your answer in the form $a + \ln b$.

Edexcel C3 Q7

7. (a) Express $\sin x + \sqrt { 3 } \cos x$ in the form $R \sin ( x + \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$.
(b) Show that the equation $\sec x + \sqrt { 3 } \operatorname { cosec } x = 4$ can be written in the form $$\sin x + \sqrt { 3 } \cos x = 2 \sin 2 x$$ (c) Deduce from parts (a) and (b) that $\sec x + \sqrt { 3 } \operatorname { cosec } x = 4$ can be written in the form $$\sin 2 x - \sin \left( x + 60 ^ { \circ } \right) = 0$$ END

Edexcel C3 Q1

\begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{909b52e5-2f16-4eab-b691-9d8fcf9bcfd9-2_679_1189_516_520}

\end{figure} Figure 2 shows part of the curve $C$ with equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = 0.5 \mathrm { e } ^ { x } - x ^ { 2 }$$ The curve $C$ cuts the $y$-axis at $A$ and there is a minimum at the point $B$.

Find an equation of the tangent to $C$ at $A$. The $x$-coordinate of $B$ is approximately 2.15 . A more exact estimate is to be made of this coordinate using iterations $x _ { n + 1 } = \ln \mathrm { g } \left( x _ { n } \right)$.
Show that a possible form for $\mathrm { g } ( x )$ is $\mathrm { g } ( x ) = 4 x$.
Using $x _ { n + 1 } = \ln 4 x _ { n }$, with $x _ { 0 } = 2.15$, calculate $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$. Give the value of $x _ { 3 }$ to 4 decimal places.

Edexcel C3 Q3

3. (a) Sketch the graph of $y = | 2 x + a | , a > 0$, showing the coordinates of the points where the graph meets the coordinate axes.
(b) On the same axes, sketch the graph of $y = \frac { 1 } { x }$.
(c) Explain how your graphs show that there is only one solution of the equation $$x | 2 x + a | - 1 = 0$$ (d) Find, using algebra, the value of $x$ for which $x | 2 x + 1 | - 1 = 0$.

Edexcel C3 Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{909b52e5-2f16-4eab-b691-9d8fcf9bcfd9-3_604_1408_868_269}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x ) , - 1 \leq x \leq 3$. 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 $B \left( \frac { 4 } { 3 } , 1 \right)$. In separate diagrams, show a sketch of the curve with equation

$y = \mathrm { f } ( x + 1 )$,
$y = | \mathrm { f } ( x ) |$,
$y = \mathrm { f } ( | x | )$,
marking on each sketch the coordinates of points at which the curve
1. has a turning point,
2. meets the $x$-axis.

Edexcel C3 Q5

5. (i) Given that $\sin x = \frac { 3 } { 5 }$, use an appropriate double angle formula to find the exact value of $\sec 2 x$.
(ii) Prove that $$\cot 2 x + \operatorname { cosec } 2 x \equiv \cot x , \quad \left( x \neq \frac { n \pi } { 2 } , n \in Z \right)$$

Edexcel C3 Q6

The function f is defined by $\mathrm { f } : x \rightarrow \frac { 3 x - 1 } { x - 3 } , x \in j , x \neq 3$.
1. Prove that $\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )$ for all $x \in j , x \neq 3$.
2. Hence find, in terms of $k , \operatorname { ff } ( k )$, where $x \neq 3$.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{909b52e5-2f16-4eab-b691-9d8fcf9bcfd9-5_864_1205_605_242}
\end{figure} Figure 3 shows a sketch of the one-one function g , defined over the domain $- 2 \leq x \leq 2$.
Find the value of $\mathrm { fg } ( - 2 )$.
Sketch the graph of the inverse function $\mathrm { g } ^ { - 1 }$ and state its domain. The function h is defined by $\mathrm { h } : x \mapsto 2 \mathrm {~g} ( x - 1 )$.
Sketch the graph of the function h and state its range.

