Questions - AQA FP1 - A-Level Maths

Write down the coordinates of the points where the ellipse $E$ intersects the coordinate axes.
The ellipse $E$ is translated by the vector $\left[ \begin{array} { c } p
0 \end{array} \right]$, where $p$ is a constant. Write down the equation of the translated ellipse.
Show that, if the translated ellipse intersects the line $L$, the $x$-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
Given that the line $L$ is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
(No credit will be given for solutions based on differentiation.)

AQA FP1 2013 June Q1

1 The equation $$x ^ { 3 } - x ^ { 2 } + 4 x - 900 = 0$$ has exactly one real root, $\alpha$. Taking $x _ { 1 } = 10$ as a first approximation to $\alpha$, use the Newton-Raphson method to find a second approximation, $x _ { 2 }$, to $\alpha$. Give your answer to four significant figures.
(3 marks)

\includegraphics[max width=\textwidth, alt={}]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-02_1659_1709_1048_153}

AQA FP1 2013 June Q2

2 The matrices $\mathbf { A }$ and $\mathbf { B }$ are defined by $$\mathbf { A } = \left[ \begin{array} { c c } p & 2
4 & p \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 3 & 1
2 & 3 \end{array} \right]$$

Find, in terms of $p$, the matrices:
1. $\mathbf { A } - \mathbf { B }$;
2. AB .
Show that there is a value of $p$ for which $\mathbf { A } - \mathbf { B } + \mathbf { A B } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix, and state the corresponding value of $k$.

AQA FP1 2013 June Q3

Find the general solution, in degrees, of the equation $$\cos \left( 5 x + 40 ^ { \circ } \right) = \cos 65 ^ { \circ }$$
Given that $$\sin \frac { \pi } { 12 } = \frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$$ express $\sin \frac { \pi } { 12 }$ in the form $\left( \cos \frac { \pi } { 4 } \right) ( \cos ( a \pi ) + \cos ( b \pi ) )$, where $a$ and $b$ are rational.
(3 marks)

AQA FP1 2013 June Q4

It is given that $z = x + y \mathrm { i }$, where $x$ and $y$ are real numbers.
1. Write down, in terms of $x$ and $y$, an expression for $( z - 2 \mathrm { i } ) ^ { * }$.
2. Solve the equation $$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$ giving your answer in the form $a + b \mathrm { i }$.
It is given that $p + q \mathrm { i }$, where $p$ and $q$ are real numbers, is a root of the equation $z ^ { 2 } + 10 \mathrm { i } z - 29 = 0$. Without finding the values of $p$ and $q$, state why $p - q$ i is not a root of the equation $z ^ { 2 } + 10 \mathrm { i } z - 29 = 0$.

AQA FP1 2013 June Q5

A curve has equation $y = 2 x ^ { 2 } - 5 x$.
The point $P$ on the curve has coordinates $( 1 , - 3 )$.
The point $Q$ on the curve has $x$-coordinate $1 + h$.
1. Show that the gradient of the line $P Q$ is $2 h - 1$.
2. Explain how the result of part (a)(i) can be used to show that the tangent to the curve at the point $P$ is parallel to the line $x + y = 0$.
For the improper integral $\int _ { 1 } ^ { \infty } x ^ { - 4 } \left( 2 x ^ { 2 } - 5 x \right) \mathrm { d } x$, either show that the integral has a finite value and state its value, or explain why the integral does not have a finite value.

AQA FP1 2013 June Q6

6 The equation $$2 x ^ { 2 } + 3 x - 6 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
Hence show that $\alpha ^ { 3 } + \beta ^ { 3 } = - \frac { 135 } { 8 }$.
Find a quadratic equation, with integer coefficients, whose roots are $\alpha + \frac { \alpha } { \beta ^ { 2 } }$ and $\beta + \frac { \beta } { \alpha ^ { 2 } }$.

