Questions - CAIE FP1 - A-Level Maths

CAIE FP1 2013 November Q1

1 The curve $C$ has polar equation $r = 2 \mathrm { e } ^ { \theta }$, for $\frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. Find

the area of the region bounded by the half-lines $\theta = \frac { 1 } { 6 } \pi , \theta = \frac { 1 } { 2 } \pi$ and $C$,
the length of $C$.

CAIE FP1 2013 November Q2

2 The cubic equation $x ^ { 3 } - p x - q = 0$, where $p$ and $q$ are constants, has roots $\alpha , \beta , \gamma$. Show that

$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 p$,
$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 q$,
$6 \left( \alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } \right) = 5 \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } \right)$.

CAIE FP1 2013 November Q3

3 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r } = 2 n ^ { 2 } + n$$ Write down the values of $S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }$. Express $u _ { r }$ in terms of $r$, justifying your answer. Find $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } .$$

CAIE FP1 2013 November Q4

4 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { x ^ { n } } { \sqrt { } ( 1 + 2 x ) } \mathrm { d } x$$ Show that, for $n \geqslant 1$, $$( 2 n + 1 ) I _ { n } = \sqrt { } 3 - n I _ { n - 1 }$$ Show that $$I _ { 3 } = \frac { 2 } { 35 } ( \sqrt { } 3 + 1 )$$

CAIE FP1 2013 November Q5

5 It is given that $y = ( 1 + x ) ^ { 2 } \ln ( 1 + x )$. Find $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$. Prove by mathematical induction that, for every integer $n \geqslant 3$, $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { 2 ( n - 3 ) ! } { ( 1 + x ) ^ { n - 2 } }$$

CAIE FP1 2013 November Q6

6 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 3 & - 1 & 2
4 & - 10 & 0 & 2
1 & - 1 & 3 & - 4
5 & - 12 & 1 & 1 \end{array} \right)$$ Find, in either order, the rank of $\mathbf { M }$ and a basis for the null space $K$ of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right)$$ and hence show that every solution of $$\mathbf { M x } = \left( \begin{array} { r } 2
16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$.

CAIE FP1 2013 November Q7

7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix.

CAIE FP1 2013 November Q8

8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right) + s \left( \begin{array} { l } 1
0
1 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
- 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$.

CAIE FP1 2013 November Q10

10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$. 7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. 8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right) + s \left( \begin{array} { l } 1
0
1 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
- 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$. 9 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$. 10 The curve $C$ has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where $p$ is a positive constant and $p \neq 3$.

Obtain the equations of the asymptotes of $C$.
Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.

CAIE FP1 2013 November Q16

16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where $\lambda$ and $\mu$ are real numbers and $\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}$ is a basis for $K$. 7 The square matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that $\mathbf { e }$ is an eigenvector of $\mathbf { A } ^ { 2 }$ and state the corresponding eigenvalue. Find the eigenvalues of the matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of $\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. 8 The plane $\Pi _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 2
3
- 1 \end{array} \right) + s \left( \begin{array} { l } 1
0
1 \end{array} \right) + t \left( \begin{array} { r } 1
- 1
- 2 \end{array} \right)$. Find a cartesian equation of $\Pi _ { 1 }$. The plane $\Pi _ { 2 }$ has equation $2 x - y + z = 10$. Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find an equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$. 9 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis. Find the coordinates of the centroid of the region bounded by $C$, the $x$-axis and the line $x = 1$. 10 The curve $C$ has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where $p$ is a positive constant and $p \neq 3$.

Obtain the equations of the asymptotes of $C$.
Find the value of $p$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
For the case $p = 1$, show that $C$ has no turning points, and sketch $C$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis. 11 Answer only one of the following two alternatives. \section*{EITHER} State the fifth roots of unity in the form $\cos \theta + \mathrm { i } \sin \theta$, where $- \pi < \theta \leqslant \pi$. Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form $\cos \theta \pm \mathrm { i } \sin \theta$. Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that $v = y ^ { 3 }$, show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for $y$ in terms of $x$, given that when $x = 0 , y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 1$.

CAIE FP1 2015 November Q1

1 The curve $C$ is defined parametrically by $$x = 2 \cos ^ { 3 } t \quad \text { and } \quad y = 2 \sin ^ { 3 } t , \quad \text { for } 0 < t < \frac { 1 } { 2 } \pi .$$ Show that, at the point with parameter $t$, $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$

CAIE FP1 2015 November Q4

4 The sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that, for all positive integers $n$, $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum $\sum _ { n = 1 } ^ { N } a _ { n }$ is denoted by $S _ { N }$. Find

the value of $S _ { 30 }$ correct to 3 decimal places,
the least value of $N$ for which $S _ { N } > 4.9$.

