Questions - Edexcel - A-Level Maths

Edexcel F3 Specimen Q4

8 marks Challenging +1.2

4. $I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a > 0 , \quad n \geqslant 0$

Show that, for $n \geqslant 2$, $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
Hence evaluate $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x d x$

Edexcel F3 Specimen Q5

9 marks Challenging +1.2

5. Given that $y = ( \operatorname { arcosh } 3 x ) ^ { 2 }$, where $3 x > 1$, show that

$\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y$,
$\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18$.

Edexcel F3 Specimen Q6

13 marks Standard +0.3

6. $\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)$, where $k$ is a constant.
Given that $\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)$ is an eigenvector of $\mathbf { M }$ ,

find the eigenvalue of $\mathbf { M }$ corresponding to $\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)$ ,
show that $k = 3$ ,
show that $\mathbf { M }$ has exactly two eigenvalues. A transformation $T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ is represented by $\mathbf { M }$ .
The transformation $T$ maps the line $l _ { 1 }$ ,with cartesian equations $\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }$ ,onto the line $l _ { 2 }$ .
6. $\mathbf { M } = \left( \begin{array} { c c c } 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)$, where $k$ is a constant. Given that $\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)$ is an eigenvector of $\mathbf { M }$ ,
1. find the eigenvalue of $\mathbf { M }$ corresponding to $\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)$
2. Taking $k = 3$ ,find cartesian equations of $l _ { 2 }$ .

Edexcel F3 Specimen Q7

14 marks Standard +0.8

The plane $\Pi$ has vector equation

$$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$

Find an equation of $\Pi$ in the form $\mathbf { r } . \mathbf { n } = p$, where $\mathbf { n }$ is a vector perpendicular to $\Pi$ and $p$ is a constant. The point $P$ has coordinates $( 6,13,5 )$. The line $l$ passes through $P$ and is perpendicular to $\Pi$. The line $l$ intersects $\Pi$ at the point $N$.
Show that the coordinates of $N$ are $( 3,1 , - 1 )$. The point $R$ lies on $\Pi$ and has coordinates $( 1,0,2 )$.
Find the perpendicular distance from $N$ to the line $P R$. Give your answer to 3 significant figures.

Edexcel F3 Specimen Q8

13 marks Challenging +1.3

The hyperbola $H$ has equation $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$.

The line $l _ { 1 }$ is the tangent to $H$ at the point $P ( 4 \sec t , 2 \tan t )$.

Use calculus to show that an equation of $l _ { 1 }$ is $$2 y \sin t = x - 4 \cos t$$ The line $l _ { 2 }$ passes through the origin and is perpendicular to $l _ { 1 }$.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $Q$.
Show that, as $t$ varies, an equation of the locus of $Q$ is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

Edexcel FP3 Q1

6 marks Standard +0.8

Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form $\ln a$, where $a$ is a rational number.

Edexcel FP3 Q2

7 marks Standard +0.3

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{045545c7-06d9-40b6-9d01-fc792ab0aa07-01_222_241_525_2042} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ respectively, relative to a fixed origin $O$, as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate

$\mathbf { b } \times \mathbf { c }$,
$\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )$,
the area of triangle $O B C$,
the volume of the tetrahedron $O A B C$.

Edexcel FP3 Q4

8 marks Challenging +1.2

4. Given that $y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0$,

find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer as a simplified fraction.
Hence, or otherwise, find $$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form $\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)$, where $a$ and $b$ are integers.

Edexcel FP3 Q5

4 marks Challenging +1.3

5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$

Find an expression for $\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5$.
Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
Find $I _ { 4 }$ in the form $k \pi$, where $k$ is a fraction.

Edexcel FP3 Q6

10 marks Challenging +1.2

6. The hyperbola $H$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $a$ and $b$ are constants. The line $L$ has equation $y = m x + c$, where $m$ and $c$ are constants.

Given that $L$ and $H$ meet, show that the $x$-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that $L$ is a tangent to $H$,
show that $a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }$. The hyperbola $H ^ { \prime }$ has equation $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1$.
Find the equations of the tangents to $H ^ { \prime }$ which pass through the point $( 1,4 )$.

Edexcel FP3 Q7

9 marks Standard +0.3

7. The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines $l _ { 1 }$ and $l _ { 2 }$ intersect, find

the value of $\alpha$,
an equation for the plane containing the lines $l _ { 1 }$ and $l _ { 2 }$, giving your answer in the form $a x + b y + c z + d = 0$, where $a , b , c$ and $d$ are constants. For other values of $\alpha$, the lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect and are skew lines.
Given that $\alpha = 2$,
find the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$.

