Questions - Edexcel F1 - A-Level Maths

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Edexcel F1 2014 June Q6

8 marks Standard +0.8

6. It is given that $\alpha$ and $\beta$ are roots of the equation $3 x ^ { 2 } + 5 x - 1 = 0$

Find the exact value of $\alpha ^ { 3 } + \beta ^ { 3 }$
Find a quadratic equation which has roots $\frac { \alpha ^ { 2 } } { \beta }$ and $\frac { \beta ^ { 2 } } { \alpha }$, giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a$, $b$ and $c$ are integers.

Edexcel F1 2014 June Q7

11 marks Moderate -0.3

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

Describe fully the single geometrical transformation $U$ represented by the matrix $\mathbf { P }$. The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a reflection in the $x$-axis.
Write down the matrix $\mathbf { Q }$. Given that $V$ followed by $U$ is the transformation $T$, which is represented by the matrix $\mathbf { R }$,
find the matrix $\mathbf { R }$.
Show that there is a real number $k$ for which the transformation $T$ maps the point $( 1 , k )$ onto itself. Give the exact value of $k$ in its simplest form.

Edexcel F1 2014 June Q8

14 marks Standard +0.3

8. The hyperbola $H$ has cartesian equation $x y = 16$ The parabola $P$ has parametric equations $x = 8 t ^ { 2 } , y = 16 t$.

Find, using algebra, the coordinates of the point $A$ where $H$ meets $P$. Another point $B ( 8,2 )$ lies on the hyperbola $H$.
Find the equation of the normal to $H$ at the point (8, 2), giving your answer in the form $y = m x + c$, where $m$ and $c$ are constants.
Find the coordinates of the points where this normal at $B$ meets the parabola $P$.

Edexcel F1 2014 June Q9

11 marks Standard +0.3

9. (i) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$ $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 ) = \frac { n ( n + 1 ) ( n + 2 ) ( n + 3 ) } { 4 }$$ (ii) Prove by induction that, $$4 ^ { n } + 6 n + 8 \text { is divisible by } 18$$ for all positive integers $n$. \includegraphics[max width=\textwidth, alt={}, center]{df5ab400-5cb1-4b51-8b0a-52dc3587f81a-16_62_44_2476_1889}

Edexcel F1 2015 June Q1

5 marks Moderate -0.8

Given that

$$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 \equiv ( 2 z - 3 ) \left( z ^ { 2 } + a z + b \right)$$ where $a$ and $b$ are real constants,

find the value of $a$ and the value of $b$.
Given that $z$ is a complex number, find the three exact roots of the equation $$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 = 0$$

Edexcel F1 2015 June Q2

5 marks Standard +0.3

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

$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { n } { 2 } \left( a n ^ { 2 } + b n + c \right)$$ where $a , b$ and $c$ are integers to be found.

Edexcel F1 2015 June Q3

6 marks Standard +0.3

3. It is given that $\alpha$ and $\beta$ are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$

Find the exact value of $\alpha ^ { 2 } + \beta ^ { 2 }$
Find a quadratic equation which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$, giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.

Edexcel F1 2015 June Q4

6 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{04f06398-ff29-4690-a6fe-825d089fba39-05_663_665_228_644} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the parabola $C$ with equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $S$ is the focus of $C$ and the point $Q$ lies on the directrix of $C$. The point $P$ lies on $C$ where $y > 0$ and the line segment $Q P$ is parallel to the $x$-axis. Given that the length of $P S$ is 13

write down the length of $P Q$. Given that the point $P$ has $x$ coordinate 9
find
the value of $a$,
the area of triangle $P S Q$.

Edexcel F1 2015 June Q5

7 marks Standard +0.3

In the interval $2 < x < 3$, the equation

$$6 - x ^ { 2 } \cos \left( \frac { x } { 5 } \right) = 0 , \text { where } x \text { is measured in radians }$$ has exactly one root $\alpha$.
[0pt]

Starting with the interval [2,3], use interval bisection twice to find an interval of width 0.25 which contains $\alpha$.
[0pt]
Use linear interpolation once on the interval [2,3] to find an approximation to $\alpha$. Give your answer to 2 decimal places.

Edexcel F1 2015 June Q6

10 marks Challenging +1.2

6. The rectangular hyperbola, $H$, has cartesian equation $$x y = 36$$ The three points $P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)$ and $R \left( 6 r , \frac { 6 } { r } \right)$, where $p , q$ and $r$ are distinct, non-zero values, lie on the hyperbola $H$.

Show that an equation of the line $P Q$ is $$p q y + x = 6 ( p + q )$$ Given that $P R$ is perpendicular to $Q R$,
show that the normal to the curve $H$ at the point $R$ is parallel to the line $P Q$.

Edexcel F1 2015 June Q7

11 marks Moderate -0.3

7. $$z = - 3 k - 2 k \mathrm { i } , \text { where } k \text { is a real, positive constant. }$$

Find the modulus and the argument of $z$, giving the argument in radians to 2 decimal places and giving the modulus as an exact answer in terms of $k$.
Express in the form $a + \mathrm { i } b$, where $a$ and $b$ are real and are given in terms of $k$ where necessary,
1. $\frac { 4 } { z + 3 k }$
2. $z ^ { 2 }$
Given that $k = 1$, plot the points $A , B , C$ and $D$ representing $z , z ^ { * } , \frac { 4 } { z + 3 k }$ and $z ^ { 2 }$ respectively on a single Argand diagram.

