Questions FP3 - A-Level Maths

AQA FP3 2015 June Q3

3

1. Write down the expansion of $\ln ( 1 + 2 x )$ in ascending powers of $x$ up to and including the term in $x ^ { 4 }$.
2. Hence, or otherwise, find the first two non-zero terms in the expansion of $$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$ in ascending powers of $x$ and state the range of values of $x$ for which the expansion is valid.
Find $\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]$.

AQA FP3 2015 June Q4

4

Explain why $\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x$ is an improper integral.
Evaluate $\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2015 June Q5

3 marks

5

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$
It is given that $y = \mathrm { f } ( x )$ is the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$ such that $\mathrm { f } ( 0 ) = 0$ and $\mathrm { f } ^ { \prime } ( 0 ) = 0$.
1. Show that $f ^ { \prime \prime } ( 0 ) = 0$.
2. Find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\mathrm { f } ( x )$.
  [0pt] [3 marks]

AQA FP3 2015 June Q6

6 A differential equation is given by $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$

Show that the substitution $x = \mathrm { e } ^ { 2 t }$ transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 4 t ^ { 2 } + 5 \mathrm { e } ^ { - t }$$
Hence find the general solution of the differential equation $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
\includegraphics[max width=\textwidth, alt={}]{7b4a1237-bb28-4cba-84b1-35fa91d87408-14_1634_1709_1071_153}

AQA FP3 2015 June Q7

7 The diagram shows the sketch of a curve $C _ { 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve $C _ { 1 }$ is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

Find the area of the region bounded by the curve $C _ { 1 }$.
The curve $C _ { 2 }$ whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve $C _ { 1 }$ at the pole $O$ and at the point $A$. The straight line drawn through $A$ parallel to the initial line intersects $C _ { 1 }$ again at the point $B$.
1. Find the polar coordinates of $A$.
2. Show that the length of $O B$ is $\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )$.
3. Find the length of $A B$, giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141}
  \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141}
  \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}

AQA FP3 2016 June Q1

3 marks

1

Find the values of the constants $a$ and $b$ for which $a x + b$ is a particular integral of the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$$
Hence find the general solution of $2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$.
[0pt] [3 marks]

AQA FP3 2016 June Q2

2

Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 5 }$.
It is given that the first non-zero term in the expansion of $$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$ in ascending powers of $x$ is $q x ^ { 5 }$.
Find the values of the rational numbers $p$ and $q$.

AQA FP3 2016 June Q3

1 marks

3

It is given that $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = ( 2 x + 1 ) \ln ( x + y )$$ and $$y ( 0 ) = 2$$ Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.1$, to obtain an approximation to $y ( 0.1 )$, giving your answer to three decimal places.
It is given that $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) \ln ( x + y )$$ and $y = 2$ when $x = 0$.
1. Use implicit differentiation to find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, giving your answer in terms of $x$ and $y$.
2. Hence find the first three non-zero terms in the expansion, in ascending powers of $x$, of $y ( x )$. Give your answer in an exact form.
3. Use your answer to part (b)(ii) to obtain an approximation to $y ( 0.1 )$, giving your answer to three decimal places.
  [0pt] [1 mark]

AQA FP3 2016 June Q4

1 marks

4

The curve with Cartesian equation $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ is mapped onto the curve with polar equation $r = \frac { 10 } { 3 - 2 \cos \theta }$ by a single geometrical transformation. By writing the polar equation as a Cartesian equation in a suitable form, find the values of the constants $c$ and $d$.
Hence describe the geometrical transformation referred to in part (a).
[0pt] [1 mark]

AQA FP3 2016 June Q5

11 marks

5

Express $\frac { 1 } { ( 1 + x ) ( 2 + x ) }$ in the form $\frac { A } { 1 + x } + \frac { B } { 2 + x }$, where $A$ and $B$ are integers.
Use the substitution $u = \frac { \mathrm { d } y } { \mathrm {~d} x }$ to solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { 1 } { ( 1 + x ) ( 2 + x ) } \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + x } { 1 + x }$$ given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 4$ when $x = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.
[0pt] [11 marks]

AQA FP3 2016 June Q6

4 marks

6

Use the substitution $a = \frac { 1 } { p }$ to find $\lim _ { p \rightarrow \infty } \left[ \frac { \ln p } { p ^ { k } } \right]$, where $k > 0$.
Evaluate the improper integral $\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 7 } } \mathrm {~d} x$, showing the limiting process used.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-16_2039_1719_671_148}