Edexcel FP3 Q8

8 marks Challenging +1.8

8. A curve, which is part of an ellipse, has parametric equations $$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The curve is rotated through $2 \pi$ radians about the $x$-axis.

Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { a } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \text { where } c = \cos \theta$$ and where $k$ and $\alpha$ are constants to be found.
Using the substitution $c = \frac { 3 } { 4 } \sinh u$, or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.

Edexcel FP3 Q9

8 marks Challenging +1.8

9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$

Show that, for $n > 0$, $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
Find $I _ { 2 }$.

Edexcel M1 Q3

Moderate -0.8

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-005_851_1073_312_456}

\end{figure} A sprinter runs a race of 200 m . Her total time for running the race is 25 s . Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in 4 s . The speed of $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is maintained for 16 s and she then decelerates uniformly to a speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the end of the race. Calculate

the distance covered by the sprinter in the first 20 s of the race,
the value of $u$,
the deceleration of the sprinter in the last 5 s of the race.

Edexcel M1 Q4

Moderate -0.8

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-007_330_675_287_644}

\end{figure} A particle $P$ of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude $X$ newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at $20 ^ { \circ }$ to the horizontal. The coefficient of friction between $P$ and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate

the normal reaction of the plane on $P$,
the value of $X$. The force of magnitude $X$ newtons is now removed.
Show that $P$ remains in equilibrium on the plane.

Edexcel M1 Q5

Standard +0.3

5. Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516} A block of wood $A$ of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley $P$ fixed at the edge of the table. The other end of the string is attached to a ball $B$ of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between $A$ and the table is $\mu$. The system is released from rest with the string taut. After release, $B$ descends a distance of 0.4 m in 0.5 s . Modelling $A$ and $B$ as particles, calculate

the acceleration of $B$,
the tension in the string,
the value of $\mu$.
State how in your calculations you have used the information that the string is inextensible.

Edexcel M1 Q7

Moderate -0.3

7. Two ships $P$ and $Q$ are travelling at night with constant velocities. At midnight, $P$ is at the point with position vector $( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }$ relative to a fixed origin $O$. At the same time, $Q$ is at the point with position vector $( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }$. Three hours later, $P$ is at the point with position vector $( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }$. The ship $Q$ travels with velocity $12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }$. At time $t$ hours after midnight, the position vectors of $P$ and $Q$ are $\mathbf { p } \mathrm { km }$ and $\mathbf { q } \mathrm { km }$ respectively. Find

the velocity of $P$, in terms of $\mathbf { i }$ and $\mathbf { j }$,
expressions for $\mathbf { p }$ and $\mathbf { q }$, in terms of $t$, i and $\mathbf { j }$. At time $t$ hours after midnight, the distance between $P$ and $Q$ is $d \mathrm {~km}$.
By finding an expression for $\overrightarrow { P Q }$, show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ Weather conditions are such that an observer on $P$ can only see the lights on $Q$ when the distance between $P$ and $Q$ is 15 km or less. Given that when $t = 1$, the lights on $Q$ move into sight of the observer,
find the time, to the nearest minute, at which the lights on $Q$ move out of sight of the observer.
1. In taking off, an aircraft moves on a straight runway $A B$ of length 1.2 km . The aircraft moves from $A$ with initial speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. It moves with constant acceleration and 20 s later it leaves the runway at $C$ with speed $74 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
  1. the acceleration of the aircraft,
  2. the distance $B C$.
  3. Two small steel balls $A$ and $B$ have mass 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of $A$ is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the speed of $B$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Immediately after the collision, the direction of motion of $A$ is unchanged and the speed of $B$ is twice the speed of $A$. Find
2. the speed of $A$ immediately after the collision,
3. the magnitude of the impulse exerted on $B$ in the collision.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
\end{figure}

Edexcel M1 Q8

Moderate -0.3

\hspace{0pt} [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are horizontal vectors due east and north respectively.]

Edexcel M2 Q3

11 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}

\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. 2.

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 M2 Q4

5 marks Moderate -0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-03_725_560_310_571} \captionsetup{labelformat=empty} \caption{Fig. 4}

\end{figure} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions $2 x \mathrm {~cm}$ by $x \mathrm {~cm}$ and height $h \mathrm {~cm}$, as shown in Fig. 4. Given that the capacity of a carton has to be $1030 \mathrm {~cm} ^ { 3 }$,

express $h$ in terms of $x$,
show that the surface area, $A \mathrm {~cm} ^ { 2 }$, of a carton is given by $$A = 4 x ^ { 2 } + \frac { 3090 } { x } .$$

Edexcel M2 Q16

13 marks Moderate -0.8

16. \section*{Figure 3}

\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-08_581_575_395_609}

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

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 M2 Q20

14 marks Moderate -0.5

20. 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 $Q$.
21. $$y = 7 + 10 x ^ { \frac { 3 } { 2 } }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Find $\int y \mathrm {~d} x$.

