Questions - Edexcel F1 - A-Level Maths

Show, by using calculus, that the normal to $H$ at the point $P$ has equation $$y = p ^ { 2 } x + \frac { 12 } { p } - 12 p ^ { 3 }$$ Given that the normal through $P$ crosses the positive $x$-axis at the point $Q$ and the negative $y$-axis at the point $R$,
find the coordinates of $Q$ and the coordinates of $R$, giving your answers in terms of $p$.
Given also that the area of triangle $O Q R$ is 512 , find the possible values of $p$.

VIUV SIHI NI JIIIM ION OC VI4V SIHI NI JINM IONOO VJYV SIHI NI GLIYM LON OO

Edexcel F1 2020 June Q1

1. $$f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval[1.4,1.5]
Determine $\mathrm { f } ^ { \prime } ( x )$ .
Using $x _ { 0 } = 1.4$ as a first approximation to $\alpha$ ,apply the Newton-Raphson procedure once to $\mathrm { f } ( x )$ to calculate a second approximation to $\alpha$ ,giving your answer to 3 decimal places.
$f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0$
Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval[1.4,1.5]
Determine $\mathrm { f } ^ { \prime } ( x )$ .

Edexcel F1 2020 June Q2

2
2. The quadratic equation $$5 x ^ { 2 } - 2 x + 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
determine, giving each answer as a simplified fraction, the value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
determine a quadratic equation that has roots $$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.

Edexcel F1 2020 June Q3

3. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where $a , b , c$ and $d$ are integers.
The complex numbers $3 + \mathrm { i }$ and $- 1 - 2 \mathrm { i }$ are roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

Write down the other roots of this equation.
Show all the roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$ on a single Argand diagram.
Determine the values of $a , b , c$ and $d$.
VILU SIHI NI JIIIM ION OC VIUV SIHI NI III M M I ON OO VIAV SIHI NI JIIIM I ION OC

Edexcel F1 2020 June Q4

4. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ for all positive integers $n$.
(b) Hence find the exact value of the sum of the squares of the odd numbers between 200 and 500 \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-13_2255_50_314_34}

VIXV SIHIANI III IM IONOO

VIAV SIHI NI JYHAM ION OO

VI4V SIHI NI JLIYM ION OO

Edexcel F1 2020 June Q5

The rectangular hyperbola $H$ has equation $x y = 64$

The point $P \left( 8 p , \frac { 8 } { p } \right)$, where $p \neq 0$, lies on $H$.

Use calculus to show that the normal to $H$ at $P$ has equation $$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$ The normal to $H$ at $P$ meets $H$ again at the point $Q$.
Determine, in terms of $p$, the coordinates of $Q$, giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-17_2255_50_314_34}

Edexcel F1 2020 June Q6

6. (i) $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0
0 & 3 \end{array} \right)$$

Describe fully the single transformation represented by the matrix $\mathbf { A }$. The matrix $\mathbf { B }$ represents a rotation of $45 ^ { \circ }$ clockwise about the origin.
Write down the matrix $\mathbf { B }$, giving each element of the matrix in exact form. The transformation represented by matrix $\mathbf { A }$ followed by the transformation represented by matrix $\mathbf { B }$ is represented by the matrix $\mathbf { C }$.
Determine $\mathbf { C }$.
(ii) The trapezium $T$ has vertices at the points $( - 2,0 ) , ( - 2 , k ) , ( 5,8 )$ and $( 5,0 )$, where $k$ is a positive constant. Trapezium $T$ is transformed onto the trapezium $T ^ { \prime }$ by the matrix $$\left( \begin{array} { r r } 5 & 1
- 2 & 3 \end{array} \right)$$ Given that the area of trapezium $T ^ { \prime }$ is 510 square units, calculate the exact value of $k$.

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel F1 2020 June Q7

7. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The line $l$ with equation $3 x - 4 y + 48 = 0$ is a tangent to $C$ at the point $P$.

Show that $a = 9$
Hence determine the coordinates of $P$. Given that the point $S$ is the focus of $C$ and that the line $l$ crosses the directrix of $C$ at the point $A$,
determine the exact area of triangle $P S A$.
\includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-25_2255_50_314_34}

VIXV SIHIANI III IM IONOO VIAV SIHI NI JYHAM ION OO VI4V SIHI NI JLIYM ION OO

Edexcel F1 2020 June Q8

(i) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$

$$\sum _ { r = 1 } ^ { n } \frac { 2 r ^ { 2 } - 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } } { ( n + 1 ) ^ { 2 } }$$ (ii) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$ $$f ( n ) = 12 ^ { n } + 2 \times 5 ^ { n - 1 }$$ is divisible by 7