AQA FP1 2013 June Q7

Show that the equation $4 x ^ { 3 } - x - 540000 = 0$ has a root, $\alpha$, in the interval $51 < \alpha < 52$.
It is given that $S _ { n } = \sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 }$.
1. Use the formulae for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that $S _ { n } = \frac { n } { 3 } \left( k n ^ { 2 } - 1 \right)$, where $k$ is an integer to be found.
2. Hence show that $6 S _ { n }$ can be written as the product of three consecutive integers.
Find the smallest value of $N$ for which the sum of the squares of the first $N$ odd numbers is greater than 180000 .

AQA FP1 2013 June Q8

8 The diagram shows two triangles, $T _ { 1 }$ and $T _ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{d74d6295-d5b8-46da-8812-c5bf7c7a35f1-09_972_967_358_589}

Find the matrix which represents the stretch that maps triangle $T _ { 1 }$ onto triangle $T _ { 2 }$.
The triangle $T _ { 2 }$ is reflected in the line $y = \sqrt { 3 } x$ to give a third triangle, $T _ { 3 }$. Find, using surd forms where appropriate:
1. the matrix which represents the reflection that maps triangle $T _ { 2 }$ onto triangle $T _ { 3 }$;
2. the matrix which represents the combined transformation that maps triangle $T _ { 1 }$ onto triangle $T _ { 3 }$.
  (2 marks)

AQA FP1 2013 June Q9

9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$

Find the equations of the three asymptotes of the curve.
1. Show that if the line $y = k$ intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
2. Given that the equation $( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$ has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
3. Hence show that the curve has only one stationary point and find its coordinates.
  (No credit will be given for solutions based on differentiation.)
Sketch the curve and its asymptotes.

AQA FP1 2014 June Q1

5 marks

1 A curve passes through the point $( 9,6 )$ and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 + \sqrt { x } }$$ Use a step-by-step method with a step length of 0.25 to estimate the value of $y$ at $x = 9.5$. Give your answer to four decimal places.
[0pt] [5 marks]

AQA FP1 2014 June Q2

2 The quadratic equation $$2 x ^ { 2 } + 8 x + 1 = 0$$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
1. Find the value of $\alpha ^ { 2 } + \beta ^ { 2 }$.
2. Hence, or otherwise, show that $\alpha ^ { 4 } + \beta ^ { 4 } = \frac { 449 } { 2 }$.
Find a quadratic equation, with integer coefficients, which has roots $$2 \alpha ^ { 4 } + \frac { 1 } { \beta ^ { 2 } } \text { and } 2 \beta ^ { 4 } + \frac { 1 } { \alpha ^ { 2 } }$$
\includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-04_2443_1707_260_153}

AQA FP1 2014 June Q4

8 marks

4 Find the complex number $z$ such that $$5 \mathrm { i } z + 3 z ^ { * } + 16 = 8 \mathrm { i }$$ Give your answer in the form $a + b \mathrm { i }$, where $a$ and $b$ are real.
[0pt] [6 marks] $5 \quad$ A curve $C$ has equation $y = x ( x + 3 )$.

Find the gradient of the line passing through the point ( $- 5,10$ ) and the point on $C$ with $x$-coordinate $- 5 + h$. Give your answer in its simplest form.
Show how the answer to part (a) can be used to find the gradient of the curve $C$ at the point $( - 5,10 )$. State the value of this gradient.
[0pt] [2 marks] $6 \quad$ A curve $C$ has equation $y = \frac { 1 } { x ( x + 2 ) }$.
Write down the equations of all the asymptotes of $C$.
The curve $C$ has exactly one stationary point. The $x$-coordinate of the stationary point is - 1 .
1. Find the $y$-coordinate of the stationary point.
2. Sketch the curve $C$.
Solve the inequality $$\frac { 1 } { x ( x + 2 ) } \leqslant \frac { 1 } { 8 }$$

