Questions FP1 - A-Level Maths

CAIE FP1 2012 November Q11 OR

The linear transformation $\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }$ is represented by the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4
3 & 4 & 6 & 1
- 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is $R$. In any order,

show that the dimension of $R$ is 2 ,
find a basis for $R$ and obtain a cartesian equation for $R$,
find a basis for the null space of T . The vector $\left( \begin{array} { l } 8
7
k \end{array} \right)$ belongs to $R$. Find the value of $k$ and, with this value of $k$, find the general solution of $$\mathbf { M x } = \left( \begin{array} { l } 8
7
k \end{array} \right)$$

CAIE FP1 2012 November Q1

1 Show that $\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = \frac { 1 } { 6 } n ( 2 n + 1 ) ( 7 n + 1 )$.

CAIE FP1 2012 November Q2

2 Find the set of values of $a$ for which the system of equations $$\begin{aligned} a x + y + 2 z & = 0
3 x - 2 y & = 4
3 x - 4 y - 6 a z & = 14 \end{aligned}$$ has a unique solution.

CAIE FP1 2012 November Q3

3 Let $S _ { N } = \frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { N } { ( N + 1 ) ! }$. Prove by mathematical induction that, for all positive integers $N$, $$S _ { N } = 1 - \frac { 1 } { ( N + 1 ) ! }$$

CAIE FP1 2012 November Q4

4 The points $A , B$ and $C$ have position vectors $\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } , 2 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$ and $2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ respectively. Find $\overrightarrow { A B } \times \overrightarrow { A C }$. Deduce, in either order, the exact value of

the area of the triangle $A B C$,
the perpendicular distance from $C$ to $A B$.

CAIE FP1 2012 November Q5

5 The curve $C$ has polar equation $r = 1 + 2 \cos \theta$. Sketch the curve for $- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi$. Find the area bounded by $C$ and the half-lines $\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi$.

CAIE FP1 2012 November Q6

6 The curve $C$ has parametric equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { 4 } t ^ { 4 } - \ln t$$ for $1 \leqslant t \leqslant 2$. Find the area of the surface generated when $C$ is rotated through $2 \pi$ radians about the $y$-axis.

CAIE FP1 2012 November Q7

7 A cubic equation has roots $\alpha , \beta$ and $\gamma$ such that $$\begin{aligned} \alpha + \beta + \gamma & = 4
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 14
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 34 \end{aligned}$$ Find the value of $\alpha \beta + \beta \gamma + \gamma \alpha$. Show that the cubic equation is $$x ^ { 3 } - 4 x ^ { 2 } + x + 6 = 0$$ and solve this equation.

CAIE FP1 2012 November Q8

8 Let $z = \cos \theta + \mathrm { i } \sin \theta$. Show that $$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$ By considering $( 1 + z ) ^ { n }$, where $n$ is a positive integer, deduce the sum of the series $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$

CAIE FP1 2012 November Q9

9 The curve $C$ has equation $y = \frac { x ^ { 2 } - 3 x + 3 } { x - 2 }$. Find the equations of the asymptotes of $C$. Show that there are no points on $C$ for which $- 1 < y < 3$. Find the coordinates of the turning points of $C$. Sketch $C$.

CAIE FP1 2012 November Q10

10 The curve $C$ has equation $x ^ { 3 } + y ^ { 3 } = 3 x y$, for $x > 0$ and $y > 0$. Find a relationship between $x$ and $y$ when $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$. Find the exact coordinates of the turning point of $C$, and determine the nature of this turning point.

CAIE FP1 2012 November Q11

11 Show that $\int x \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = - \frac { 1 } { 3 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } + c$, where $c$ is a constant. Given that $I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$, prove that, for $n \geqslant 2$, $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 }$$ Use the substitution $x = \sin u$ to show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \pi$$ Find $I _ { 4 }$.

CAIE FP1 2012 November Q12 EITHER

The vector $\mathbf { e }$ is an eigenvector of each of the $n \times n$ matrices $\mathbf { A }$ and $\mathbf { B }$, with corresponding eigenvalues $\lambda$ and $\mu$ respectively. Prove that $\mathbf { e }$ is an eigenvector of the matrix $\mathbf { A B }$ with eigenvalue $\lambda \mu$. It is given that the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 2 & 2
- 2 & - 2 & - 2
1 & 2 & 2 \end{array} \right) ,$$ has eigenvectors $\left( \begin{array} { r } 0
1
- 1 \end{array} \right)$ and $\left( \begin{array} { r } 1
0
- 1 \end{array} \right)$. Find the corresponding eigenvalues. Given that 2 is also an eigenvalue of $\mathbf { A }$, find a corresponding eigenvector. The matrix $\mathbf { B }$, where $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 2 & 2
2 & 2 & 2
- 3 & - 6 & - 6 \end{array} \right) ,$$ has the same eigenvectors as $\mathbf { A }$. Given that $\mathbf { A B } = \mathbf { C }$, find a non-singular matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { P } ^ { - 1 } \mathbf { C } ^ { 2 } \mathbf { P } = \mathbf { D }$$

CAIE FP1 2012 November Q12 OR

Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 75 \cos 2 t$$ Given that $x = 5$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 0$ when $t = 0$, find $x$ in terms of $t$. Show that, for large positive values of $t$ and for any initial conditions, $$x \approx 5 \cos ( 2 t - \phi ) ,$$ where the constant $\phi$ is such that $\tan \phi = \frac { 4 } { 3 }$.

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 $t$, $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$

Questions FP1 (1385 questions)

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