Questions FP1 - A-Level Maths

Show that $\lambda ^ { 2 } - 5 \lambda + 4 = 0$.
Find the acute angle between the planes through $A , B , D$ corresponding to the values of $\lambda$ satisfying the equation in part (i).

CAIE FP1 2008 June Q12 OR

Challenging +1.8

The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 2 & - 1 & - 1 \\ 1 & 3 & - 1 & 0 \\ 1 & 0 & 3 & 1 \\ 0 & 3 & - 4 & - 1 \end{array} \right) .$$ The range space of T is denoted by $V$.

Determine the dimension of $V$.
Show that the vectors $\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 2 \\ 3 \\ 0 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \\ - 4 \end{array} \right)$ are linearly independent.
Write down a basis of $V$. The set of elements of $\mathbb { R } ^ { 4 }$ which do not belong to $V$ is denoted by $W$.
State, with a reason, whether $W$ is a vector space.
Show that if the vector $\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)$ belongs to $W$ then $y - z - t \neq 0$.

CAIE FP1 2009 June Q1

Standard +0.8

1 The equation $$x ^ { 4 } - x ^ { 3 } - 1 = 0$$ has roots $\alpha , \beta , \gamma , \delta$. By using the substitution $y = x ^ { 3 }$, or by any other method, find the exact value of $\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } + \delta ^ { 6 }$.

CAIE FP1 2009 June Q2

Standard +0.8

2 Verify that, for all positive values of $n$, $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) } .$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find

the sum to $N$ terms,
the sum to infinity.

CAIE FP1 2009 June Q3

Challenging +1.2

3 The equation of a curve is $y = \lambda x ^ { 2 }$, where $\lambda > 0$. The region bounded by the curve, the $x$-axis and the line $x = a$, where $a > 0$, is denoted by $R$. The $y$-coordinate of the centroid of $R$ is $a$. Show that $\lambda = \frac { 10 } { 3 a }$.

CAIE FP1 2009 June Q4

Standard +0.8

4 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where $x = 0$ to the point where $x = 1$ is denoted by $s$. Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$ The surface area generated when this arc is rotated through one complete revolution about the $x$-axis is denoted by $S$. Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$ [Do not attempt to evaluate $s$ or $S$.]

CAIE FP1 2009 June Q5

Standard +0.8

5 Draw a sketch of the curve $C$ whose polar equation is $r = \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. On the same diagram draw the line $\theta = \alpha$, where $0 < \alpha < \frac { 1 } { 2 } \pi$. The region bounded by $C$ and the line $\theta = \frac { 1 } { 2 } \pi$ is denoted by $R$. Find the exact value of $\alpha$ for which the line $\theta = \alpha$ divides $R$ into two regions of equal area.

CAIE FP1 2009 June Q6

Standard +0.8

6 A curve has equation $$( x + y ) \left( x ^ { 2 } + y ^ { 2 } \right) = 1$$ Find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the point $( 0,1 )$.

CAIE FP1 2009 June Q7

Challenging +1.2

7 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where $n \geqslant 0$. Show that, for all $n \geqslant 1$, $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 }$$ Hence prove by induction that, for all positive integers $n$, $$I _ { n } < n ! .$$

CAIE FP1 2009 June Q8

Challenging +1.2

8 Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73$$ Show that, whatever the initial conditions, $\frac { y } { x ^ { 2 } } \rightarrow 1$ as $x \rightarrow \infty$.

CAIE FP1 2009 June Q9

Standard +0.8

9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4 \\ 1 & 5 & - 1 \\ 2 & 1 & 5 \end{array} \right)$$ has eigenvalues $1,5,7$. Find a set of corresponding eigenvectors. Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }$.
[0pt] [The evaluation of $\mathbf { P } ^ { - 1 }$ is not required.]
Determine the set of values of the real constant $k$ such that $k ^ { n } \mathbf { A } ^ { n }$ tends to the zero matrix as $n \rightarrow \infty$.