AQA FP3 2016 June Q7

10 marks

7 Find the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 10 \mathrm { e } ^ { 4 x } + 8 \sin 2 x + 4 \cos 2 x$$ given that $y = 2.5$ when $x = 0$ and $y = \frac { \pi } { 4 }$ when $x = \frac { \pi } { 4 }$.
[0pt] [10 marks]

AQA FP3 2016 June Q8

7 marks

8 The diagram shows the sketch of part of a curve, the pole $O$ and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is $r = 1 + \tan \theta$.
The point $A$ is the point on the curve at which $\theta = \frac { \pi } { 3 }$.
The perpendicular, $A N$, from $A$ to the initial line intersects the curve at the point $B$.

Find the exact length of $O A$.
1. Given that, at the point $B , \theta = \alpha$, show that $( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }$.
2. Hence, or otherwise, find $\alpha$ in terms of $\pi$.
Show that the area of triangle $O A B$ is $\frac { 3 + 2 \sqrt { 3 } } { 6 }$.
Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment $A B$.
[0pt] [7 marks]
\includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

OCR FP3 2011 January Q5

Find the general solution of the differential equation $$3 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = - 2 x + 13 .$$
Find the particular solution for which $y = - \frac { 7 } { 2 }$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.
Write down the function to which $y$ approximates when $x$ is large and positive.
$6 Q$ is a multiplicative group of order 12.
Two elements of $Q$ are $a$ and $r$. It is given that $r$ has order 6 and that $a ^ { 2 } = r ^ { 3 }$. Find the orders of the elements $a , a ^ { 2 } , a ^ { 3 }$ and $r ^ { 2 }$. The table below shows the number of elements of $Q$ with each possible order.
Order of element 1 2 3 4 6
Number of elements 1 1 2 6 2
$G$ and $H$ are the non-cyclic groups of order 4 and 6 respectively.
Construct two tables, similar to the one above, to show the number of elements with each possible order for the groups $G$ and $H$. Hence explain why there are no non-cyclic proper subgroups of $Q$.

OCR FP3 2007 June Q7

Show that $\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1$.
Write down the seven roots of the equation $z ^ { 7 } = 1$ in the form $\mathrm { e } ^ { \mathrm { i } \theta }$ and show their positions in an Argand diagram.
Hence express $z ^ { 7 } - 1$ as the product of one real linear factor and three real quadratic factors.

OCR FP3 2013 June Q2

Write down the operation table and, assuming associativity, show that $G$ is a group.
State the order of each element.
Find all the proper subgroups of $G$. The group $H$ consists of the set $\{ 1,3,7,9 \}$ with the operation of multiplication modulo 10 .
Explaining your reasoning, determine whether $H$ is isomorphic to $G$.

OCR FP3 2016 June Q7

Use de Moivre's theorem to show that $$\sin 6 \theta \equiv \cos \theta \left( 6 \sin \theta - 32 \sin ^ { 3 } \theta + 32 \sin ^ { 5 } \theta \right)$$
Hence show that, for $\sin 2 \theta \neq 0$, $$- 1 \leqslant \frac { \sin 6 \theta } { \sin 2 \theta } < 3$$

Edexcel FP3 Q1

Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form $\ln a$, where $a$ is a rational number.

Edexcel FP3 Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{045545c7-06d9-40b6-9d01-fc792ab0aa07-01_222_241_525_2042} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The points $A , B$ and $C$ have position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ respectively, relative to a fixed origin $O$, as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate

$\mathbf { b } \times \mathbf { c }$,
$\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )$,
the area of triangle $O B C$,
the volume of the tetrahedron $O A B C$.

Edexcel FP3 Q4

4. Given that $y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0$,

find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer as a simplified fraction.
Hence, or otherwise, find $$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form $\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)$, where $a$ and $b$ are integers.

Edexcel FP3 Q5

5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$

Find an expression for $\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5$.
Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
Find $I _ { 4 }$ in the form $k \pi$, where $k$ is a fraction.

Edexcel FP3 Q6

6. The hyperbola $H$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, where $a$ and $b$ are constants. The line $L$ has equation $y = m x + c$, where $m$ and $c$ are constants.

Given that $L$ and $H$ meet, show that the $x$-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that $L$ is a tangent to $H$,
show that $a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }$. The hyperbola $H ^ { \prime }$ has equation $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1$.
Find the equations of the tangents to $H ^ { \prime }$ which pass through the point $( 1,4 )$.

