Questions FP1 - A-Level Maths

Prove by induction that $u _ { n } < 2$ for all $n \geqslant 1$.
Show that if $u _ { n } = 2 - \varepsilon$, where $\varepsilon$ is small, then $$u _ { n + 1 } \approx 2 - \frac { 1 } { 6 } \varepsilon$$

CAIE FP1 2005 November Q3

3 The curve $C$ has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where $\lambda$ is a non-zero constant. Obtain the equations of the asymptotes of $C$. In separate diagrams, sketch $C$ for the cases where

$\lambda > 0$,
$\lambda < 0$.

CAIE FP1 2005 November Q4

4 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 24 \mathrm { e } ^ { 2 x }$$ given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 9$ when $x = 0$.

CAIE FP1 2005 November Q5

5 In the equation $$x ^ { 3 } + a x ^ { 2 } + b x + c = 0$$ the coefficients $a , b$ and $c$ are real. It is given that all the roots are real and greater than 1 .

Prove that $a < - 3$.
By considering the sum of the squares of the roots, prove that $a ^ { 2 } > 2 b + 3$.
By considering the sum of the cubes of the roots, prove that $a ^ { 3 } < - 9 b - 3 c - 3$.

CAIE FP1 2005 November Q6

6 Let $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x$$ where $n \geqslant 1$. By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 1 + x ^ { 2 } \right) ^ { - n } \right)$, or otherwise, prove that $$2 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 ^ { - n }$$ Deduce that $I _ { 3 } = \frac { 3 } { 32 } \pi + \frac { 1 } { 4 }$.
$7 \quad$ Write down an expression in terms of $z$ and $N$ for the sum of the series $$\sum _ { n = 1 } ^ { N } 2 ^ { - n } z ^ { n }$$ Use de Moivre's theorem to deduce that $$\sum _ { n = 1 } ^ { 10 } 2 ^ { - n } \sin \left( \frac { 1 } { 10 } n \pi \right) = \frac { 1025 \sin \left( \frac { 1 } { 10 } \pi \right) } { 2560 - 2048 \cos \left( \frac { 1 } { 10 } \pi \right) }$$

CAIE FP1 2005 November Q8

8 Find the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is $$y = x ^ { 2 } ( 1 - x )$$ Deduce the coordinates of the centroid of the finite region bounded by the $x$-axis and the curve whose equation is $$y = x ( 1 - x ) ^ { 2 }$$

CAIE FP1 2005 November Q9

9 The planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$ have vector equations $$\mathbf { r } = \lambda _ { 1 } ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) + \mu _ { 1 } ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \lambda _ { 2 } ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) + \mu _ { 2 } ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )$$ respectively. The line $l$ passes through the point with position vector $4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }$ and is parallel to both $\Pi _ { 1 }$ and $\Pi _ { 2 }$. Find a vector equation for $l$. Find also the shortest distance between $l$ and the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.

CAIE FP1 2005 November Q10

10 It is given that the eigenvalues of the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r } 4 & 1 & - 1
- 4 & - 1 & 4
0 & - 1 & 5 \end{array} \right)$$ are $1,3,4$. Find a set of corresponding eigenvectors. Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { M } ^ { n } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where $n$ is a positive integer. Find $\mathbf { P } ^ { - 1 }$ and deduce that $$\lim _ { n \rightarrow \infty } 4 ^ { - n } \mathbf { M } ^ { n } = \left( \begin{array} { r r r } - \frac { 1 } { 3 } & 0 & - \frac { 1 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \end{array} \right)$$

CAIE FP1 2005 November Q11

11 Find the rank of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 1 & 2 & 3
4 & 3 & 5 & 16
6 & 6 & 13 & 13
14 & 12 & 23 & 45 \end{array} \right)$$ Find vectors $\mathbf { x } _ { 0 }$ and $\mathbf { e }$ such that any solution of the equation $$\mathbf { A x } = \left( \begin{array} { r } 0
2
- 1
3 \end{array} \right)$$ can be expressed in the form $\mathbf { x } _ { 0 } + \lambda \mathbf { e }$, where $\lambda \in \mathbb { R }$. Hence show that there is no vector which satisfies (*) and has all its elements positive.

