Questions - Edexcel C1 - A-Level Maths

Find an equation for $l _ { 1 }$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers. The points $C$ and $D$ lie on the straight line $l _ { 2 }$ which has the equation $x - 4 y - 21 = 0$.
Show that the distance between $l _ { 1 }$ and $l _ { 2 }$ is $k \sqrt { 17 }$, where $k$ is an integer to be found.
Find the area of parallelogram $A B C D$.

Edexcel C1 Q1

The points $A , B$ and $C$ have coordinates $( - 3,0 ) , ( 5 , - 2 )$ and $( 4,1 )$ respectively.

Find an equation for the straight line which passes through $C$ and is parallel to $A B$.
Give your answer in the form $a x + b y = c$, where $a$, $b$ and $c$ are integers.

Edexcel C1 Q2

2. Express $\sqrt { 22.5 }$ in the form $k \sqrt { 10 }$.

Edexcel C1 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{01488c70-db95-43cb-9216-23d7dbaaf9fe-2_549_944_708_347} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve has a maximum at $( - 3,4 )$ and a minimum at $( 1 , - 2 )$. Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

$y = 2 \mathrm { f } ( x )$,
$y = - \mathrm { f } ( x )$.

Edexcel C1 Q4

4. (a) Solve the inequality $$4 ( x - 2 ) < 2 x + 5$$ (b) Find the value of $y$ such that $$4 ^ { y + 1 } = 8 ^ { 2 y - 1 } .$$

Edexcel C1 Q5

A sequence of terms $\left\{ t _ { n } \right\}$ is defined for $n \geq 1$ by the recurrence relation

$$t _ { n + 1 } = k t _ { n } - 7 , \quad t _ { 1 } = 3$$ where $k$ is a constant.

Find expressions for $t _ { 2 }$ and $t _ { 3 }$ in terms of $k$. Given that $t _ { 3 } = 13$,
find the possible values of $k$.

Edexcel C1 Q6

6. The curve with equation $y = \sqrt { 8 x }$ passes through the point $A$ with $x$-coordinate 2. Find an equation for the tangent to the curve at $A$.

Edexcel C1 Q7

7. As part of a new training programme, Habib decides to do sit-ups every day. He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.

Find the number of sit-ups that Habib will do in the fifth week.
Show that he will do a total of 1512 sit-ups during the first eight weeks. In the $n$th week of training, the number of sit-ups that Habib does is greater than 300 for the first time.
Find the value of $n$.

Edexcel C1 Q8

8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, $A \mathrm {~cm} ^ { 2 }$, after $t$ seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ where $p$ and $q$ are positive constants.
Given that when $t = 0 , A = 4$ and that when $t = 5 , A = 9$,

find the value of $p$ and show that $q = \frac { 1 } { 5 }$,
find $\frac { \mathrm { d } A } { \mathrm {~d} t }$ in terms of $t$,
find the rate at which the area of the stain is increasing when $t = 15$.

Edexcel C1 Q9

9. The curve $C$ has the equation $y = x ^ { 2 } + 2 x + 4$.

Express $x ^ { 2 } + 2 x + 4$ in the form $a ( x + b ) ^ { 2 } + c$ and hence state the coordinates of the minimum point of $C$. The straight line $l$ has the equation $x + y = 8$.
Sketch $l$ and $C$ on the same set of axes.
Find the coordinates of the points where $I$ and $C$ intersect.

Edexcel C1 Q10

10. The curve $C$ has the equation $y = \mathrm { f } ( x )$. Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0 ,$$ and that the point $A$ on $C$ has coordinates (2, 6),

find an equation for $C$,
find an equation for the tangent to $C$ at $A$, giving your answer in the form $a x + b y + c = 0$ where $a , b$ and $c$ are integers,
show that the line $y = x + 3$ is also a tangent to $C$.

