Questions - Edexcel FP3 - A-Level Maths

Edexcel FP3 2018 June Q5

5. Given that $$I _ { n } = \int x ^ { n } \sqrt { ( x + 8 ) } \mathrm { d } x , \quad n \geqslant 0 , x \geqslant 0$$

show that, for $n \geqslant 1$ $$I _ { n } = \frac { p x ^ { n } ( x + 8 ) ^ { \frac { 3 } { 2 } } } { 2 n + 3 } - \frac { q n } { 2 n + 3 } I _ { n - 1 }$$ where $p$ and $q$ are constants to be found.
Use part (a) to find the exact value of $$\int _ { 0 } ^ { 10 } x ^ { 2 } \sqrt { ( x + 8 ) } d x$$ giving your answer in the form $k \sqrt { 2 }$, where $k$ is rational.

Edexcel FP3 2018 June Q6

6. The line $l _ { 1 }$ has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )$$ where $\lambda$ is a scalar parameter. The line $l _ { 2 }$ has equation $$\frac { x + 1 } { 1 } = \frac { y - 4 } { 1 } = \frac { z - 1 } { 3 }$$

Prove that the lines $l _ { 1 }$ and $l _ { 2 }$ are skew.
Find the shortest distance between the lines $l _ { 1 }$ and $l _ { 2 }$ The plane $\Pi$ contains $l _ { 1 }$ and intersects $l _ { 2 }$ at the point $( 3,8,13 )$.
Find a cartesian equation for the plane $\Pi$.

Edexcel FP3 2018 June Q7

7. The ellipse $E$ has foci at the points $( \pm 3,0 )$ and has directrices with equations $x = \pm \frac { 25 } { 3 }$

Find a cartesian equation for the ellipse $E$. The straight line $l$ has equation $y = m x + c$, where $m$ and $c$ are positive constants.
Show that the $x$ coordinates of any points of intersection of $l$ and $E$ satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line $l$ is a tangent to $E$,
show that $c ^ { 2 } = p m ^ { 2 } + q$, where $p$ and $q$ are constants to be found. The line $l$ intersects the $x$-axis at the point $A$ and intersects the $y$-axis at the point $B$.
Show that the area of triangle $O A B$, where $O$ is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
Find the minimum area of triangle $O A B$.
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Q7

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Edexcel FP3 Specimen Q1

Find the eigenvalues of the matrix $\left( \begin{array} { l l } 7 & 6
6 & 2 \end{array} \right)$
Find the values of $x$ for which

$$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms.
(Total 6 marks)

Edexcel FP3 Specimen Q3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6c256e1b-455d-42fb-81f2-a9a8ed1148bc-2_503_801_998_566}

\end{figure} The parametric equations of the curve $C$ shown in Figure 1 are $$x = a ( t - \sin t ) , \quad y = a ( 1 - \cos t ) , \quad 0 \leq t \leq 2 \pi$$ Find, by using integration, the length of $C$.

Edexcel FP3 Specimen Q4

4. Find $\int \sqrt { } \left( x ^ { 2 } + 4 \right) \mathrm { d } x$.

Edexcel FP3 Specimen Q5

5. Given that $y = \arcsin x$ prove that

$\quad \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { \left( 1 - x ^ { 2 } \right) } }$
$\quad \left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0$

Edexcel FP3 Specimen Q6

6. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \sin x \mathrm {~d} x$$

Show that for $n \geq 2$ $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
Hence obtain $I _ { 3 }$, giving your answers in terms of $\pi$.

Edexcel FP3 Specimen Q7

7. $$\mathbf { A } ( x ) = \left( \begin{array} { c c c } 1 & x & - 1
3 & 0 & 2
1 & 1 & 0 \end{array} \right) , x \neq \frac { 5 } { 2 }$$

Calculate the inverse of $\mathbf { A } ( x )$. $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & - 1
3 & 0 & 2
1 & 1 & 0 \end{array} \right)$$ The image of the vector $\left( \begin{array} { c } p
q
r \end{array} \right)$ when transformed by $\mathbf { B }$ is $\left( \begin{array} { l } 2
3
4 \end{array} \right)$
Find the values of $p , q$ and $r$.

Edexcel FP3 Specimen Q8

8. The points $A , B , C$, and $D$ have position vectors $$\mathbf { a } = 2 \mathbf { i } + \mathbf { k } , \mathrm { b } = \mathbf { i } + 3 \mathbf { j } , \mathbf { c } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \mathbf { d } = 4 \mathbf { j } + \mathbf { k }$$ respectively.

