Questions FP1 - A-Level Maths

$\mathbf { e }$ is an eigenvector of $\mathbf { G } + k \mathbf { I }$, where $k$ is a scalar and $\mathbf { I }$ is an identity matrix,
$\mathbf { e }$ is an cigenvector of $\mathbf { G } ^ { 2 }$. Find the eigenvalues, and corresponding eigenvectors, of the matrices $\mathbf { A }$ and $\mathbf { B } ^ { 2 }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0
1 & 0 & 1
- 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0
1 & - 8 & 1
- 1 & 3 & - 6 \end{array} \right)$$

CAIE FP1 2002 November Q11 OR

The curve $C$ has equation $$y = \frac { ( x - a ) ( x - b ) } { x - c }$$ where $a , b , c$ are constants, and it is given that $0 < a < b < c$.

Express $y$ in the form $$x + P + \frac { Q } { x - c }$$ giving the constants $P$ and $Q$ in terms of $a , b$ and $c$.
Find the equations of the asymptotes of $C$.
Show that $C$ has two stationary points.
Given also that $a + b > c$, sketch $C$, showing the asymptotes and the coordinates of the points of intersection of $C$ with the axes.

CAIE FP1 2003 November Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{653d57aa-7775-4063-a8c9-11c8bc964fae-2_566_606_264_772} The curve $C$ has polar equation $$r = \theta ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { \theta ^ { 2 } / \pi }$$ where $0 \leqslant \theta \leqslant \pi$. The area of the finite region bounded by $C$ and the line $\theta = \beta$ is $\pi$ (see diagram). Show that $$\beta = ( \pi \ln 3 ) ^ { \frac { 1 } { 2 } }$$

CAIE FP1 2003 November Q2

2 Given that $$u _ { n } = \frac { 1 } { n ^ { 2 } - n + 1 } - \frac { 1 } { n ^ { 2 } + n + 1 } ,$$ find $S _ { N } = \sum _ { n = N + 1 } ^ { 2 N } u _ { n }$ in terms of $N$. Find a number $M$ such that $S _ { N } < 10 ^ { - 20 }$ for all $N > M$.

CAIE FP1 2003 November Q3

3 Three $n \times 1$ column vectors are denoted by $\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }$, and $\mathbf { M }$ is an $n \times n$ matrix. Show that if $\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }$ are linearly dependent then the vectors $\mathbf { M x } _ { 1 } , \mathbf { M x } _ { 2 } , \mathbf { M x } _ { 3 }$ are also linearly dependent. The vectors $\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }$ and the matrix $\mathbf { P }$ are defined as follows: $$\begin{gathered} \mathbf { y } _ { 1 } = \left( \begin{array} { l } 1
5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3

CAIE FP1 2003 November Q5

5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) , \quad \mathbf { y } _ { 3 } = \left( \begin{array} { r } 5
51
55 \end{array} \right)
\mathbf { P } = \left( \begin{array} { r r r } 1 & - 4 & 3
0 & 2 & 5
0 & 0 & - 7 \end{array} \right) . \end{gathered}$$

Show that $\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }$ are linearly dependent.
Find a basis for the linear space spanned by the vectors $\mathbf { P y } _ { 1 } , \mathbf { P y } _ { 2 } , \mathbf { P y } _ { 3 }$. 4 Given that $y = x \sin x$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$, simplifying your results as far as possible, and show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = - x \sin x + 6 \cos x$$ Use induction to establish an expression for $\frac { \mathrm { d } ^ { 2 n } y } { \mathrm {~d} x ^ { 2 n } }$, where $n$ is a positive integer. 5 The integral $I _ { n }$ is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$$ By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( \tan x \sec ^ { n } x \right)$, or otherwise, show that $$( n + 1 ) I _ { n + 2 } = 2 ^ { \frac { 1 } { 2 } n } + n I _ { n }$$ Find the value of $I _ { 6 }$.

CAIE FP1 2003 November Q7

7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) , \quad \mathbf { y } _ { 3 } = \left( \begin{array} { r } 5
51
55 \end{array} \right)
\mathbf { P } = \left( \begin{array} { r r r } 1 & - 4 & 3
0 & 2 & 5
0 & 0 & - 7 \end{array} \right) . \end{gathered}$$