AQA FP1 2014 June Q7

5 marks

Write down the $2 \times 2$ matrix corresponding to each of the following transformations:
1. a reflection in the line $y = - x$;
2. a stretch parallel to the $y$-axis of scale factor 7 .
Hence find the matrix corresponding to the combined transformation of a reflection in the line $y = - x$ followed by a stretch parallel to the $y$-axis of scale factor 7 .
The matrix $\mathbf { A }$ is defined by $\mathbf { A } = \left[ \begin{array} { c c } - 3 & - \sqrt { 3 }
- \sqrt { 3 } & 3 \end{array} \right]$.
1. Show that $\mathbf { A } ^ { 2 } = k \mathbf { I }$, where $k$ is a constant and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
2. Show that the matrix $\mathbf { A }$ corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form $y = ( \tan \theta ) x$.
  [0pt] [5 marks]

AQA FP1 2014 June Q8

4 marks

Find the general solution of the equation $$\cos \left( \frac { 5 } { 4 } x - \frac { \pi } { 3 } \right) = \frac { \sqrt { 2 } } { 2 }$$ giving your answer for $x$ in terms of $\pi$.
Use your general solution to find the sum of all the solutions of the equation $\cos \left( \frac { 5 } { 4 } x - \frac { \pi } { 3 } \right) = \frac { \sqrt { 2 } } { 2 }$ that lie in the interval $0 \leqslant x \leqslant 20 \pi$. Give your answer in the form $k \pi$, stating the exact value of $k$.
[0pt] [4 marks]

AQA FP1 2014 June Q9

15 marks

9 An ellipse $E$ has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

Sketch the ellipse $E$, showing the values of the intercepts on the coordinate axes.
[0pt] [2 marks]
Given that the line with equation $y = x + k$ intersects the ellipse $E$ at two distinct points, show that $- 5 < k < 5$.
[0pt] [5 marks]
The ellipse $E$ is translated by the vector $\left[ \begin{array} { l } a
b \end{array} \right]$ to form another ellipse whose equation is $9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c$. Find the values of the constants $a , b$ and $c$.
[0pt] [5 marks]
Hence find an equation for each of the two tangents to the ellipse $9 x ^ { 2 } + 16 y ^ { 2 } + 18 x - 64 y = c$ that are parallel to the line $y = x$.
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}]{2eaee88a-9e08-4392-8a4c-79fc9861603e-10_1438_1707_1265_153}

AQA FP1 2015 June Q1

1 The quadratic equation $2 x ^ { 2 } + 6 x + 7 = 0$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
Find a quadratic equation, with integer coefficients, which has roots $\alpha ^ { 2 } - 1$ and $\beta ^ { 2 } - 1$.
Hence find the values of $\alpha ^ { 2 }$ and $\beta ^ { 2 }$.

AQA FP1 2015 June Q2

4 marks

Explain why $\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x$ is an improper integral.
Either find the value of the integral $\int _ { 0 } ^ { 4 } \frac { x - 4 } { x ^ { 1.5 } } \mathrm {~d} x$ or explain why it does not have a finite value.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-04_1970_1712_737_150}

AQA FP1 2015 June Q3

Show that $( 2 + \mathrm { i } ) ^ { 3 }$ can be expressed in the form $2 + b \mathrm { i }$, where $b$ is an integer.
It is given that $2 + \mathrm { i }$ is a root of the equation $$z ^ { 3 } + p z + q = 0$$ where $p$ and $q$ are real numbers.
1. Show that $p = - 11$ and find the value of $q$.
2. Given that $2 - \mathrm { i }$ is also a root of $z ^ { 3 } + p z + q = 0$, find a quadratic factor of $z ^ { 3 } + p z + q$ with real coefficients.
3. Find the real root of the equation $z ^ { 3 } + p z + q = 0$.
  \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-06_1568_1707_1139_155}

AQA FP1 2015 June Q4

1 marks

Find the general solution, in degrees, of the equation $$2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1$$
Use your general solution to find the solution of $2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1$ that is closest to $200 ^ { \circ }$.
[0pt] [1 mark]