CAIE FP1 2015 November Q5

5 The cubic equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers, has roots $\alpha , \beta$ and $\gamma$, such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 ,
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 . \end{aligned}$$ Write down the value of $p$ and find the value of $q$. Given that $\alpha , \beta$ and $\gamma$ are all real and that $\alpha \beta + \alpha \gamma = 36$, find $\alpha$ and hence find the value of $r$.

CAIE FP1 2015 November Q6

6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0
10 & - 7 & 10
7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that $\left( \begin{array} { l } 0
2
1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$. Find the corresponding eigenvalue. Find a diagonal matrix $\mathbf { D }$ and matrices $\mathbf { P }$ and $\mathbf { P } ^ { - 1 }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$.

CAIE FP1 2015 November Q7

7 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1
3 & - 5 & - 7 & 7
5 & - 9 & - 13 & 9
7 & - 13 & - 19 & 11 \end{array} \right)$$ Find the rank of $\mathbf { M }$ and a basis for the null space of T . The vector $\left( \begin{array} { l } 1
2
3
4 \end{array} \right)$ is denoted by $\mathbf { e }$. Show that there is a solution of the equation $\mathbf { M x } = \mathbf { M e }$ of the form $\mathbf { x } = \left( \begin{array} { c } a
b
- 1
- 1 \end{array} \right)$, where the constants $a$ and $b$ are to be found.

CAIE FP1 2015 November Q8

8 The curve $C$ has equation $y = \frac { 2 x ^ { 2 } + k x } { x + 1 }$, where $k$ is a constant. Find the set of values of $k$ for which $C$ has no stationary points. For the case $k = 4$, find the equations of the asymptotes of $C$ and sketch $C$, indicating the coordinates of the points where $C$ intersects the coordinate axes.

CAIE FP1 2015 November Q9

9 It is given that $I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x$ for $n \geqslant 0$. Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2$$ Hence find, in an exact form, the mean value of $( \ln x ) ^ { 3 }$ with respect to $x$ over the interval $1 \leqslant x \leqslant \mathrm { e }$.

CAIE FP1 2015 November Q10

10 Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$ Hence show that the equation $x ^ { 2 } - 10 x + 5 = 0$ has roots $\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)$ and $\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)$. Deduce a quadratic equation, with integer coefficients, having roots $\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)$ and $\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)$.
[0pt] [Question 11 is printed on the next page.]

CAIE FP1 2015 November Q11 EITHER

The points $A , B$ and $C$ have position vectors $\mathbf { i } , 2 \mathbf { j }$ and $4 \mathbf { k }$ respectively, relative to an origin $O$. The point $N$ is the foot of the perpendicular from $O$ to the plane $A B C$. The point $P$ on the line-segment $O N$ is such that $O P = \frac { 3 } { 4 } O N$. The line $A P$ meets the plane $O B C$ at $Q$. Find a vector perpendicular to the plane $A B C$ and show that the length of $O N$ is $\frac { 4 } { \sqrt { } ( 21 ) }$. Find the position vector of the point $Q$. Show that the acute angle between the planes $A B C$ and $A B Q$ is $\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)$.

CAIE FP1 2015 November Q11 OR

The curve $C$ has polar equation $r = a ( 1 - \cos \theta )$ for $0 \leqslant \theta < 2 \pi$. Sketch $C$. Find the area of the region enclosed by the arc of $C$ for which $\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$, the half-line $\theta = \frac { 1 } { 2 } \pi$ and the half-line $\theta = \frac { 3 } { 2 } \pi$. Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) ,$$ where $s$ denotes arc length, and find the length of the arc of $C$ for which $\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

CAIE FP1 2015 November Q5

5 The cubic equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers, has roots $\alpha , \beta$ and $\gamma$, such that $$\begin{aligned} \alpha + \beta + \gamma & = 15
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$ Write down the value of $p$ and find the value of $q$. Given that $\alpha , \beta$ and $\gamma$ are all real and that $\alpha \beta + \alpha \gamma = 36$, find $\alpha$ and hence find the value of $r$.

CAIE FP1 2015 November Q6

6 The matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0
10 & - 7 & 10
7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that $\left( \begin{array} { l } 0
2
1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$. Find the corresponding eigenvalue. Find a diagonal matrix $\mathbf { D }$ and matrices $\mathbf { P }$ and $\mathbf { P } ^ { - 1 }$ such that $\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }$.

CAIE FP1 2017 November Q8

Find the value of $I _ { 2 }$.
Show that, for $n > 2$, $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
The curve $C$ has equation $y = \sec ^ { 3 } x$ for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$. The region $R$ is bounded by $C$, the $x$-axis, the $y$-axis and the line $x = \frac { 1 } { 4 } \pi$. Find the volume of revolution generated when $R$ is rotated through $2 \pi$ radians about the $x$-axis.

CAIE FP1 2019 November Q5

Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
Use the method of differences to express $T _ { N }$ in terms of $N$.
Find $\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)$.

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CAIE FP1 2015 November Q11 EITHER

CAIE FP1 2015 November Q11 OR

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