Show that $\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }$ are linearly dependent.
Find a basis for the linear space spanned by the vectors $\mathbf { P y } _ { 1 } , \mathbf { P y } _ { 2 } , \mathbf { P y } _ { 3 }$. 4 Given that $y = x \sin x$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$, simplifying your results as far as possible, and show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = - x \sin x + 6 \cos x$$ Use induction to establish an expression for $\frac { \mathrm { d } ^ { 2 n } y } { \mathrm {~d} x ^ { 2 n } }$, where $n$ is a positive integer. 5 The integral $I _ { n }$ is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$$ By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( \tan x \sec ^ { n } x \right)$, or otherwise, show that $$( n + 1 ) I _ { n + 2 } = 2 ^ { \frac { 1 } { 2 } n } + n I _ { n }$$ Find the value of $I _ { 6 }$. 6 Find the sum of the squares of the roots of the equation $$x ^ { 3 } + x + 12 = 0$$ and deduce that only one of the roots is real. The real root of the equation is denoted by $\alpha$. Prove that $- 3 < \alpha < - 2$, and hence prove that the modulus of each of the other roots lies between 2 and $\sqrt { } 6$. 7 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 y = \mathrm { e } ^ { - \alpha t }$$ where $\alpha$ is a constant and $\alpha \neq 2$. Show that if $\alpha < 2$ then, whatever the initial conditions, $y \mathrm { e } ^ { \alpha t } \rightarrow \frac { 1 } { ( 2 - \alpha ) ^ { 2 } }$ as $t \rightarrow \infty$.

CAIE FP1 2003 November Q8

8 Given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$ and $n$ is a positive integer, show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ Hence express $\sin ^ { 6 } \theta$ in the form $$p \cos 6 \theta + q \cos 4 \theta + r \cos 2 \theta + s$$ where the constants $p , q , r , s$ are to be determined. Hence find the mean value of $\sin ^ { 6 } \theta$ with respect to $\theta$ over the interval $0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$.

CAIE FP1 2003 November Q9

9 The line $l _ { 1 }$ passes through the point $A$ with position vector $\mathbf { i } - \mathbf { j } - 2 \mathbf { k }$ and is parallel to the vector $3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }$. The variable line $l _ { 2 }$ passes through the point $( 1 + 5 \cos t ) \mathbf { i } - ( 1 + 5 \sin t ) \mathbf { j } - 14 \mathbf { k }$, where $0 \leqslant t < 2 \pi$, and is parallel to the vector $15 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }$. The points $P$ and $Q$ are on $l _ { 1 }$ and $l _ { 2 }$ respectively, and $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$.

Find the length of $P Q$ in terms of $t$.
Hence show that the lines $l _ { 1 }$ and $l _ { 2 }$ do not intersect, and find the maximum length of $P Q$ as $t$ varies.
The plane $\Pi _ { 1 }$ contains $l _ { 1 }$ and $P Q$; the plane $\Pi _ { 2 }$ contains $l _ { 2 }$ and $P Q$. Find the angle between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$, correct to the nearest tenth of a degree.

CAIE FP1 2003 November Q10

10 Find the eigenvalues and corresponding eigenvectors of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 4 & 1
- 6 & - 1 & 3
8 & 8 & 4 \end{array} \right)$$ Hence find a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } + \mathbf { A } ^ { 2 } + \mathbf { A } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$.

CAIE FP1 2003 November Q11 EITHER

The curve $C$ has equation $y = \frac { 5 ( x - 1 ) ( x + 2 ) } { ( x - 2 ) ( x + 3 ) }$.

Express $y$ in the form $P + \frac { Q } { x - 2 } + \frac { R } { x + 3 }$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ for exactly one value of $x$ and find the corresponding value of $y$.
Write down the equations of all the asymptotes of $C$.
Find the set of values of $k$ for which the line $y = k$ does not intersect $C$.

CAIE FP1 2003 November Q11 OR

A curve has equation $y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } }$, for $x \geqslant 0$. The arc of the curve joining the origin to the point where $x = 3$ is denoted by $R$.

Find the length of $R$.
Find the $y$-coordinate of the centroid of the region bounded by the $x$-axis, the line $x = 3$ and $R$.
Show that the area of the surface generated when $R$ is rotated through one revolution about the $y$-axis is $\frac { 232 } { 15 } \pi$.

CAIE FP1 2004 November Q1

1 The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 5 & 2 & 6
2 & 0 & - 1 & 7
3 & - 1 & - 2 & 10
4 & 10 & 13 & 29 \end{array} \right)$$ Find the dimension of the null space of T .

CAIE FP1 2004 November Q2

2 The curve $C$ is defined parametrically by $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ where $a$ is a positive constant. Find the area of the surface generated when $C$ is rotated through one complete revolution about the $x$-axis.

CAIE FP1 2004 November Q3

3 Given that $$\alpha + \beta + \gamma = 0 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 14 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 18$$ find a cubic equation whose roots are $\alpha , \beta , \gamma$. Hence find possible values for $\alpha , \beta , \gamma$.

CAIE FP1 2004 November Q4

4 The curve $C$ has polar equation $$r = \mathrm { e } ^ { \frac { 1 } { 5 } \theta } , \quad 0 \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$$

Draw a sketch of $C$.
Find the length of $C$, correct to 3 significant figures.

CAIE FP1 2004 November Q5

5 Let $$S _ { N } = \sum _ { n = 1 } ^ { N } ( - 1 ) ^ { n - 1 } n ^ { 3 }$$ Find $S _ { 2 N }$ in terms of $N$, simplifying your answer as far as possible. Hence write down an expression for $S _ { 2 N + 1 }$ and find the limit, as $N \rightarrow \infty$, of $\frac { S _ { 2 N + 1 } } { N ^ { 3 } }$.