VILU SIHI NI JIIIM ION OC

VIUV SIHI NI III M M I ON OO

VIAV SIHI NI JIIIM I ION OC

\includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-29_2255_50_314_34}

VIXV SIHIANI III IM IONOO

VIAV SIHI NI JYHAM ION OO

VI4V SIHI NI JLIYM ION OO

\includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-31_2255_50_314_34}

END

Edexcel F1 2022 June Q1

1. $$z _ { 1 } = 3 + 3 i \quad z _ { 2 } = p + q i \quad p , q \in \mathbb { R }$$ Given that $\left| z _ { 1 } z _ { 2 } \right| = 15 \sqrt { 2 }$

determine $\left| z _ { 2 } \right|$ Given also that $p = - 4$
determine the possible values of $q$
Show $z _ { 1 }$ and the possible positions for $z _ { 2 }$ on the same Argand diagram.

Edexcel F1 2022 June Q2

2. $$f ( x ) = 10 - 2 x - \frac { 1 } { 2 \sqrt { x } } - \frac { 1 } { x ^ { 3 } } \quad x > 0$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [0.4, 0.5]
Determine $\mathrm { f } ^ { \prime } ( x )$.
Using $x _ { 0 } = 0.5$ as a first approximation to $\alpha$, apply the Newton-Raphson procedure once to $\mathrm { f } ( x )$ to find a second approximation to $\alpha$, giving your answer to 3 decimal places. The equation $\mathrm { f } ( x ) = 0$ has another root $\beta$ in the interval [4.8, 4.9]
[0pt]
Use linear interpolation once on the interval [4.8, 4.9] to find an approximation to $\beta$, giving your answer to 3 decimal places.

Edexcel F1 2022 June Q3

$\mathbf { M } = \left( \begin{array} { c c } k & k
3 & 5 \end{array} \right) \quad$ where $k$ is a non-zero constant
1. Determine $\mathbf { M } ^ { - 1 }$, giving your answer in simplest form in terms of $k$.
Hence, given that $\mathbf { N } ^ { - 1 } = \left( \begin{array} { c c } k & k
4 & - 1 \end{array} \right)$
determine $( \mathbf { M N } ) ^ { - 1 }$, giving your answer in simplest form in terms of $k$.

Edexcel F1 2022 June Q4

4. $$f ( z ) = 2 z ^ { 4 } - 19 z ^ { 3 } + A z ^ { 2 } + B z - 156$$ where $A$ and $B$ are constants.
The complex number $5 - \mathrm { i }$ is a root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

Write down another complex root of this equation.
Solve the equation $\mathrm { f } ( \mathrm { z } ) = 0$ completely.
Determine the value of $A$ and the value of $B$.

Edexcel F1 2022 June Q5

The quadratic equation

$$2 x ^ { 2 } - 3 x + 5 = 0$$ has roots $\alpha$ and $\beta$
Without solving the equation,

write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
determine the value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
find a quadratic equation which has roots $$\left( \alpha ^ { 3 } - \beta \right) \text { and } \left( \beta ^ { 3 } - \alpha \right)$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2022 June Q6

The parabola $C$ has equation $y ^ { 2 } = 36 x$

The point $P \left( 9 t ^ { 2 } , 18 t \right)$, where $t \neq 0$, lies on $C$

Use calculus to show that the normal to $C$ at $P$ has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
Hence find the equations of the two normals to $C$ which pass through the point (54, 0), giving your answers in the form $y = p x + q$ where $p$ and $q$ are constants to be determined. Given that
- the normals found in part (b) intersect the directrix of $C$ at the points $A$ and $B$
- the point $F$ is the focus of $C$
- determine the area of triangle $A F B$

Edexcel F1 2022 June Q7

7. $$A = \left( \begin{array} { c c } - \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

Determine the matrix $\mathbf { A } ^ { 2 }$
Describe fully the single geometrical transformation represented by the matrix $\mathbf { A } ^ { 2 }$
Hence determine the smallest positive integer value of $n$ for which $\mathbf { A } ^ { n } = \mathbf { I }$ The matrix $\mathbf { B }$ represents a stretch scale factor 4 parallel to the $x$-axis.
Write down the matrix $\mathbf { B }$ The transformation represented by matrix $\mathbf { A }$ followed by the transformation represented by matrix $\mathbf { B }$ is represented by the matrix $\mathbf { C }$
Determine the matrix $\mathbf { C }$ The parallelogram $P$ is transformed onto the parallelogram $P ^ { \prime }$ by the matrix $\mathbf { C }$
Given that the area of parallelogram $P ^ { \prime }$ is 20 square units, determine the area of parallelogram $P$