Edexcel C3 Q7

7. (i) (a) Express $( 12 \cos \theta - 5 \sin \theta )$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$.
(b) Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4$$ for $0 < \theta < 90 ^ { \circ }$, giving your answer to 1 decimal place.
(ii) Solve $$8 \cot \theta - 3 \tan \theta = 2$$ for $0 < \theta < 90 ^ { \circ }$, giving your answer to 1 decimal place.

Edexcel C3 Q8

8. The curve $C$ has equation $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = 3 \ln x + \frac { 1 } { x } , \quad x > 0$$ The point $P$ is a stationary point on $C$.

Calculate the $x$-coordinate of $P$.
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. The point $Q$ on $C$ has $x$-coordinate 1 .
Find an equation for the normal to $C$ at $Q$. The normal to $C$ at $Q$ meets $C$ again at the point $R$.
Show that the $x$-coordinate of $R$
1. satisfies the equation $6 \ln x + x + \frac { 2 } { x } - 3 = 0$,
2. lies between 0.13 and 0.14 .

Edexcel C3 Q1

The curve $C$ has equation $y = 2 \mathrm { e } ^ { x } + 3 x ^ { 2 } + 2$. The point $A$ with coordinates $( 0,4 )$ lies on $C$. Find the equation of the tangent to $C$ at $A$.
Express $\frac { x } { ( x + 1 ) ( x + 3 ) } + \frac { x + 12 } { x ^ { 2 } - 9 }$ as a single fraction in its simplest form.
The functions f and g are defined by

$$\begin{aligned} & \mathrm { f } : x \propto x ^ { 2 } - 2 x + 3 , x \in \mathbb { R } , 0 \leq x \leq 4
& \mathrm {~g} : x \propto \lambda x ^ { 2 } + 1 , \text { where } \lambda \text { is a constant, } x \in \mathbb { R } . \end{aligned}$$

Find the range of f.
Given that $\operatorname { gf } ( 2 ) = 16$, find the value of $\lambda$.

Edexcel C3 Q4

4. (a) Sketch, on the same set of axes, the graphs of $$y = 2 - \mathrm { e } ^ { - x } \text { and } y = \sqrt { } x$$ [It is not necessary to find the coordinates of any points of intersection with the axes.]
Given that $\mathrm { f } ( x ) = \mathrm { e } ^ { - x } + \sqrt { } x - 2 , x \geq 0$,
(b) explain how your graphs show that the equation $\mathrm { f } ( x ) = 0$ has only one solution,
(c) show that the solution of $\mathrm { f } ( x ) = 0$ lies between $x = 3$ and $x = 4$. The iterative formula $x _ { n + 1 } = \left( 2 - \mathrm { e } ^ { - x _ { n } } \right) ^ { 2 }$ is used to solve the equation $\mathrm { f } ( x ) = 0$.
(d) Taking $x _ { 0 } = 4$, write down the values of $x _ { 1 } , x _ { 2 } , x _ { 3 }$ and $x _ { 4 }$, and hence find an approximation to the solution of $\mathrm { f } ( x ) = 0$, giving your answer to 3 decimal places.

Edexcel C3 Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{bdb2b50d-0816-4ef2-adfa-aee3afe18582-3_515_739_228_534}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { e } ^ { - x } - 1$.

Copy Fig. 1 and on the same axes sketch the graph of $y = \frac { 1 } { 2 } | x - 1 |$. Show the coordinates of the points where the graph meets the axes. The $x$-coordinate of the point of intersection of the graph is $\alpha$.
Show that $x = \alpha$ is a root of the equation $x + 2 \mathrm { e } ^ { - x } - 3 = 0$.
Show that $- 1 < \alpha < 0$. The iterative formula $x _ { \mathrm { n } + 1 } = - \ln \left[ \frac { 1 } { 2 } \left( 3 - x _ { n } \right) \right]$ is used to solve the equation $x + 2 \mathrm { e } ^ { - x } - 3 = 0$.
Starting with $x _ { 0 } = - 1$, find the values of $x _ { 1 }$ and $x _ { 2 }$.
Show that, to 2 decimal places, $\alpha = - 0.58$.

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