Questions FP1 - A-Level Maths

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]
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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]
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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]

AQA FP1 2016 June Q4

2 marks

Given that $\sin \frac { \pi } { 3 } = \cos \frac { \pi } { k }$, state the value of the integer $k$.
Hence, or otherwise, find the general solution of the equation $$\cos \left( 2 x - \frac { 5 \pi } { 6 } \right) = \sin \frac { \pi } { 3 }$$ giving your answer, in its simplest form, in terms of $\pi$.
Hence, given that $\cos \left( 2 x - \frac { 5 \pi } { 6 } \right) = \sin \frac { \pi } { 3 }$, show that there is only one finite value for $\tan x$ and state its exact value.
[0pt] [2 marks]

AQA FP1 2016 June Q5

4 marks

Use the formulae for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r$ to show that $\sum _ { r = 1 } ^ { n } ( 6 r - 3 ) ^ { 2 } = 3 n \left( 4 n ^ { 2 } - 1 \right)$.
Hence express $\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } ( 6 r - 3 ) ^ { 2 }$ as a product of four linear factors in terms of $n$.
[0pt] [4 marks]

AQA FP1 2016 June Q6

6 marks

6 A parabola with equation $y ^ { 2 } = 4 a x$, where $a$ is a constant, is translated by the vector $\left[ \begin{array} { l } 2
3 \end{array} \right]$ to give the curve $C$. The curve $C$ passes through the point (4, 7).

Show that $a = 2$.
Find the values of $k$ for which the line $k y = x$ does not meet the curve $C$.
[0pt] [6 marks]

AQA FP1 2016 June Q7

11 marks

Solve the equation $x ^ { 2 } + 4 x + 20 = 0$, giving your answers in the form $c + d \mathrm { i }$, where $c$ and $d$ are integers.
The roots of the quadratic equation $$z ^ { 2 } + ( 4 + i + q i ) z + 20 = 0$$ are $w$ and $w ^ { * }$.
1. In the case where $q$ is real, explain why $q$ must be - 1 .
2. In the case where $w = p + 2 \mathrm { i }$, where $p$ is real, find the possible values of $q$.
  [0pt] [5 marks] $8 \quad$ The matrix $\mathbf { A }$ is defined by $\mathbf { A } = \left[ \begin{array} { l l } 2 & 0
  0 & 1 \end{array} \right]$.
1. Find the matrix $\mathbf { A } ^ { 2 }$.
2. Describe fully the single geometrical transformation represented by the matrix $\mathbf { A } ^ { 2 }$.
Given that the matrix $\mathbf { B }$ represents a reflection in the line $x + \sqrt { 3 } y = 0$, find the matrix $\mathbf { B }$, giving the exact values of any trigonometric expressions.
Hence find the coordinates of the point $P$ which is mapped onto $( 0 , - 4 )$ under the transformation represented by $\mathbf { A } ^ { 2 }$ followed by a reflection in the line $x + \sqrt { 3 } y = 0$.
[0pt] [6 marks] $9 \quad$ A curve $C$ has equation $y = \frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) }$.
The line $L$ has equation $y = \frac { 1 } { 2 } ( x - 1 )$.
Write down the equations of the asymptotes of $C$.
By forming and solving a suitable cubic equation, find the $x$-coordinates of the points of intersection of $L$ and $C$.
Given that $C$ has no stationary points, sketch $C$ and $L$ on the same axes.
Hence solve the inequality $\frac { x - 1 } { ( x - 2 ) ( 2 x - 1 ) } \geqslant \frac { 1 } { 2 } ( x - 1 )$.

Edexcel FP1 2019 June Q1

Use Simpson's rule with 4 intervals to estimate

$$\int _ { 0.4 } ^ { 2 } e ^ { x ^ { 2 } } d x$$

Edexcel FP1 2019 June Q2

Given that $k$ is a real non-zero constant and that

$$y = x ^ { 3 } \sin k x$$ use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = \left( k ^ { 2 } x ^ { 2 } + A \right) k ^ { 3 } x \cos k x + B \left( k ^ { 2 } x ^ { 2 } + C \right) k ^ { 2 } \sin k x$$ where $A$, $B$ and $C$ are integers to be determined.

