Questions F1 - A-Level Maths

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval $[ 3,4 ]$.
Taking 3 as a first approximation to $\alpha$, apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to obtain a second approximation to $\alpha$. Give your answer to 3 decimal places.
[0pt]
Use linear interpolation once on the interval [3,4] to find another approximation to $\alpha$. Give your answer to 3 decimal places.

Edexcel F1 2014 January Q2

2. The quadratic equation $$5 x ^ { 2 } - 4 x + 2 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
Find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$.
Find a quadratic equation which has roots $$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers.
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Edexcel F1 2014 January Q3

3. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 4
1 & 1 \end{array} \right)$$

Show that $\mathbf { A }$ is non-singular. The triangle $R$ is transformed to the triangle $S$ by the matrix $\mathbf { A }$.
Given that the area of triangle $R$ is 10 square units,
find the area of triangle $S$. Given that $$\mathbf { B } = \mathbf { A } ^ { 4 }$$ and that the triangle $R$ is transformed to the triangle $T$ by the matrix $\mathbf { B }$,
find, without evaluating $\mathbf { B }$, the area of triangle $T$.

Edexcel F1 2014 January Q4

4. $$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 19 x - 60$$

Given that $x = - 4$ and $x = 3$ are roots of the equation $\mathrm { f } ( x ) = 0$, use algebra to solve $\mathrm { f } ( x ) = 0$ completely.
Show the four roots of $\mathrm { f } ( x ) = 0$ on a single Argand diagram.

Edexcel F1 2014 January Q5

5. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ to show that $$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$ for all positive integers $n$. Given that $$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$ (b) find the exact value of the constant $k$.

Edexcel F1 2014 January Q6

$$\mathbf { B } = \left( \begin{array} { r r } - 1 & 2
3 & - 4 \end{array} \right) , \quad \mathbf { Y } = \left( \begin{array} { r r } 4 & - 2
1 & 0 \end{array} \right)$$ (a) Find $\mathbf { B } ^ { - 1 }$. The transformation represented by $\mathbf { Y }$ is equivalent to the transformation represented by $\mathbf { B }$ followed by the transformation represented by the matrix $\mathbf { A }$.
(b) Find $\mathbf { A }$.
$$\mathbf { M } = \left( \begin{array} { r r } - \sqrt { 3 } & - 1
1 & - \sqrt { 3 } \end{array} \right)$$ The matrix $\mathbf { M }$ represents an enlargement scale factor $k$, centre ( 0,0 ), where $k > 0$, followed by a rotation anti-clockwise through an angle $\theta$ about $( 0,0 )$.
(a) Find the value of $k$.
(b) Find the value of $\theta$.

Edexcel F1 2014 January Q7

7. (i) Given that $$\frac { 2 w - 3 } { 10 } = \frac { 4 + 7 i } { 4 - 3 i }$$ find $w$, giving your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real constants. You must show your working.
(ii) Given that $$z = ( 2 + \lambda i ) ( 5 + i )$$ where $\lambda$ is a real constant, and that $$\arg z = \frac { \pi } { 4 }$$ find the value of $\lambda$.
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Edexcel F1 2014 January Q8

8. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $P \left( a p ^ { 2 } , 2 a p \right)$ lies on the parabola $C$.

Show that an equation of the tangent to $C$ at $P$ is $$p y = x + a p ^ { 2 }$$ The tangent to $C$ at the point $P$ intersects the directrix of $C$ at the point $B$ and intersects the $x$-axis at the point $D$. Given that the $y$-coordinate of $B$ is $\frac { 5 } { 6 } a$ and $p > 0$,
find, in terms of $a$, the $x$-coordinate of $D$. Given that $O$ is the origin,
find, in terms of $a$, the area of the triangle $O P D$, giving your answer in its simplest form.

Edexcel F1 2014 January Q9

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

$$f ( n ) = 7 ^ { n } - 2 ^ { n } \text { is divisible by } 5$$

Edexcel F1 2015 January Q1

1. $$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$ Given that $x = 1 + 2 \mathrm { i }$ is a root of the equation $\mathrm { f } ( x ) = 0$, use algebra to find the three other roots of the equation $\mathrm { f } ( x ) = 0$

Edexcel F1 2015 January Q2

2. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + \frac { 1 } { 2 \sqrt { x ^ { 5 } } } + 2 , \quad x > 0$$

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval $[ 2,3 ]$.
Taking 3 as a first approximation to $\alpha$, apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to find a second approximation to $\alpha$. Give your answer to 3 decimal places.

Edexcel F1 2015 January Q3

3. Given that $z = x + \mathrm { i } y$, where $x$ and $y$ are real numbers, solve the equation $$( z - 2 i ) \left( z ^ { * } - 2 i \right) = 21 - 12 i$$ where $z ^ { * }$ is the complex conjugate of $z$.

Edexcel F1 2015 January Q4

4. The parabola $C$ has cartesian equation $y ^ { 2 } = 12 x$ The point $P \left( 3 p ^ { 2 } , 6 p \right)$ lies on $C$, where $p \neq 0$

Show that the equation of the normal to the curve $C$ at the point $P$ is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve $C$ again at the point $Q$.
Given that $p = 2$ and that $S$ is the focus of the parabola, find
the coordinates of the point $Q$,
the area of the triangle $P Q S$.