CAIE FP1 2005 November Q12 EITHER

Show that $\left( n + \frac { 1 } { 2 } \right) ^ { 3 } - \left( n - \frac { 1 } { 2 } \right) ^ { 3 } \equiv 3 n ^ { 2 } + \frac { 1 } { 4 }$. Use this result to prove that $\sum _ { n = 1 } ^ { N } n ^ { 2 } = \frac { 1 } { 6 } N ( N + 1 ) ( 2 N + 1 )$. The sums $S , T$ and $U$ are defined as follows: $$\begin{aligned} & S = 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } ,
& T = 1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 2 N - 1 ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } ,
& U = 1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } . \end{aligned}$$ Find and simplify expressions in terms of $N$ for each of $S , T$ and $U$. Hence

describe the behaviour of $\frac { S } { T }$ as $N \rightarrow \infty$,
prove that if $\frac { S } { U }$ is an integer then $\frac { T } { U }$ is an integer.

CAIE FP1 2005 November Q12 OR

The curves $C _ { 1 }$ and $C _ { 2 }$ have polar equations $$r = 4 \cos \theta \quad \text { and } \quad r = 1 + \cos \theta$$ respectively, where $- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

Show that $C _ { 1 }$ and $C _ { 2 }$ meet at the points $A \left( \frac { 4 } { 3 } , \alpha \right)$ and $B \left( \frac { 4 } { 3 } , - \alpha \right)$, where $\alpha$ is the acute angle such that $\cos \alpha = \frac { 1 } { 3 }$.
In a single diagram, draw sketch graphs of $C _ { 1 }$ and $C _ { 2 }$.
Show that the area of the region bounded by the arcs $O A$ and $O B$ of $C _ { 1 }$, and the $\operatorname { arc } A B$ of $C _ { 2 }$, is $$4 \pi - \frac { 1 } { 3 } \sqrt { } 2 - \frac { 13 } { 2 } \alpha .$$

CAIE FP1 2006 November Q1

1 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & - 2
0 & 2 & 1
0 & 0 & - 3 \end{array} \right)$$ Write down the eigenvalues of $\mathbf { A }$ and find corresponding eigenvectors.

CAIE FP1 2006 November Q2

2 The integral $I _ { n }$, where $n$ is a non-negative integer, is defined by $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 3 } } \mathrm {~d} x$$ By considering $\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { n + 1 } \mathrm { e } ^ { - x ^ { 3 } } \right)$ or otherwise, show that $$3 I _ { n + 3 } = ( n + 1 ) I _ { n } - \mathrm { e } ^ { - 1 }$$ Hence find $I _ { 6 }$ in terms of e and $I _ { 0 }$.

CAIE FP1 2006 November Q3

3 Verify that if $$v _ { n } = n ( n + 1 ) ( n + 2 ) \ldots ( n + m )$$ then $$v _ { n + 1 } - v _ { n } = ( m + 1 ) ( n + 1 ) ( n + 2 ) \ldots ( n + m ) .$$ Given now that $$u _ { n } = ( n + 1 ) ( n + 2 ) \ldots ( n + m ) ,$$ find $\sum _ { n = 1 } ^ { N } u _ { n }$ in terms of $m$ and $N$.

CAIE FP1 2006 November Q4

4 Prove by mathematical induction that, for all positive integers $n , 10 ^ { 3 n } + 13 ^ { n + 1 }$ is divisible by 7 .

CAIE FP1 2006 November Q5

5 Show that if $a \neq 3$ then the system of equations $$\begin{aligned} & 2 x + 3 y + 4 z = - 5
& 4 x + 5 y - z = 5 a + 15
& 6 x + 8 y + a z = b - 2 a + 21 \end{aligned}$$ has a unique solution. Given that $a = 3$, find the value of $b$ for which the equations are consistent.

CAIE FP1 2006 November Q6

6 The roots of the equation $$x ^ { 3 } + x + 1 = 0$$ are $\alpha , \beta , \gamma$. Show that the equation whose roots are $$\frac { 4 \alpha + 1 } { \alpha + 1 } , \quad \frac { 4 \beta + 1 } { \beta + 1 } , \quad \frac { 4 \gamma + 1 } { \gamma + 1 }$$ is of the form $$y ^ { 3 } + p y + q = 0$$ where the numbers $p$ and $q$ are to be determined. Hence find the value of $$\left( \frac { 4 \alpha + 1 } { \alpha + 1 } \right) ^ { n } + \left( \frac { 4 \beta + 1 } { \beta + 1 } \right) ^ { n } + \left( \frac { 4 \gamma + 1 } { \gamma + 1 } \right) ^ { n }$$ for $n = 2$ and for $n = 3$.

