Questions - Edexcel C1 - A-Level Maths

Use integration to find $y$ in terms of $x$.
Given that $y = 7$ when $x = 1$, find the value of $y$ at $x = 2$.

Edexcel C1 Q7

7. Each year, for 40 years, Anne will pay money into a savings scheme. In the first year she pays $\pounds 500$. Her payments then increase by $\pounds 50$ each year, so that she pays $\pounds 550$ in the second year, $\pounds 600$ in the third year, and so on.

Find the amount that Anne will pay in the 40th year.
Find the total amount that Anne will pay in over the 40 years. Over the same 40 years, Brian will also pay money into the savings scheme. In the first year he pays in $\pounds 890$ and his payments then increase by $\pounds d$ each year. Given that Brian and Anne will pay in exactly the same amount over the 40 years,
find the value of $d$.

Edexcel C1 Q8

8. The points $A$ and $B$ have coordinates $( 4,6 )$ and $( 12,2 )$ respectively. The straight line $l _ { 1 }$ passes through $A$ and $B$.

Find an equation for $l _ { 1 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integers. The straight line $l _ { 2 }$ passes through the origin and has gradient - 4 .
Write down an equation for $l _ { 2 }$. The lines $l _ { 1 }$ and $l _ { 2 }$ intercept at the point $C$.
Find the exact coordinates of the mid-point of $A C$.

Edexcel C1 Q9

9. A curve $C$ has equation $y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$. The points $P$ and $Q$ lie on $C$. The gradient of $C$ at both $P$ and $Q$ is 2 . The $x$-coordinate of $P$ is 3 .
Find the $x$-coordinate of $Q$.
Find an equation for the tangent to $C$ at $P$, giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants. This tangent intersects the coordinate axes at the points $R$ and $S$.
Find the length of $R S$, giving your answer as a surd.

Edexcel C1 Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{50145adb-ef84-47d4-ad27-294c141d3822-3_625_1000_1546_264} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A ( 3,0 )$ and $B ( 0,4 )$ are two vertices of the rectangle $A B C D$, as shown in Fig. 1.

Write down the gradient of $A B$ and hence the gradient of $B C$. The point $C$ has coordinates $( 8 , k )$, where $k$ is a positive constant.
Find the length of $B C$ in terms of $k$. Given that the length of $B C$ is 10 and using your answer to part (b),
find the value of $k$,
find the coordinates of $D$.

Edexcel C1 Q1

Given that $2 ^ { x } = \frac { 1 } { \sqrt { 2 } }$ and $2 ^ { y } = 4 \sqrt { } 2$,
1. find the exact value of $x$ and the exact value of $y$,
2. calculate the exact value of $2 ^ { y - x }$.
3. $f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$.
4. Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.
5. Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$.
6. The sum of an arithmetic series is $\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$.
7. Write down the first two terms of the series.
8. Find the common difference of the series.
Given that $n = 50$,
find the sum of the series.

Edexcel C1 Q4

4. Find the set of values for $x$ for which

$6 x - 7 < 2 x + 3$,
$\quad 2 x ^ { 2 } - 11 x + 5 < 0$,
both $6 x - 7 < 2 x + 3$ and $2 x ^ { 2 } - 11 x + 5 < 0$.

Edexcel C1 Q5

5. The equation $x ^ { 2 } + 5 k x + 2 k = 0$, where $k$ is a constant, has real roots.

Prove that $k ( 25 k - 8 ) \geq 0$.
Hence find the set of possible values of $k$.
Write down the values of $k$ for which the equation $x ^ { 2 } + 5 k x + 2 k = 0$ has equal roots.

Edexcel C1 Q6

6. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation $\quad u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000$.
In this relation $u _ { n }$ is the number of fish in the lake after $n$ years and $d$ is the number of fish which are caught each year.
Given that $d = 15000$,

calculate $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$ and comment briefly on your results. Given that $d = 100000$,
show that the population of fish dies out during the sixth year.
Find the value of $d$ which would leave the population each year unchanged.

Edexcel C1 Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{813612f1-92c8-456d-84a2-aa6bb91b8a6a-3_689_1077_927_484}

\end{figure} Fig. 1 shows the curve with equation $y ^ { 2 } = 4 ( x - 2 )$ and the line with equation $2 x - 3 y = 12$.
The curve crosses the $x$-axis at the point $A$, and the line intersects the curve at the points $P$ and $Q$.

Write down the coordinates of $A$.
Find, using algebra, the coordinates of $P$ and $Q$.
Show that $\angle P A Q$ is a right angle.

Edexcel C1 Q8

8. The points $A ( - 1 , - 2 ) , B ( 7,2 )$ and $C ( k , 4 )$, where $k$ is a constant, are the vertices of $\triangle A B C$. Angle $A B C$ is a right angle.