Edexcel F1 2015 January Q5

5. The quadratic equation $$4 x ^ { 2 } + 3 x + 1 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$.
Find the value of $\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$.
Find a quadratic equation which has roots $$( 4 \alpha - \beta ) \text { and } ( 4 \beta - \alpha )$$ 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 2015 January Q6

$$\mathbf { A } = \left( \begin{array} { l l } 3 & 0
0 & 1 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 }
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$ (a) Describe fully the single transformation represented by the matrix $\mathbf { A }$.
(b) Describe fully the single transformation represented by the matrix $\mathbf { B }$. The transformation represented by $\mathbf { A }$ followed by the transformation represented by $\mathbf { B }$ is equivalent to the transformation represented by the matrix $\mathbf { C }$.
(c) Find $\mathbf { C }$.
$\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4
1 & k \end{array} \right)$, where $k$ is a real number. Show that $\operatorname { det } \mathbf { M } \neq 0$ for all values of $k$.

Edexcel F1 2015 January Q7

7. Given that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } ( r + a ) ( r + b ) = \frac { 1 } { 6 } n ( 2 n + 11 ) ( n - 1 )$$ where $a$ and $b$ are constants and $a > b$,

find the value of $a$ and the value of $b$.
Find the value of $$\sum _ { r = 9 } ^ { 20 } ( r + a ) ( r + b )$$

Edexcel F1 2015 January Q8

(i) A sequence of numbers is defined by

$$\begin{gathered} u _ { 1 } = 5 \quad u _ { 2 } = 13
u _ { n + 2 } = 5 u _ { n + 1 } - 6 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for $n \in \mathbb { Z } ^ { + }$, $$u _ { n } = 2 ^ { n } + 3 ^ { n }$$ (ii) Prove by induction that for $n \geqslant 2$, where $n \in \mathbb { Z }$, $$f ( n ) = 7 ^ { 2 n } - 48 n - 1$$ is divisible by 2304
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Edexcel F1 2016 January Q1

1. $$z = 3 + 2 \mathrm { i } , \quad w = 1 - \mathrm { i }$$ Find in the form $a + b \mathrm { i }$, where $a$ and $b$ are real constants,

$z w$
$\frac { z } { w ^ { * } }$, showing clearly how you obtained your answer. Given that $$| z + k | = \sqrt { 53 } \text {, where } k \text { is a real constant }$$
find the possible values of $k$.

Edexcel F1 2016 January Q2

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

Show that the equation $\mathrm { f } ( x ) = 0$ has a root $\alpha$ in the interval [1.6,1.7]
Taking 1.6 as a first approximation to $\alpha$, apply the Newton-Raphson process once to $\mathrm { f } ( x )$ to find a second approximation to $\alpha$. Give your answer to 3 decimal places.

Edexcel F1 2016 January Q3

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

1. write down the value of $( \alpha + \beta )$ and the value of $\alpha \beta$
2. show that $\alpha ^ { 2 } + \beta ^ { 2 } = - 2$
3. find the value of $\alpha ^ { 3 } + \beta ^ { 3 }$
1. show that $\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }$
2. find a quadratic equation which has roots $$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$ giving your answer in the form $p x ^ { 2 } + q x + r = 0$ where $p , q$ and $r$ are integers.

Edexcel F1 2016 January Q4

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

Describe fully the single geometrical transformation represented by the matrix $\mathbf { A }$.
Hence find the smallest positive integer value of $n$ for which $$\mathbf { A } ^ { n } = \mathbf { I }$$ where $\mathbf { I }$ is the $2 \times 2$ identity matrix. The transformation represented by the matrix $\mathbf { A }$ followed by the transformation represented by the matrix $\mathbf { B }$ is equivalent to the transformation represented by the matrix $\mathbf { C }$. Given that $\mathbf { C } = \left( \begin{array} { r r } 2 & 4
- 3 & - 5 \end{array} \right)$,
find the matrix $\mathbf { B }$.

Edexcel F1 2016 January Q5

5. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 3 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 3 ) ( a n + b )$$ where $a$ and $b$ are integers to be found. Given that $$\sum _ { r = 5 } ^ { 10 } \left( 8 r ^ { 3 } - 3 r + k r ^ { 2 } \right) = 22768$$ (b) find the exact value of the constant $k$.

Edexcel F1 2016 January Q6

6. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a non-zero constant. The point $P \left( c p , \frac { c } { p } \right)$, where $p \neq 0$, lies on $H$.

Show that the normal to $H$ at $P$ has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to $H$ at $P$ meets $H$ again at the point $Q$.
Find, in terms of $c$ and $p$, the coordinates of $Q$.

Edexcel F1 2016 January Q7

7. $$f ( x ) = x ^ { 4 } - 3 x ^ { 3 } - 15 x ^ { 2 } + 99 x - 130$$

Given that $x = 3 + 2 \mathrm { i }$ is a root of the equation $\mathrm { f } ( x ) = 0$, use algebra to find the three other roots of the equation $\mathrm { f } ( x ) = 0$
Show the four roots of $\mathrm { f } ( x ) = 0$ on a single Argand diagram.

Edexcel F1 2016 January Q8

8. The parabola $P$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $S$ is the focus of $P$. The point $B$, which does not lie on the parabola, has coordinates ( $q , r$ ) where $q$ and $r$ are positive constants and $q > a$. The line $l$ passes through $B$ and $S$.

Show that an equation of the line $l$ is $$( q - a ) y = r ( x - a )$$ The line $l$ intersects the directrix of $P$ at the point $C$. Given that the area of triangle $O C S$ is three times the area of triangle $O B S$, where $O$ is the origin,
show that the area of triangle $O B C$ is $\frac { 6 } { 5 } \mathrm { qr }$

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