Edexcel F1 2022 June Q8

(a) Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that for all positive integers $n$

$$\sum _ { r = 0 } ^ { n } ( r + 1 ) ( r + 2 ) = \frac { 1 } { 3 } ( n + 1 ) ( n + 2 ) ( n + 3 )$$ (b) Hence determine the value of $$10 \times 11 + 11 \times 12 + 12 \times 13 + \ldots + 100 \times 101$$

Edexcel F1 2022 June Q9

(i) A sequence of numbers is defined by

$$\begin{gathered} u _ { 1 } = 3
u _ { n + 1 } = 2 u _ { n } - 2 ^ { n + 1 } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for $n \in \mathbb { N }$ $$u _ { n } = 5 \times 2 ^ { n - 1 } - n \times 2 ^ { n }$$ (ii) Prove by induction that, for $n \in \mathbb { N }$ $$f ( n ) = 5 ^ { n + 2 } - 4 n - 9$$ is divisible by 16

Edexcel F1 2023 June Q1

Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where $a$, $b$ and $c$ are integers to be determined.

Edexcel F1 2023 June Q2

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Given that $x = 2 + 3 \mathrm { i }$ is a root of the equation $$2 x ^ { 4 } - 8 x ^ { 3 } + 29 x ^ { 2 } - 12 x + 39 = 0$$

write down another complex root of this equation.
Use algebra to determine the other 2 roots of the equation.
Show all 4 roots on a single Argand diagram.

Edexcel F1 2023 June Q3

The rectangular hyperbola $H$ has Cartesian equation $x y = 9$

The point $P$ with coordinates $\left( 3 t , \frac { 3 } { t } \right)$, where $t \neq 0$, lies on $H$

Use calculus to determine an equation for the normal to $H$ at the point $P$ Give your answer in the form $t y - t ^ { 3 } x = \mathrm { f } ( t )$ Given that $t = 2$
determine the coordinates of the point where the normal meets $H$ again. Give your answer in simplest form.

Edexcel F1 2023 June Q4

(i) $\mathbf { A } = \left( \begin{array} { c c } - 3 & 8
- 3 & k \end{array} \right) \quad$ where $k$ is a constant The transformation represented by $\mathbf { A }$ transforms triangle $T$ to triangle $T ^ { \prime }$ The area of triangle $T ^ { \prime }$ is three times the area of triangle $T$

Determine the possible values of $k$
(ii) $\mathbf { B } = \left( \begin{array} { r r } a & - 4
2 & 3 \end{array} \right)$ and $\mathbf { B C } = \left( \begin{array} { l l l } 2 & 5 & 1
1 & 4 & 2 \end{array} \right)$ where $a$ is a constant Determine, in terms of $a$, the matrix $\mathbf { C }$

Edexcel F1 2023 June Q5

5. $$f ( x ) = x ^ { 2 } - 6 x + 3$$ The equation $\mathrm { f } ( x ) = 0$ has roots $\alpha$ and $\beta$
Without solving the equation,

determine the value of $$\left( \alpha ^ { 2 } + 1 \right) \left( \beta ^ { 2 } + 1 \right)$$
find a quadratic equation which has roots $$\frac { \alpha } { \left( \alpha ^ { 2 } + 1 \right) } \text { and } \frac { \beta } { \left( \beta ^ { 2 } + 1 \right) }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers to be determined.

Edexcel F1 2023 June Q6

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$z _ { 1 } = 3 + 2 i \quad z _ { 2 } = 2 + 3 i \quad z _ { 3 } = a + b i \quad a , b \in \mathbb { R }$$

Determine the exact value of $\left| z _ { 1 } + z _ { 2 } \right|$ Given that $w = \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }$
determine $w$ in terms of $a$ and $b$, giving your answer in the form $x + \mathrm { i } y$, where $x , y \in \mathbb { R }$ Given also that $w = \frac { 4 } { 13 } + \frac { 58 } { 13 } \mathrm { i }$
determine the value of $a$ and the value of $b$
determine arg $w$, giving your answer in radians to 4 significant figures.

Edexcel F1 2023 June Q7

7. $$f ( x ) = x ^ { \frac { 3 } { 2 } } + x - 3$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root, $\alpha$, in the interval $[ 1,2 ]$
[0pt]
Starting with the interval [1, 2], use interval bisection twice to show that $\alpha$ lies in the interval [1.25, 1.5]
1. Determine $\mathrm { f } ^ { \prime } ( x )$
2. Using 1.375 as a first approximation for $\alpha$, apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to determine a second approximation for $\alpha$, giving your answer to 3 decimal places.
  [0pt]
Use linear interpolation once on the interval [1.25,1.5] to obtain a different approximation for $\alpha$, giving your answer to 3 decimal places.

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