Edexcel C1 Q1

Find the value of $y$ such that

$$4 ^ { y + 3 } = 8 .$$

Edexcel C1 Q2

Find

$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$

Edexcel C1 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c4ae1bec-12f4-492d-8027-bba4840ff545-2_337_1235_781_383} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the rectangles $A B C D$ and $E F G H$ which are similar.
Given that $A B = ( 3 - \sqrt { 5 } ) \mathrm { cm } , A D = \sqrt { 5 } \mathrm {~cm}$ and $E F = ( 1 + \sqrt { 5 } ) \mathrm { cm }$, find the length $E H$ in cm, giving your answer in the form $a + b \sqrt { 5 }$ where $a$ and $b$ are integers.

Edexcel C1 Q4

4. (a) Sketch on the same diagram the curves $y = x ^ { 2 } - 4 x$ and $y = - \frac { 1 } { x }$.
(b) State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

Edexcel C1 Q5

(a) By completing the square, find in terms of the constant $k$ the roots of the equation

$$x ^ { 2 } + 2 k x + 4 = 0 .$$ (b) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$

Edexcel C1 Q6

(a) Evaluate

$$\sum _ { r = 1 } ^ { 50 } ( 80 - 3 r )$$ (b) Show that $$\sum _ { r = 1 } ^ { n } \frac { r + 3 } { 2 } = k n ( n + 7 )$$ where $k$ is a rational constant to be found.

Edexcel C1 Q8

Given that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 3 } - 4 } { x ^ { 3 } } , \quad x \neq 0$$

find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$. Given also that $y = 0$ when $x = - 1$,
find the value of $y$ when $x = 2$.

Edexcel C1 Q9

9. A curve has the equation $y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0$.

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }$. The point $P$ on the curve has $x$-coordinate 4 .
Find an equation for the normal to the curve at $P$ in the form $y = m x + c$.
Show that the normal to the curve at $P$ does not intersect the curve again.

Edexcel C1 Q10

10. The straight line $l$ has gradient 3 and passes through the point $A ( - 6,4 )$.

Find an equation for $l$ in the form $y = m x + c$. The straight line $m$ has the equation $x - 7 y + 14 = 0$.
Given that $m$ crosses the $y$-axis at the point $B$ and intersects $l$ at the point $C$,
find the coordinates of $B$ and $C$,
show that $\angle B A C = 90 ^ { \circ }$,
find the area of triangle $A B C$.

Edexcel C1 Q1

Evaluate $49 ^ { \frac { 1 } { 2 } } + 8 ^ { \frac { 2 } { 3 } }$.
A sequence is defined by the recurrence relation

$$u _ { n + 1 } = \frac { u _ { n } + 1 } { 3 } , \quad n = 1,2,3 , \ldots$$ Given that $u _ { 3 } = 5$,

find the value of $u _ { 4 }$,
find the value of $u _ { 1 }$.

Edexcel C1 Q3

3. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9$$

Determine the number of real roots that exist for the equation $\mathrm { f } ( x ) = 0$.
Solve the equation $\mathrm { f } ( x ) = 8$, giving your answers in the form $a + b \sqrt { 2 }$ where $a$ and $b$ are rational.

Edexcel C1 Q4

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

$6 x - 11 > x + 4$,
$x ^ { 2 } - 6 x - 16 < 0$,
both $6 x - 11 > x + 4$ and $x ^ { 2 } - 6 x - 16 < 0$.

Edexcel C1 Q5

5. $$f ( x ) = ( 2 - \sqrt { x } ) ^ { 2 } , \quad x > 0$$

Solve the equation $\mathrm { f } ( x ) = 0$.
Find $\mathrm { f } ( 3 )$, giving your answer in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.
Find $$\int \mathrm { f } ( x ) \mathrm { d } x$$

Edexcel C1 Q6

The straight line $l$ passes through the point $P ( - 3,6 )$ and the point $Q ( 1 , - 4 )$.
1. Find an equation for $l$ in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
The straight line $m$ has the equation $2 x + k y + 7 = 0$, where $k$ is a constant.
Given that $l$ and $m$ are perpendicular,
find the value of $k$.

Questions — Edexcel C1 (490 questions)

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