CAIE FP1 2009 June Q10

Standard +0.8

10 The curve $C$ has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where $\lambda$ is a non-zero constant. Obtain the equation of each of the asymptotes of $C$. In separate diagrams, sketch $C$ for the cases $\lambda > 0$ and $\lambda < 0$. In both cases the coordinates of the turning points must be indicated.

CAIE FP1 2009 June Q11

Challenging +1.8

11 The line $l _ { 1 }$ is parallel to the vector $4 \mathbf { j } - \mathbf { k }$ and passes through the point $A$ whose position vector is $2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }$. The variable line $l _ { 2 }$ is parallel to the vector $\mathbf { i } - ( 2 \sin t ) \mathbf { j }$, where $0 \leqslant t < 2 \pi$, and passes through the point $B$ whose position vector is $\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }$. 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 find the values of $t$ for which $l _ { 1 }$ and $l _ { 2 }$ intersect.
For the case $t = \frac { 1 } { 4 } \pi$, find the perpendicular distance from $A$ to the plane $B P Q$, giving your answer correct to 3 decimal places.

CAIE FP1 2009 June Q12 EITHER

Challenging +1.8

By considering $\sum _ { k = 0 } ^ { n - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { k }$, show that $$\sum _ { k = 0 } ^ { n - 1 } \cos k \theta \sec ^ { k } \theta = \cot \theta \sin n \theta \sec ^ { n } \theta$$ provided $\theta$ is not an integer multiple of $\frac { 1 } { 2 } \pi$. Hence or otherwise show that $$\sum _ { k = 0 } ^ { n - 1 } 2 ^ { k } \cos \left( \frac { 1 } { 3 } k \pi \right) = \frac { 2 ^ { n } } { \sqrt { 3 } } \sin \left( \frac { 1 } { 3 } n \pi \right)$$ Given that $0 < x < 1$, show that $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$

CAIE FP1 2009 June Q12 OR

Challenging +1.8

The linear transformations $\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ and $\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ are represented by the matrices $\mathbf { M } _ { 1 }$ and $\mathbf { M } _ { 2 }$, respectively, where $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 2 \\ 1 & 4 & 7 & 8 \\ 1 & 7 & 11 & 13 \\ 1 & 2 & 5 & 5 \end{array} \right) , \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 2 & 0 & - 1 & - 1 \\ 5 & 1 & - 3 & - 3 \\ 3 & - 1 & - 1 & - 1 \\ 13 & - 1 & - 6 & - 6 \end{array} \right) .$$

Find a basis for $R _ { 1 }$, the range space of $\mathrm { T } _ { 1 }$.
Find a basis for $K _ { 2 }$, the null space of $\mathrm { T } _ { 2 }$, and hence show that $K _ { 2 }$ is a subspace of $R _ { 1 }$. The set of vectors which belong to $R _ { 1 }$ but do not belong to $K _ { 2 }$ is denoted by $W$.
State whether $W$ is a vector space, justifying your answer. The linear transformation $\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }$ is the result of applying $\mathrm { T } _ { 1 }$ and then $\mathrm { T } _ { 2 }$, in that order.
Find the dimension of the null space of $\mathrm { T } _ { 3 }$.

CAIE FP1 2010 June Q1

Standard +0.8

1 The variables $x$ and $y$ are such that $y = - 1$ when $x = 1$ and $$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$ Find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 1$.

CAIE FP1 2010 June Q2

Standard +0.8

2 The curve $C$ has polar equation $$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$ where $a$ is a positive constant and $0 \leqslant \theta < 2 \pi$.

Draw a sketch of $C$.
Show that the area of the region bounded by $C$ and the lines $\theta = \ln 2$ and $\theta = \ln 4$ is $$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$

CAIE FP1 2010 June Q3

Challenging +1.8

3 At any point $( x , y )$ on the curve $C$, $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$ where the parameter $t$ is such that $0 \leqslant t \leqslant 2$. Show that the length of $C$ is $4 \sqrt { } 2$. Given that $y = 0$ when $t = 2$, determine the area of the surface generated when $C$ is rotated through one complete revolution about the $x$-axis, leaving your answer in an exact form.