22.

Given that $3 ^ { x } = 9 ^ { y - 1 }$, show that $x = 2 y - 2$.
Solve the simultaneous equations $$\begin{gathered} x = 2 y - 2 \\ x ^ { 2 } = y ^ { 2 } + 7 \end{gathered}$$
1. The straight line $l _ { 1 }$ with equation $y = \frac { 3 } { 2 } x - 2$ crosses the $y$-axis at the point $P$. The point $Q$ has coordinates $( 5 , - 3 )$.
The straight line $l _ { 2 }$ is perpendicular to $l _ { 1 }$ and passes through $Q$.

Calculate the coordinates of the mid-point of $P Q$.
Find an equation for $l _ { 2 }$ in the form $a x + b y = c$, where $a$, b and $c$ are integer constants. The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $R$.
Calculate the exact coordinates of $R$.
24. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 + \frac { 1 } { x ^ { 2 } } .$$

Use integration to find $y$ in terms of $x$.
Given that $y = 7$ when $x = 1$, find the value of $y$ at $x = 2$.
25. Find the set of values for $x$ for which

$6 x - 7 < 2 x + 3$,
$2 x ^ { 2 } - 11 x + 5 < 0$,
both $6 x - 7 < 2 x + 3$ and $2 x ^ { 2 } - 11 x + 5 < 0$.
[0pt] [P1 June 2003 Question 2]
26. In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by $x$ phones per month for the next 35 months, so that $( 280 + x )$ phones will be sold in the second month, $( 280 + 2 x )$ in the third month, and so on. Using this model with $x = 5$, calculate

1. the number of phones sold in the 36th month,
2. the total number of phones sold over the 36 months. The shop sets a sales target of 17000 phones to be sold over the 36 months.
  Using the same model,
find the least value of $x$ required to achieve this target.
[0pt] [P1 June 2003 Question 3]
27. 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 $A C$.
28. 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*]
29. The sum of an arithmetic series is $$\sum _ { r = 1 } ^ { n } ( 80 - 3 r )$$

Write down the first two terms of the series.
Find the common difference of the series. Given that $n = 50$,
find the sum of the series.

30.

Solve the equation $4 x ^ { 2 } + 12 x = 0$. $$f ( x ) = 4 x ^ { 2 } + 12 x + c$$ where $c$ is a constant.
Given that $\mathrm { f } ( x ) = 0$ has equal roots, find the value of $c$ and hence solve $\mathrm { f } ( x ) = 0$.
31. Solve the simultaneous equations $$\begin{aligned} & x - 3 y + 1 = 0 \\ & x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{aligned}$$
1. A container made from thin metal is in the shape of a right circular cylinder with height $h \mathrm {~cm}$ and base radius $r \mathrm {~cm}$. The container has no lid. When full of water, the container holds $500 \mathrm {~cm} ^ { 3 }$ of water.
Show that the exterior surface area, $A \mathrm {~cm} ^ { 2 }$, of the container is given by $$A = \pi r ^ { 2 } + \frac { 1000 } { r } .$$ 33. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-15_668_748_358_699}
The points $A$ and $B$ have coordinates $( 2 , - 3 )$ and $( 8,5 )$ respectively, and $A B$ is a chord of a circle with centre $C$, as shown in Fig. 1.

Find the gradient of $A B$. The point $M$ is the mid-point of $A B$.
Find an equation for the line through $C$ and $M$. Given that the $x$-coordinate of $C$ is 4 ,
find the $y$-coordinate of $C$,
show that the radius of the circle is $\frac { 5 \sqrt { } 17 } { 4 }$.
34. The first three terms of an arithmetic series are $p , 5 p - 8$, and $3 p + 8$ respectively.