Edexcel FP1 2019 June Q3

3. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x - y ^ { 2 }$$

Show that $$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = a y \frac { \mathrm {~d} ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } + b \frac { \mathrm {~d} y } { \mathrm {~d} x } \frac { \mathrm {~d} ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + c \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) ^ { 2 }$$ where $a$, $b$ and $c$ are integers to be determined.
Hence find a series solution, in ascending powers of $x$ as far as the term in $x ^ { 5 }$, of the differential equation (I), given that $y = 1$ at $x = 0$

Edexcel FP1 2019 June Q4

The parabola $C$ has equation

$$y ^ { 2 } = 16 x$$ The distinct points $P \left( p ^ { 2 } , 4 p \right)$ and $Q \left( q ^ { 2 } , 4 q \right)$ lie on $C$, where $p \neq 0 , q \neq 0$
The tangent to $C$ at $P$ and the tangent to $C$ at $Q$ meet at the point $R ( - 28,6 )$.
Show that the area of triangle $P Q R$ is 1331

Edexcel FP1 2019 June Q5

5. $$I = \int \frac { 1 } { 4 \cos x - 3 \sin x } \mathrm {~d} x \quad 0 < x < \frac { \pi } { 4 }$$ Use the substitution $t = \tan \left( \frac { x } { 2 } \right)$ to show that $$I = \frac { 1 } { 5 } \ln \left( \frac { 2 + \tan \left( \frac { x } { 2 } \right) } { 1 - 2 \tan \left( \frac { x } { 2 } \right) } \right) + k$$ where $k$ is an arbitrary constant.

Edexcel FP1 2019 June Q6

The concentration of a drug in the bloodstream of a patient, $t$ hours after the drug has been administered, where $t \leqslant 6$, is modelled by the differential equation

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } C } { \mathrm {~d} t ^ { 2 } } - 5 t \frac { \mathrm {~d} C } { \mathrm {~d} t } + 8 C = t ^ { 3 }$$ where $C$ is measured in micrograms per litre.

Show that the transformation $t = \mathrm { e } ^ { x }$ transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} C } { \mathrm {~d} x } + 8 C = \mathrm { e } ^ { 3 x }$$
Hence find the general solution for the concentration $C$ at time $t$ hours. Given that when $t = 6 , C = 0$ and $\frac { \mathrm { d } C } { \mathrm {~d} t } = - 36$
find the maximum concentration of the drug in the bloodstream of the patient.

Edexcel FP1 2019 June Q7

With respect to a fixed origin $O$, the points $A$, $B$ and $C$ have coordinates $( 3,4,5 ) , ( 10 , - 1,5 )$ and ( $4,7 , - 9$ ) respectively.

The plane $\Pi$ has equation $4 x - 8 y + z = 2$
The line segment $A B$ meets the plane $\Pi$ at the point $P$ and the line segment $B C$ meets the plane $\Pi$ at the point $Q$.

Show that, to 3 significant figures, the area of quadrilateral $A P Q C$ is 38.5 The point $D$ has coordinates $( k , 4 , - 1 )$, where $k$ is a constant.
Given that the vectors $\overrightarrow { A B } , \overrightarrow { A C }$ and $\overrightarrow { A D }$ form three edges of a parallelepiped of volume 226
find the possible values of the constant $k$.

Edexcel FP1 2019 June Q8

The hyperbola $H$ has equation

$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The line $l _ { 1 }$ is the tangent to $H$ at the point $P ( 4 \cosh \theta , 3 \sinh \theta )$.
The line $l _ { 1 }$ meets the $x$-axis at the point $A$.
The line $l _ { 2 }$ is the tangent to $H$ at the point $( 4,0 )$.
The lines $l _ { 1 }$ and $l _ { 2 }$ meet at the point $B$ and the midpoint of $A B$ is the point $M$.

Show that, as $\theta$ varies, a Cartesian equation for the locus of $M$ is $$y ^ { 2 } = \frac { 9 ( 4 - x ) } { 4 x } \quad p < x < q$$ where $p$ and $q$ are values to be determined. Let $S$ be the focus of $H$ that lies on the positive $x$-axis.
Show that the distance from $M$ to $S$ is greater than 1

Edexcel FP1 2020 June Q1

Use l'Hospital's Rule to show that

$$\lim _ { x \rightarrow \frac { \pi } { 2 } } \frac { \left( e ^ { \sin x } - \cos ( 3 x ) - e \right) } { \tan ( 2 x ) } = - \frac { 3 } { 2 }$$

Edexcel FP1 2020 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9f127ab1-0e03-4f9f-87c2-01c553c54ee9-04_807_649_251_708} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the vertical cross-section of the entrance to a tunnel. The width at the base of the tunnel entrance is 2 metres and its maximum height is 3 metres. The shape of the cross-section can be modelled by the curve with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = 3 \cos \left( \frac { \pi } { 2 } x ^ { 2 } \right) \quad x \in [ - 1,1 ]$$ A wooden door of uniform thickness 85 mm is to be made to seal the tunnel entrance.
Use Simpson's rule with 6 intervals to estimate the volume of wood required for this door, giving your answer in $\mathrm { m } ^ { 3 }$ to 4 significant figures.

Questions FP1 (1385 questions)

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