Questions FP1 - A-Level Maths

$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $t$,
the mean value of $y$ with respect to $x$ over the interval $0 \leqslant x \leqslant 4$,
the $y$-coordinate of the centroid of the region enclosed by $C$, the $x$-axis and the $y$-axis.

CAIE FP1 2014 June Q12 OR

Challenging +1.2

The curve $C$ has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$ where $a , b , c$ and $d$ are constants. The curve cuts the $y$-axis at $( 0 , - 2 )$ and has asymptotes $x = 2$ and $y = x + 1$.

Write down the value of $d$.
Determine the values of $a , b$ and $c$.
Show that, at all points on $C$, either $y \leqslant 3 - 2 \sqrt { 6 }$ or $y \geqslant 3 + 2 \sqrt { 6 }$.

CAIE FP1 2014 June Q1

5 marks Moderate -0.5

1 The vectors $\mathbf { a } , \mathbf { b } , \mathbf { c }$ and $\mathbf { d }$ in $\mathbb { R } ^ { 3 }$ are given by $$\mathbf { a } = \left( \begin{array} { r }

CAIE FP1 2014 June Q2

6 marks Standard +0.8

2
- 1
1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 1
1
1 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 0
1
- 1 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r }

CAIE FP1 2014 June Q3

7 marks Standard +0.3

3
- 2
0 \end{array} \right) .$$ Show that $\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}$ is a basis for $\mathbb { R } ^ { 3 }$. Express $\mathbf { d }$ in terms of $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$. 2 Show that the difference between the squares of consecutive integers is an odd integer. Find the sum to $n$ terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } + \ldots$$ and deduce the sum to infinity of the series. 3 It is given that $\phi ( n ) = 5 ^ { n } ( 4 n + 1 ) - 1$, for $n = 1,2,3 , \ldots$. Prove, by mathematical induction, that $\phi ( n )$ is divisible by 8 , for every positive integer $n$.

CAIE FP1 2014 June Q4

7 marks Standard +0.8

4 The curve $C$ has cartesian equation $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 a ^ { 2 } x y$, where $a$ is a positive constant. Show that the polar equation of $C$ is $r ^ { 2 } = a ^ { 2 } \sin 2 \theta$. Sketch $C$ for $- \pi < \theta \leqslant \pi$. Find the area enclosed by one loop of $C$.

CAIE FP1 2014 June Q5

8 marks Challenging +1.2

5 State the sum of the series $z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }$, for $z \neq 1$. By letting $z = \cos \theta + \mathrm { i } \sin \theta$, show that $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos n \theta = \frac { \sin \frac { 1 } { 2 } n \theta } { \sin \frac { 1 } { 2 } \theta } \cos \frac { 1 } { 2 } ( n + 1 ) \theta$$ where $\sin \frac { 1 } { 2 } \theta \neq 0$.

CAIE FP1 2014 June Q6

10 marks Challenging +1.2

6 The curve $C$ has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 , \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { for } 0 \leqslant t \leqslant 2$$

Find, in terms of e , the length of $C$.
Find, in terms of $\pi$ and e , the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis.

CAIE FP1 2014 June Q7

10 marks Standard +0.3

7 The curve $C$ has parametric equations $$x = \sin t , \quad y = \sin 2 t , \quad \text { for } 0 \leqslant t \leqslant \pi .$$ Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ in terms of $t$. Hence, or otherwise, find the coordinates of the stationary points on $C$ and determine their nature.

CAIE FP1 2014 June Q8

11 marks

8 It is given that $\lambda$ is an eigenvalue of the non-singular square matrix $\mathbf { A }$, with corresponding eigenvector
e. Show that $\lambda ^ { - 1 }$ is an eigenvalue of $\mathbf { A } ^ { - 1 }$ for which $\mathbf { e }$ is a corresponding eigenvector. Deduce that $\lambda + \lambda ^ { - 1 }$ is an eigenvalue of $\mathbf { A } + \mathbf { A } ^ { - 1 }$. It is given that 1 is an eigenvalue of the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ Find a corresponding eigenvector. It is also given that $\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$ and $\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)$ are eigenvectors of the matrix $\mathbf { A }$. Find the corresponding eigenvalues.
Hence find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$

CAIE FP1 2014 June Q9

10 marks Challenging +1.2

9 Using the substitution $u = \cos \theta$, or any other method, find $\int \sin \theta \cos ^ { 2 } \theta d \theta$. It is given that $I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$, for $n \geqslant 0$. Show that, for $n \geqslant 2$, $$I _ { n } = \frac { n - 1 } { n + 2 } I _ { n - 2 }$$ Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta$.