CAIE FP1 2004 November Q6

6 Write down all the 8th roots of unity. Verify that $$\left( z - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 } - ( 2 \cos \theta ) z + 1$$ Hence express $z ^ { 8 } - 1$ as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form.

CAIE FP1 2004 November Q7

7 The curve $C$ has equation $$x y + ( x + y ) ^ { 5 } = 1$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 5 } { 6 }$ at the point $A ( 1,0 )$ on $C$.
Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.

CAIE FP1 2004 November Q8

8 The sequence of real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that $a _ { 1 } = 1$ and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { \lambda }$$ where $\lambda$ is a constant greater than 1 . Prove by mathematical induction that, for $n \geqslant 2$, $$a _ { n } \geqslant 2 ^ { \mathrm { g } ( n ) }$$ where $g ( n ) = \lambda ^ { n - 1 }$. Prove also that, for $n \geqslant 2 , \frac { a _ { n + 1 } } { a _ { n } } > 2 ^ { ( \lambda - 1 ) \mathrm { g } ( n ) }$.

CAIE FP1 2004 November Q9

9 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 3 } \right) ^ { - n } \mathrm {~d} x$$ where $n > 0$.

Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x \left( 1 + x ^ { 3 } \right) ^ { - n } \right] = - ( 3 n - 1 ) \left( 1 + x ^ { 3 } \right) ^ { - n } + 3 n \left( 1 + x ^ { 3 } \right) ^ { - n - 1 }$$ and hence, or otherwise, show that $$I _ { n + 1 } = \frac { 2 ^ { - n } } { 3 n } + \left( 1 - \frac { 1 } { 3 n } \right) I _ { n }$$
By considering the graph of $y = \frac { 1 } { 1 + x ^ { 3 } }$, show that $I _ { 1 } < 1$.
Deduce that $I _ { 3 } < \frac { 53 } { 72 }$.

CAIE FP1 2004 November Q10

10 The curve $C$ has equation $$y = \frac { x ^ { 2 } + 2 x - 3 } { ( \lambda x + 1 ) ( x + 4 ) }$$ where $\lambda$ is a constant.

Find the equations of the asymptotes of $C$ for the case where $\lambda = 0$.
Find the equations of the asymptotes of $C$ for the case where $\lambda$ is not equal to any of $- 1,0 , \frac { 1 } { 4 } , \frac { 1 } { 3 }$.
Sketch $C$ for the case where $\lambda = - 1$. Show, on your diagram, the equations of the asymptotes and the coordinates of the points of intersection of $C$ with the coordinate axes.

CAIE FP1 2004 November Q11

11 The line $l _ { 1 }$ passes through the point $A$, whose position vector is $3 \mathbf { i } - 5 \mathbf { j } - 4 \mathbf { k }$, and is parallel to the vector $3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }$. The line $l _ { 2 }$ passes through the point $B$, whose position vector is $2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }$, and is parallel to the vector $\mathbf { i } - \mathbf { j } - 4 \mathbf { k }$. The point $P$ on $l _ { 1 }$ and the point $Q$ on $l _ { 2 }$ are such that $P Q$ is perpendicular to both $l _ { 1 }$ and $l _ { 2 }$. The plane $\Pi _ { 1 }$ contains $P Q$ and $l _ { 1 }$, and the plane $\Pi _ { 2 }$ contains $P Q$ and $l _ { 2 }$.

Find the length of $P Q$.
Find a vector perpendicular to $\Pi _ { 1 }$.
Find the perpendicular distance from $B$ to $\Pi _ { 1 }$.
Find the angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.

CAIE FP1 2004 November Q12 EITHER

The variable $y$ depends on $x$, and the variables $x$ and $t$ are related by $x = \mathrm { e } ^ { t }$. Show that $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t } \quad \text { and } \quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } .$$

Given that $y$ satisfies the differential equation $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 50 ( \ln x ) - 1$$ find a differential equation involving only $t$ and $y$.
Show that the complementary function of the differential equation in $t$ and $y$ may be written in the form $$R \mathrm { e } ^ { - \frac { 3 } { 2 } t } \sin ( 2 t + \phi )$$ where $R$ and $\phi$ are arbitrary constants.
Find a particular integral of the differential equation in $t$ and $y$.
Hence find the general solution of the differential equation in $x$ and $y$.

CAIE FP1 2004 November Q12 OR

The matrix $\mathbf { A }$ has $\lambda$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. Show that if $\mathbf { A }$ is non-singular then

$\lambda \neq 0$,
the matrix $\mathbf { A } ^ { - 1 }$ has $\lambda ^ { - 1 }$ as an eigenvalue with $\mathbf { e }$ as a corresponding eigenvector. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 2
0 & - 2 & 4
0 & 0 & - 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 4 \mathbf { I } ) ^ { - 1 }$$ Find a non-singular matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { B } = \mathbf { P D P } ^ { - 1 }$.

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