Find $\overrightarrow { A B } \times \overrightarrow { A C }$ and hence find the area of triangle $A B C$.
Find the volume of the tetrahedron $A B C D$.
Find the perpendicular distance of $D$ from the plane containing $A , B$ and $C$.

Edexcel FP3 Specimen Q9

9. The hyperbola $C$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$

Show that an equation of the normal to $C$ at $P ( a \sec \theta , b \tan \theta )$ is $$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The normal at $P$ cuts the coordinate axes at $A$ and $B$. The mid-point of $A B$ is $M$.
Find, in cartesian form, an equation of the locus of $M$ as $\theta$ varies.
(Total 13 marks)

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 $I _ { 2 }$.

Edexcel FP3 Q1

An ellipse has equation $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$.
1. Sketch the ellipse.
2. Find the value of the eccentricity $e$.
3. State the coordinates of the foci of the ellipse.
4. Solve the equation
$$10 \cosh x + 2 \sinh x = 11 .$$ Give each answer in the form $\ln a$ where $a$ is a rational number.
[0pt] [P5 June 2002 Qn 3]

Edexcel FP3 Q4

4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \cos x \mathrm {~d} x , \quad n \geq 0$$

Prove that $I _ { n } = \left( \frac { \pi } { 2 } \right) ^ { n } - n ( n - 1 ) I _ { n - 2 } , n \geq 2$.
Find an exact expression for $I _ { 6 }$.
[0pt] [P5 June 2002 Qn 4]

Edexcel FP3 Q5

5. (a) Given that $y = \arctan 3 x$, and assuming the derivative of $\tan x$, prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 1 + 9 x ^ { 2 } }$$ (b) Show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 3 } } 6 x \arctan 3 x \mathrm {~d} x = \frac { 1 } { 9 } ( 4 \pi - 3 \sqrt { } 3 )$$ (6)
[0pt] [P5 June 2002 Qn 6] \section*{6.} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6706ed7f-4575-4898-b757-aee8475b2a30-04_815_431_351_913}

\end{figure} The curve $C$ shown in Fig. 1 has equation $y ^ { 2 } = 4 x , 0 \leq x \leq 1$.
The part of the curve in the first quadrant is rotated through $2 \pi$ radians about the $x$-axis.
(a) Show that the surface area of the solid generated is given by $$4 \pi \int _ { 0 } ^ { 1 } \sqrt { ( 1 + x ) } d x$$ (b) Find the exact value of this surface area.
(c) Show also that the length of the curve $C$, between the points $( 1 , - 2 )$ and $( 1,2 )$, is given by $$2 \int _ { 0 } ^ { 1 } \sqrt { \left( \frac { x + 1 } { x } \right) } \mathrm { d } x$$ (d) Use the substitution $x = \sinh ^ { 2 } \theta$ to show that the exact value of this length is $$2 [ \sqrt { } 2 + \ln ( 1 + \sqrt { } 2 ) ]$$ [P5 June 2002 Qn 8]

Edexcel FP3 Q7

7. Prove that $\sinh ( \mathrm { i } \pi - \theta ) = \sinh \theta$.
[0pt] [P6 June 2002 Qn 1]

Edexcel FP3 Q8

8. $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 0 & 4
0 & 5 & 4
4 & 4 & 3 \end{array} \right)$$

Verify that $\left( \begin{array} { r } 2
- 2
1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$ and find the corresponding eigenvalue.
Show that 9 is another eigenvalue of $\mathbf { A }$ and find the corresponding eigenvector.
Given that the third eigenvector of $\mathbf { A }$ is $\left( \begin{array} { r } 2
1
- 2 \end{array} \right)$, write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { A } \mathbf { P } = \mathbf { D } .$$

Edexcel FP3 Q9

The plane $\Pi$ passes through the points

$$A ( - 1 , - 1,1 ) , B ( 4,2,1 ) \text { and } C ( 2,1,0 )$$

Find a vector equation of the line perpendicular to $\Pi$ which passes through the point $D ( 1,2,3 )$.
Find the volume of the tetrahedron $A B C D$.
Obtain the equation of $\Pi$ in the form r.n $= p$. The perpendicular from $D$ to the plane $\Pi$ meets $\Pi$ at the point $E$.
Find the coordinates of $E$.
Show that $D E = \frac { 11 \sqrt { 35 } } { 35 }$. The point $D ^ { \prime }$ is the reflection of $D$ in $\Pi$.
Find the coordinates of $D ^ { \prime }$.
(3)
[0pt] [P6 June 2002 Qn 7]

Questions — Edexcel FP3 (136 questions)

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Edexcel FP3 2018 June Q5

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