AQA FP1 2015 June Q5

7 marks

The matrix $\mathbf { A }$ is defined by $\mathbf { A } = \left[ \begin{array} { c c } - 2 & c
d & 3 \end{array} \right]$.
Given that the image of the point $( 5,2 )$ under the transformation represented by $\mathbf { A }$ is $( - 2,1 )$, find the value of $c$ and the value of $d$.
[0pt] [4 marks]
The matrix $\mathbf { B }$ is defined by $\mathbf { B } = \left[ \begin{array} { c c } \sqrt { 2 } & \sqrt { 2 }
- \sqrt { 2 } & \sqrt { 2 } \end{array} \right]$.
1. Show that $\mathbf { B } ^ { 4 } = k \mathbf { I }$, where $k$ is an integer and $\mathbf { I }$ is the $2 \times 2$ identity matrix.
2. Describe the transformation represented by the matrix $\mathbf { B }$ as a combination of two geometrical transformations.
3. Find the matrix $\mathbf { B } ^ { 17 }$. $6 \quad \mathrm {~A}$ curve $C _ { 1 }$ has equation $$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$$

AQA FP1 2015 June Q7

4 marks

The equation $2 x ^ { 3 } + 5 x ^ { 2 } + 3 x - 132000 = 0$ has exactly one real root $\alpha$.
1. Show that $\alpha$ lies in the interval $39 < \alpha < 40$.
2. Taking $x _ { 1 } = 40$ as a first approximation to $\alpha$, use the Newton-Raphson method to find a second approximation, $x _ { 2 }$, to $\alpha$. Give your answer to two decimal places.
Use the formulae for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that $$\sum _ { r = 1 } ^ { n } 2 r ( 3 r + 2 ) = n ( n + p ) ( 2 n + q )$$ where $p$ and $q$ are integers.
1. Express $\log _ { 8 } 4 ^ { r }$ in the form $\lambda r$, where $\lambda$ is a rational number.
2. By first finding a suitable cubic inequality for $k$, find the greatest value of $k$ for which $\sum _ { r = k + 1 } ^ { 60 } ( 3 r + 2 ) \log _ { 8 } 4 ^ { r }$ is greater than 106060.
  [0pt] [4 marks]

AQA FP1 2015 June Q8

8 A curve $C$ has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$

State the equation of the asymptote of $C$.
The line $y = k$ intersects the curve $C$. Show that $4 k ^ { 2 } - 4 k - 3 \leqslant 0$.
Hence find the coordinates of the stationary points of the curve $C$. (No credit will be given for solutions based on differentiation.)
\includegraphics[max width=\textwidth, alt={}, center]{e45b07a3-e303-4caf-8f3a-5341bad7560a-24_2488_1728_219_141}

AQA FP1 2016 June Q1

7 marks

1 The quadratic equation $x ^ { 2 } - 6 x + 14 = 0$ has roots $\alpha$ and $\beta$.

Write down the value of $\alpha + \beta$ and the value of $\alpha \beta$.
Find a quadratic equation, with integer coefficients, which has roots $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$.
[0pt] [5 marks] $2 \quad$ A curve $C$ has equation $y = ( 2 - x ) ( 1 + x ) + 3$.
A line passes through the point $( 2,3 )$ and the point on $C$ with $x$-coordinate $2 + h$. Find the gradient of the line, giving your answer in its simplest form.
Show how your answer to part (a) can be used to find the gradient of the curve $C$ at the point $( 2,3 )$. State the value of this gradient.
[0pt] [2 marks]

AQA FP1 2016 June Q3

4 marks

3 The variables $y$ and $x$ are related by an equation of the form $$y = a \left( b ^ { x } \right)$$ where $a$ and $b$ are positive constants.
Let $Y = \log _ { 10 } y$.

Show that there is a linear relationship between $Y$ and $x$.
The graph of $Y$ against $x$, shown below, passes through the points ( $0,2.5$ ) and (5, 0.5).
\includegraphics[max width=\textwidth, alt={}, center]{7e7eaea5-22ca-4418-8ac6-351ce9ac09ea-06_433_506_904_776}
1. Find the gradient of the line.
2. Find the value of $a$ and the value of $b$, giving each answer to three significant figures. [4 marks]

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