Edexcel F1 2015 June Q8

13 marks Standard +0.3

8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a \\ 4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$

Find the matrix $\mathbf { P } ^ { - 1 }$ in terms of $a$.
(3) The matrix $\mathbf { P }$ represents the transformation $U$ which transforms a triangle $T _ { 1 }$ onto the triangle $T _ { 2 }$.
The triangle $T _ { 2 }$ has vertices at the points ( $- 3 a , - 4 a$ ), ( $6 a , 8 a$ ), and ( $- 20 a , 15 a$ ).
Find the coordinates of the vertices of $T _ { 1 }$
Hence, or otherwise, find the area of triangle $T _ { 2 }$ in terms of $a$. The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a rotation through an angle $\alpha$ clockwise about the origin, where $\tan \alpha = \frac { 4 } { 3 }$ and $0 < \alpha < \frac { \pi } { 2 }$
Write down the matrix $\mathbf { Q }$, giving each element as an exact value. The transformation $U$ followed by the transformation $V$ is the transformation $W$. The matrix $\mathbf { R }$ represents the transformation $W$.
Find the matrix $\mathbf { R }$.

Edexcel F1 2015 June Q9

12 marks Standard +0.3

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

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + n - 1 \right)$$ (ii) Prove by induction that, for $n \in \mathbb { Z } ^ { + }$, $$\left( \begin{array} { c c } 7 & - 12 \\ 3 & - 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 6 n + 1 & - 12 n \\ 3 n & 1 - 6 n \end{array} \right)$$

Edexcel F1 2016 June Q1

4 marks Standard +0.3

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

$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { n } { 4 } ( n + a ) ( n + b ) ( n + c )$$ where $a$, $b$ and $c$ are integers to be found.

\includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-03_2673_1710_84_116} \includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-03_24_21_109_2042}

Edexcel F1 2016 June Q2

5 marks Standard +0.8

2. A parabola $P$ has cartesian equation $y ^ { 2 } = 28 x$. The point $S$ is the focus of the parabola $P$.

Write down the coordinates of the point $S$. Points $A$ and $B$ lie on the parabola $P$. The line $A B$ is parallel to the directrix of $P$ and cuts the $x$-axis at the midpoint of $O S$, where $O$ is the origin.
Find the exact area of triangle $A B S$.
VILM SIHI NITIIIUMI ON OC
VILV SIHI NI III HM ION OC
VALV SIHI NI JIIIM ION OO \includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-05_2264_53_315_36}

Edexcel F1 2016 June Q3

7 marks Standard +0.3

3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, $\alpha$, of the equation $\mathrm { f } ( x ) = 0$ lies in the interval $[ - 2 , - 1 ]$.

Taking - 1.5 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 2 decimal places.
Show that your answer to part (a) gives $\alpha$ correct to 2 decimal places.

tion 3continued -

Edexcel F1 2016 June Q4

5 marks Moderate -0.8

4. Given that $$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

show that $\operatorname { det } ( \mathbf { A } ) > 0$ for all real values of $k$,
find $\mathbf { A } ^ { - 1 }$ in terms of $k$.

Edexcel F1 2016 June Q5

5 marks Moderate -0.3

5. $$2 z + z ^ { * } = \frac { 3 + 4 i } { 7 + i }$$ Find $z$, giving your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real constants. You must show all your working.

Edexcel F1 2016 June Q6

10 marks Standard +0.8

6. The rectangular hyperbola $H$ has equation $x y = 25$

Verify that, for $t \neq 0$, the point $P \left( 5 t , \frac { 5 } { t } \right)$ is a general point on $H$. The point $A$ on $H$ has parameter $t = \frac { 1 } { 2 }$
Show that the normal to $H$ at the point $A$ has equation $$8 y - 2 x - 75 = 0$$ This normal at $A$ meets $H$ again at the point $B$.
Find the coordinates of $B$.

Edexcel F1 2016 June Q7

10 marks Standard +0.3

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

Describe fully the single geometrical transformation $U$ represented by the matrix $\mathbf { P }$. The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a reflection in the line with equation $y = x$
Write down the matrix $\mathbf { Q }$. Given that the transformation $V$ followed by the transformation $U$ is the transformation $T$, which is represented by the matrix $\mathbf { R }$,
find the matrix $\mathbf { R }$.
Show that there is a value of $k$ for which the transformation $T$ maps each point on the straight line $y = k x$ onto itself, and state the value of $k$. \section*{II}

Edexcel F1 2016 June Q8

9 marks Standard +0.3

8. $$f ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where $a$ and $b$ are real constants. Given that $- 3 + 8 \mathrm { i }$ is a complex root of the equation $\mathrm { f } ( \mathrm { z } ) = 0$

write down another complex root of this equation.
Hence, or otherwise, find the other roots of the equation $\mathrm { f } ( \mathrm { z } ) = 0$
Show on a single Argand diagram all four roots of the equation $f ( z ) = 0$

Edexcel F1 2016 June Q9

9 marks Standard +0.8

9. The quadratic equation $$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots $\alpha$ and $\beta$.
Without solving the quadratic equation,

find the exact value of
1. $\alpha ^ { 2 } + \beta ^ { 2 }$
2. $\alpha ^ { 3 } + \beta ^ { 3 }$
Find a quadratic equation which has roots ( $\alpha ^ { 2 } + \beta$ ) and ( $\beta ^ { 2 } + \alpha$ ), giving your answer in the form $a x ^ { 2 } + b x + c = 0$, where $a , b$ and $c$ are integers.
\includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-27_99_332_2622_1466}