Find the gradient of $A B$.
Calculate the value of $k$.
Show that the length of $A B$ may be written in the form $p \sqrt { 5 }$, where $p$ is an integer to be found.
Find the exact value of the area of $\triangle A B C$.
Find an equation for the straight line $l$ passing through $B$ and $C$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

Edexcel C1 Q9

9. The curve $C$ has equation $y = \mathrm { f } ( x )$. Given that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 29$ and that $C$ passes through the point $P ( 2,6 )$,

find $y$ in terms of $x$.
Verify that $C$ passes through the point $( 4,0 )$.
Find an equation of the tangent to $C$ at $P$. The tangent to $C$ at the point $Q$ is parallel to the tangent at $P$.
Calculate the exact $x$-coordinate of $Q$.

Edexcel C1 Q1

(a) Solve the equation $4 x ^ { 2 } + 12 x = 0$.

You are given that $\mathrm { f } ( x ) = 4 x ^ { 2 } + 12 x + c$, where $c$ is a constant.
(b) Given that $\mathrm { f } ( x ) = 0$ has equal roots, find the value of $c$ and hence solve $\mathrm { f } ( x ) = 0$.

Edexcel C1 Q2

2. A sequence is defined by the recurrence relation $u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , n = 1,2,3 , \ldots$, where $a$ is a constant.

Given that $a = 20$ and $u _ { 1 } = 3$, find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$, giving your answers to 2 decimal places.
Given instead that $u _ { 1 } = u _ { 2 } = 3$,
1. calculate the value of $a$,
2. write down the value of $u _ { 5 }$.

Edexcel C1 Q3

3. For the curve $C$ with equation $y = x ^ { 4 } - 8 x ^ { 2 } + 3$,

find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, The point $A$, on the curve $C$, has $x$-coordinate 1 .
Find an equation for the normal to $C$ at $A$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
[0pt] [P1 June 2003 Question 8*]

Edexcel C1 Q4

4. The width of a rectangular sports pitch is $x$ metres, $x > 0$. The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,

form a linear inequality in $x$. Given that the area of the pitch must be greater than $4800 \mathrm {~m} ^ { 2 }$,
form a quadratic inequality in $x$.
by solving your inequalities, find the set of possible values of $x$.

Edexcel C1 Q5

5. The curve $C$ with equation $y = \mathrm { f } ( x )$ is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , x > 0$.

Show that, when $x = 8$, the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ is $9 \sqrt { } 2$. The curve $C$ passes through the point $( 4,30 )$.
Using integration, find $\mathrm { f } ( x )$.

Edexcel C1 Q6

6. (a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$. A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.
(b) Find the value of $d$. Using your value of $d$,
(c) find the predicted profit for the year 2011.

Edexcel C1 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8bae58f7-c53a-43ed-9a1d-2f718bd1e539-3_563_570_785_561} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A ( - 3 , - 2 )$ and $B ( 8,4 )$ are at the ends of a diameter of the circle shown in Fig. 1.

Find the coordinates of the centre of the circle.
Find an equation of the diameter $A B$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Find an equation of tangent to the circle at $B$. The line $l$ passes through $A$ and the origin.
Find the coordinates of the point at which $l$ intersects the tangent to the circle at $B$, giving your answer as exact fractions.

Edexcel C1 Q8

8. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

Write down the maximum value of $\mathrm { f } ( x )$.
Sketch the graph of $y = \mathrm { f } ( x )$, showing the coordinates of the points at which the graph meets the coordinate axes. The points $A$ and $B$ on the graph of $y = \mathrm { f } ( x )$ have coordinates $( - 2 , - 7 )$ and $( 3,8 )$ respectively.
Find, in the form $y = m x + c$, an equation of the straight line through $A$ and $B$.
Find the coordinates of the point at which the line $A B$ crosses the $x$-axis. The mid-point of $A B$ lies on the line with equation $y = k x$, where $k$ is a constant.
Find the value of $k$.

Edexcel C1 Q1

(a) Express $\frac { 21 } { \sqrt { 7 } }$ in the form $k \sqrt { 7 }$.
(b) Express $8 ^ { - \frac { 1 } { 3 } }$ as an exact fraction in its simplest form.
Evaluate

$$\sum _ { r = 10 } ^ { 30 } ( 7 + 2 r ) .$$

Edexcel C1 Q3

Differentiate with respect to $x$

$$\frac { 6 x ^ { 2 } - 1 } { 2 \sqrt { x } } .$$

Edexcel C1 Q4

(a) Solve the inequality

$$x ^ { 2 } + 3 x > 10$$ (b) Find the set of values of $x$ which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3
& x ^ { 2 } + 3 x > 10 \end{aligned}$$

Edexcel C1 Q5

The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by the recurrence relation

$$u _ { n + 1 } = \left( u _ { n } \right) ^ { 2 } - 1 , \quad n \geq 1 .$$ Given that $u _ { 1 } = k$, where $k$ is a constant,

find expressions for $u _ { 2 }$ and $u _ { 3 }$ in terms of $k$. Given also that $u _ { 2 } + u _ { 3 } = 11$,
find the possible values of $k$.

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