CAIE FP1 2014 June Q10

12 marks Challenging +1.2

10 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 0.16 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 0.0064 x = 8.64 + 0.32 t$$ given that when $t = 0 , x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$. Show that, for large positive $t , \frac { \mathrm {~d} x } { \mathrm {~d} t } \approx 50$.

CAIE FP1 2014 June Q11 EITHER

Standard +0.8

Express $\frac { 2 x ^ { 2 } - x + 5 } { x ^ { 2 } - 1 }$ in the form $2 + \frac { A } { x - 1 } + \frac { B } { x + 1 }$, where $A$ and $B$ are integers to be found. The curve $C$ has equation $y = \frac { 2 x ^ { 2 } - x + 5 } { x ^ { 2 } - 1 }$. Show that there are two distinct values of $x$ for which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$. Sketch $C$, stating the equations of the asymptotes and giving the coordinates of any points of intersection with the coordinate axes and with the asymptotes. You do not need to find the coordinates of the turning points.

CAIE FP1 2014 June Q11 OR

Standard +0.8

With respect to an origin $O$, the point $A$ has position vector $4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }$ and the plane $\Pi _ { 1 }$ has equation $$\mathbf { r } = ( 4 + \lambda + 3 \mu ) \mathbf { i } + ( - 2 + 7 \lambda + \mu ) \mathbf { j } + ( 2 + \lambda - \mu ) \mathbf { k } ,$$ where $\lambda$ and $\mu$ are real. The point $L$ is such that $\overrightarrow { O L } = 3 \overrightarrow { O A }$ and $\Pi _ { 2 }$ is the plane through $L$ which is parallel to $\Pi _ { 1 }$. The point $M$ is such that $\overrightarrow { A M } = 3 \overrightarrow { M L }$.

Show that $A$ is in $\Pi _ { 1 }$.
Find a vector perpendicular to $\Pi _ { 2 }$.
Find the position vector of the point $N$ in $\Pi _ { 2 }$ such that $O N$ is perpendicular to $\Pi _ { 2 }$.
Show that the position vector of $M$ is $10 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }$ and find the perpendicular distance of $M$ from the line through $O$ and $N$, giving your answer correct to 3 significant figures.

CAIE FP1 2015 June Q1

4 marks Moderate -0.8

1 Use the List of Formulae (MF10) to show that $\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)$ and $\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)$ have the same numerical value.

CAIE FP1 2015 June Q2

6 marks Standard +0.8

2 Find the value of the constant $k$ for which the system of equations $$\begin{aligned} 2 x - 3 y + 4 z & = 1 \\ 3 x - y & = 2 \\ x + 2 y + k z & = 1 \end{aligned}$$ does not have a unique solution. For this value of $k$, solve the system of equations.

CAIE FP1 2015 June Q3

7 marks

3 The sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that $a _ { 1 } > 5$ and $a _ { n + 1 } = \frac { 4 a _ { n } } { 5 } + \frac { 5 } { a _ { n } }$ for every positive integer $n$.
Prove by mathematical induction that $a _ { n } > 5$ for every positive integer $n$. Prove also that $a _ { n } > a _ { n + 1 }$ for every positive integer $n$.

CAIE FP1 2015 June Q4

8 marks Challenging +1.2

4 The roots of the cubic equation $x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0$ are $\alpha , \beta$ and $\gamma$. Find the values of

$\frac { 1 } { ( \alpha \beta ) ( \beta \gamma ) ( \gamma \alpha ) }$,
$\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }$,
$\frac { 1 } { \alpha ^ { 2 } \beta \gamma } + \frac { 1 } { \alpha \beta ^ { 2 } \gamma } + \frac { 1 } { \alpha \beta \gamma ^ { 2 } }$. Deduce a cubic equation, with integer coefficients, having roots $\frac { 1 } { \alpha \beta } , \frac { 1 } { \beta \gamma }$ and $\frac { 1 } { \gamma \alpha }$.