Edexcel FP3 Q7

7. The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $$\mathbf { r } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha
- 4
0 \end{array} \right) + \mu \left( \begin{array} { l } 0
3
2 \end{array} \right) .$$ If the lines $l _ { 1 }$ and $l _ { 2 }$ intersect, find

the value of $\alpha$,
an equation for the plane containing the lines $l _ { 1 }$ and $l _ { 2 }$, giving your answer in the form $a x + b y + c z + d = 0$, where $a , b , c$ and $d$ are constants. For other values of $\alpha$, the lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect and are skew lines.
Given that $\alpha = 2$,
find the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$.

Edexcel FP3 Q8

8. A curve, which is part of an ellipse, has parametric equations $$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The curve is rotated through $2 \pi$ radians about the $x$-axis.

Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { a } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \text { where } c = \cos \theta$$ and where $k$ and $\alpha$ are constants to be found.
Using the substitution $c = \frac { 3 } { 4 } \sinh u$, or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures. \section*{END} \section*{TOTAL FOR PAPER: 75 MARKS} Mathematical Formulae (Pink) Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP3), the paper reference (6669), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
There are 8 questions in this question paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner.
Answers without working may not gain full credit. N35389RA ${ } _ { \text {This publication may only be reproduced in accordance with Edexcel Limited copyright policy. } }$ ©2010 Edexcel Limited.
1. The line $x = 8$ is a directrix of the ellipse with equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > 0 , \quad b > 0$$ and the point $( 2,0 )$ is the corresponding focus.
Find the value of $a$ and the value of $b$.
2. Use calculus to find the exact value of $\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x$.
3. (a) Starting from the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, prove that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$
Solve the equation $$\cosh 2 x - 3 \sinh x = 15$$ giving your answers as exact logarithms.
4. $$I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a \geq 0 , \quad n \geq 0$$
Show that, for $n \geq 2$, $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
Hence evaluate $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x \mathrm {~d} x$.
5. Given that $y = ( \operatorname { arcosh } 3 x ) ^ { 2 }$, where $3 x > 1$, show that
$\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y$,
$\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18$.
6. $\mathbf { M } = \left( \begin{array} { r r r } 1 & 0 & 3
0 & - 2 & 1
k & 0 & 1 \end{array} \right)$, where $k$ is a constant.
Given that $\left( \begin{array} { l } 6
1
6 \end{array} \right)$ is an eigenvector of $\mathbf { M }$,
find the eigenvalue of $\mathbf { M }$ corresponding to $\left( \begin{array} { l } 6
1
6 \end{array} \right)$,
show that $k = 3$,
show that $\mathbf { M }$ has exactly two eigenvalues. A transformation $T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }$ is represented by $\mathbf { M }$.
The transformation $T$ maps the line $l _ { 1 }$, with cartesian equations $\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }$, onto the line $l _ { 2 }$.
Taking $k = 3$, find cartesian equations of $l _ { 2 }$.
7. The plane $\Pi$ has vector equation $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
Find an equation of $\Pi$ in the form $\mathbf { r } \cdot \mathbf { n } = p$, where $\mathbf { n }$ is a vector perpendicular to $\Pi$ and $p$ is a constant. The point $P$ has coordinates $( 6,13,5 )$. The line $l$ passes through $P$ and is perpendicular to $\Pi$. The line $l$ intersects $\Pi$ at the point $N$.
Show that the coordinates of $N$ are $( 3,1 , - 1 )$. The point $R$ lies on $\Pi$ and has coordinates $( 1,0,2 )$.
Find the perpendicular distance from $N$ to the line $P R$. Give your answer to 3 significant figures.
8. The hyperbola $H$ has equation $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$. The line $l _ { 1 }$ is the tangent to $H$ at the point $P ( 4 \sec t , 2 \tan t )$.
Use calculus to show that an equation of $l _ { 1 }$ is $$2 y \sin t = x - 4 \cos t$$ The line $l _ { 2 }$ passes through the origin and is perpendicular to $l _ { 1 }$.
The lines $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $Q$.
Show that, as $t$ varies, an equation of the locus of $Q$ is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$

Edexcel FP3 Q9

9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$

Show that, for $n > 0$, $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
Find $I _ { 2 }$.

Questions FP3 (473 questions)

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