CAIE FP1 2006 November Q7

7 The curve $C$ has equation $$r = 10 \ln ( 1 + \theta )$$ where $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$. Draw a sketch of $C$. Use the substitution $w = \ln ( 1 + \theta )$ to show that the area of the sector bounded by the line $\theta = \frac { 1 } { 2 } \pi$ and the arc of $C$ joining the origin to the point where $\theta = \frac { 1 } { 2 } \pi$ is $$50 \left( b ^ { 2 } - 2 b + 2 \right) \mathrm { e } ^ { b } - 100$$ where $b = \ln \left( 1 + \frac { 1 } { 2 } \pi \right)$.

CAIE FP1 2006 November Q8

8 Given that $$2 y ^ { 3 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 12 y ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y ^ { 2 } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 17 y ^ { 4 } = 13 \mathrm { e } ^ { - 4 x }$$ and that $v = y ^ { 4 }$, show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 34 v = 26 \mathrm { e } ^ { - 4 x }$$ Hence find the general solution for $y$ in terms of $x$.

CAIE FP1 2006 November Q9

9 With $O$ as origin, the points $A , B , C$ have position vectors $$\mathbf { i } , \quad \mathbf { i } + \mathbf { j } , \quad \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$ respectively. Find a vector equation of the common perpendicular of the lines $A B$ and $O C$. Show that the shortest distance between the lines $A B$ and $O C$ is $\frac { 2 } { 5 } \sqrt { } 5$. Find, in the form $a x + b y + c z = d$, an equation for the plane containing $A B$ and the common perpendicular of the lines $A B$ and $O C$.

CAIE FP1 2006 November Q10

10 The curve $C$ has equation $$y = x ^ { 2 } + \lambda \sin ( x + y ) ,$$ where $\lambda$ is a constant, and passes through the point $A \left( \frac { 1 } { 4 } \pi , \frac { 1 } { 4 } \pi \right)$. Show that $C$ has no tangent which is parallel to the $y$-axis. Show that, at $A$, $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 - \frac { 1 } { 64 } \pi ( 4 - \pi ) ( \pi + 2 ) ^ { 2 }$$

CAIE FP1 2006 November Q11

11 Prove de Moivre's theorem for a positive integral exponent: $$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$ Use de Moivre's theorem to show that $$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ Hence obtain the roots of the equation $$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$ in the form $\cos q \pi$, where $q$ is a rational number.

CAIE FP1 2006 November Q12 EITHER

The curve $C$ has equation $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where $q$ is a positive constant.

Obtain the equations of the asymptotes of $C$.
Find the value of $q$ for which the $x$-axis is a tangent to $C$, and sketch $C$ in this case.
Sketch $C$ for the case $q = 3$, giving the exact coordinates of the points of intersection of $C$ with the $x$-axis.
It is given that, for all values of the constant $\lambda$, the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of $C$. Use this result, with the diagrams you have drawn, to show that if $\lambda < \frac { 1 } { 2 }$ then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if $q$ has the value found in part (ii), but has 2 real distinct solutions if $q = 3$.

CAIE FP1 2006 November Q12 OR

The curve $C$ has equation $$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$ where $\lambda > 0$ and $0 \leqslant x \leqslant 3$. The length of $C$ is denoted by $s$. Prove that $s = 2 \sqrt { } 3$. The area of the surface generated when $C$ is rotated through one revolution about the $x$-axis is denoted by $S$. Find $S$ in terms of $\lambda$. The $y$-coordinate of the centroid of the region bounded by $C$, the axes and the line $x = 3$ is denoted by h. Given that $\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }$, show that $$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$

Questions FP1 (1385 questions)

Browse by module

Browse by board

CAIE FP1 2005 November Q1

CAIE FP1 2005 November Q2

CAIE FP1 2005 November Q3

CAIE FP1 2005 November Q4

CAIE FP1 2005 November Q5

CAIE FP1 2005 November Q6

CAIE FP1 2005 November Q8

CAIE FP1 2005 November Q9

CAIE FP1 2005 November Q10

CAIE FP1 2005 November Q11

CAIE FP1 2005 November Q12 EITHER

CAIE FP1 2005 November Q12 OR

CAIE FP1 2006 November Q1

CAIE FP1 2006 November Q2

CAIE FP1 2006 November Q3

CAIE FP1 2006 November Q4

CAIE FP1 2006 November Q5

CAIE FP1 2006 November Q6

CAIE FP1 2006 November Q7

CAIE FP1 2006 November Q8

CAIE FP1 2006 November Q9

CAIE FP1 2006 November Q10

CAIE FP1 2006 November Q11

CAIE FP1 2006 November Q12 EITHER

CAIE FP1 2006 November Q12 OR