Show that $p = 4$.
Find the value of the 40th term of this series.
Prove that the sum of the first $n$ terms of the series is a perfect square.
35. $$\mathrm { f } ( x ) = x ^ { 2 } - k x + 9 , \text { where } k \text { is a constant. }$$

Find the set of values of $k$ for which the equation $\mathrm { f } ( x ) = 0$ has no real solutions. Given that $k = 4$,
express $\mathrm { f } ( x )$ in the form $( x - p ) ^ { 2 } + q$, where $p$ and $q$ are constants to be found,
36. The curve $C$ with equation $y = \mathrm { f } ( x )$ is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 \sqrt { } x + \frac { 12 } { \sqrt { } x } , \quad x > 0 .$$

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 $\mathrm { f } ( x )$.
37. \section*{Figure 2}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-17_687_1074_351_539}
Figure 2 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.
38. A sequence is defined by the recurrence relation $$u _ { n + 1 } = \sqrt { \left( \frac { u _ { n } } { 2 } + \frac { a } { u _ { n } } \right) } , \quad n = 1,2,3 , \ldots ,$$ 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 $u _ { 5 }$.
  [0pt] [P2 January 2004 Question 2]
  39. The points $A$ and $B$ have coordinates $( 1,2 )$ and $( 5,8 )$ respectively.

Find the coordinates of the mid-point of $A B$.
Find, in the form $y = m x + c$, an equation for the straight line through $A$ and $B$.
40. Giving your answers in the form $a + b \sqrt { 2 }$, where $a$ and $b$ are rational numbers, find

$( 3 - \sqrt { } 8 ) ^ { 2 }$,
$\frac { 1 } { 4 - \sqrt { 8 } }$.
41. 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 $x$.
42. The curve $C$ has equation $y = x ^ { 2 } - 4$ and the straight line $l$ has equation $y + 3 x = 0$.

In the space below, sketch $C$ and $l$ on the same axes.
Write down the coordinates of the points at which $C$ meets the coordinate axes.
Using algebra, find the coordinates of the points at which $l$ intersects $C$.
43. $$f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0$$

Show that $\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }$.
Hence, or otherwise, differentiate $\mathrm { f } ( x )$ with respect to $x$.

Edexcel S1 2022 January Q1

11 marks Easy -1.2

A factory produces shoes.

A quality control inspector at the factory checks a sample of 120 shoes for each of three types of defect. The Venn diagram represents the inspector's results. A represents the event that a shoe has defective stitching $B$ represents the event that a shoe has defective colouring $C$ represents the event that a shoe has defective soles \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-02_684_935_607_566} One of the shoes in the sample is selected at random.

Find the probability that it does not have defective soles.
Find $\mathrm { P } \left( A \cap B \cap C ^ { \prime } \right)$
Find $\mathrm { P } \left( A \cup B \cup C ^ { \prime } \right)$
Find the probability that the shoe has at most one type of defect.
Given the selected shoe has at most one type of defect, find the probability it has defective stitching. The random variable $X$ is the number of the events $A , B , C$ that occur for a randomly selected shoe.
Find $\mathrm { E } ( X )$ \section*{This is a copy of the Venn diagram for this question.} \includegraphics[max width=\textwidth, alt={}, center]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-05_684_940_388_566}

Edexcel S1 2022 January Q2

6 marks Moderate -0.8

2. Tom's car holds 50 litres of petrol when the fuel tank is full. For each of 10 journeys, each starting with 50 litres of petrol in the fuel tank, Tom records the distance travelled, $d$ kilometres, and the amount of petrol used, $p$ litres. The summary statistics for the 10 journeys are given below. $$\sum d = 1029 \quad \sum p = 50.8 \quad \sum d p = 5240.8 \quad \mathrm {~S} _ { d d } = 344.9 \quad \mathrm {~S} _ { p p } = 0.576$$

Calculate the product moment correlation coefficient between $d$ and $p$ The amount of petrol remaining in the fuel tank for each journey, $w$ litres, is recorded.
1. Write down an equation for $w$ in terms of $p$
2. Hence, write down the value of the product moment correlation coefficient between $w$ and $p$
Write down the value of the product moment correlation coefficient between $d$ and $w$

Edexcel S1 2022 January Q3

10 marks Moderate -0.8

The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days

\begin{table}[h]

\captionsetup{labelformat=empty} \caption{Key: 10 $\mathbf { 8 }$ represents 108 deliveries}

10	8	9										(2)
11	0	3	6	6	6	8	8	9	9	9	9	(11)
12	4	5	5	5	5	5	5	8				(8)
13	$a$	$b$	$c$									(3)

\end{table} where $a$, $b$ and $c$ are positive integers with $a < b < c$ An outlier is defined as any value greater than $1.5 \times$ interquartile range above the upper quartile. Given that there is only one outlier for these data,

show that $c = 9$ The number of deliveries made by Pat each day is represented by $d$ The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
Find the mean number of deliveries.
Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable $D$ represents the number of deliveries made by Pat on this day. The random variable $X = D - 125$
Find $\mathrm { P } ( D > 118 \mid X < 0 )$

Questions — Edexcel (10514 questions)

Browse by module

Browse by board