CAIE FP1 2010 June Q4

Challenging +1.2

4 The sum $S _ { N }$ is defined by $S _ { N } = \sum _ { n = 1 } ^ { N } n ^ { 5 }$. Using the identity $$\left( n + \frac { 1 } { 2 } \right) ^ { 6 } - \left( n - \frac { 1 } { 2 } \right) ^ { 6 } \equiv 6 n ^ { 5 } + 5 n ^ { 3 } + \frac { 3 } { 8 } n$$ find $S _ { N }$ in terms of $N$. [You need not simplify your result.] Hence find $\lim _ { N \rightarrow \infty } N ^ { - \lambda } S _ { N }$, for each of the two cases

$\lambda = 6$,
$\lambda > 6$.

CAIE FP1 2010 June Q5

Challenging +1.2

5 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { n } \mathrm {~d} x$$ where $n \geqslant 1$. Show that $$I _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } ( n + 1 ) I _ { n }$$ Hence prove by induction that, for all positive integers $n , I _ { n }$ is of the form $A _ { n } \mathrm { e } ^ { 2 } + B _ { n }$, where $A _ { n }$ and $B _ { n }$ are rational numbers.

CAIE FP1 2010 June Q6

Challenging +1.2

6 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots $\alpha , \beta , \gamma$. Use the relation $x = \sqrt { } y$ to show that the equation $$y ^ { 3 } + 2 y ^ { 2 } + y - 1 = 0$$ has roots $\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }$. Let $S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }$.

Write down the value of $S _ { 2 }$ and show that $S _ { 4 } = 2$.
Find the values of $S _ { 6 }$ and $S _ { 8 }$.

CAIE FP1 2010 June Q7

Standard +0.8

7 The lines $l _ { 1 }$ and $l _ { 2 }$ have vector equations $$\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$$ respectively.

Show that $l _ { 1 }$ and $l _ { 2 }$ intersect.
Find the perpendicular distance from the point $P$ whose position vector is $3 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }$ to the plane containing $l _ { 1 }$ and $l _ { 2 }$.
Find the perpendicular distance from $P$ to $l _ { 1 }$.

CAIE FP1 2010 June Q8

Standard +0.3

8 The matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 1 & - 1 \\ - 4 & - 1 & 4 \\ 0 & - 1 & 5 \end{array} \right)$$ Given that one eigenvector of $\mathbf { A }$ is $\left( \begin{array} { r } 1 \\ - 2 \\ - 1 \end{array} \right)$, find the corresponding eigenvalue. Given also that another eigenvalue of $\mathbf { A }$ is 4, find a corresponding eigenvector. Given further that $\left( \begin{array} { r } 1 \\ - 4 \\ - 1 \end{array} \right)$ is an eigenvector of $\mathbf { A }$, with corresponding eigenvalue 1 , find matrices $\mathbf { P }$ and $\mathbf { Q }$, together with a diagonal matrix $\mathbf { D }$, such that $\mathbf { A } ^ { 5 } = \mathbf { P D Q }$.

CAIE FP1 2010 June Q9

Standard +0.3

Write down the five fifth roots of unity.
Hence find all the roots of the equation $$z ^ { 5 } + 16 + ( 16 \sqrt { } 3 ) i = 0$$ giving answers in the form $r \mathrm { e } ^ { \mathrm { i } q \pi }$, where $r > 0$ and $q$ is a rational number. Show these roots on an Argand diagram. Let $w$ be a root of the equation in part (ii).
Show that $$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
Identify the root for which $| 2 - w |$ is least.

CAIE FP1 2010 June Q10

Challenging +1.2

10 Find the set of values of $a$ for which the system of equations $$\begin{aligned} x + 4 y + 12 z & = 5 \\ 2 x + a y + 12 z & = a - 1 \\ 3 x + 12 y + 2 a z & = 10 \end{aligned}$$ has a unique solution. Show that the system does not have any solution in the case $a = 18$. Given that $a = 8$, show that the number of solutions is infinite and find the solution for which $x + y + z = 1$.

Questions FP1 (1385 questions)

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