Questions - OCR FP1 - A-Level Maths

OCR FP1 2014 June Q5

5 The cubic equation $2 x ^ { 3 } + 3 x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.

Use the substitution $x = u + 2$ to find a cubic equation in $u$.
Hence find the value of $\frac { 1 } { \alpha - 2 } + \frac { 1 } { \beta - 2 } + \frac { 1 } { \gamma - 2 }$.

OCR FP1 2014 June Q6

6

Show that $\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 2 ) ^ { 2 } } \equiv \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }$.
Hence find an expression, in terms of $n$, for $\sum _ { r = 1 } ^ { n } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }$.
Find $\sum _ { r = 5 } ^ { \infty } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }$, giving your answer in the form $\frac { p } { q }$ where $p$ and $q$ are integers.

OCR FP1 2014 June Q7

7 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $\arg ( z - 2 - 2 \mathrm { i } ) = \frac { 1 } { 4 } \pi$ and $| z | = | z - 10 |$ respectively.

Sketch on a single Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$.
Indicate, by shading, the region of the Argand diagram for which $$0 \leqslant \arg ( z - 2 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 4 } \pi \text { and } | z | \geqslant | z - 10 | .$$

OCR FP1 2014 June Q8

8

Show that $\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$.
Hence find $\sum _ { r = n } ^ { 2 n } r \left( r ^ { 2 } - 2 \right)$, giving your answer in a fully factorised form.

OCR FP1 2014 June Q9

9 The roots of the equation $x ^ { 3 } - k x ^ { 2 } - 2 = 0$ are $\alpha , \beta$ and $\gamma$, where $\alpha$ is real and $\beta$ and $\gamma$ are complex.

Show that $k = \alpha - \frac { 2 } { \alpha ^ { 2 } }$.
Given that $\beta = u + \mathrm { i } v$, where $u$ and $v$ are real, find $u$ in terms of $\alpha$.
Find $v ^ { 2 }$ in terms of $\alpha$.

OCR FP1 2014 June Q10

10 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = 5 ^ { n } + 2 ^ { n - 1 }$.

Find $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$.
Hence suggest a positive integer, other than 1 , which divides exactly into every term of the sequence.
By considering $u _ { n + 1 } + u _ { n }$, prove by induction that your suggestion in part (ii) is correct. \section*{OCR}

OCR FP1 2015 June Q1

1 The complex number $x + \mathrm { i } y$ is denoted by $z$. Express $3 z z ^ { * } - | z | ^ { 2 }$ in terms of $x$ and $y$.

OCR FP1 2015 June Q2

2 Find $\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 5 \right)$, expressing your answer in a fully factorised form.

OCR FP1 2015 June Q3

3 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 2 & a
0 & 1 \end{array} \right)$, where $a$ is a constant.

Find $\mathbf { A } ^ { - 1 }$. The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { l l } 2 & a
4 & 1 \end{array} \right)$.
Given that $\mathbf { P A } = \mathbf { B }$, find the matrix $\mathbf { P }$.

OCR FP1 2015 June Q4

4 Prove by induction that, for $n \geqslant 1 , \sum _ { r = 1 } ^ { n } r ( 3 r + 1 ) = n ( n + 1 ) ^ { 2 }$.

OCR FP1 2015 June Q5

5 The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z + 2 | = 2$ and $\arg ( z + 2 ) = \frac { 5 } { 6 } \pi$ respectively.

Sketch, on a single Argand diagram, the loci $C _ { 1 }$ and $C _ { 2 }$.
Find the complex number represented by the intersection of $C _ { 1 }$ and $C _ { 2 }$.
Indicate, by shading, the region of the Argand diagram for which $$| z + 2 | \leqslant 2 \text { and } \frac { 5 } { 6 } \pi \leqslant \arg ( z + 2 ) \leqslant \pi .$$

OCR FP1 2015 June Q6

6 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r } 0 & 2
- 1 & 0 \end{array} \right)$.

The diagram in the Printed Answer Book shows the unit square $O A B C$. The image of the unit square under the transformation represented by $\mathbf { M }$ is $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$. Draw and label $O A ^ { \prime } B ^ { \prime } C ^ { \prime }$, indicating clearly the coordinates of $A ^ { \prime } , B ^ { \prime }$ and $C ^ { \prime }$.
The transformation represented by $\mathbf { M }$ is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a possible pair of transformations P and Q and state the matrices that represent them.

OCR FP1 2015 June Q7

7

Use an algebraic method to find the square roots of the complex number $5 + 12 \mathrm { i }$. You must show sufficient working to justify your answers.
Hence solve the quadratic equation $x ^ { 2 } - 4 x - 1 - 12 \mathrm { i } = 0$.

OCR FP1 2015 June Q8

8

Show that $\frac { 3 } { r - 1 } - \frac { 2 } { r } - \frac { 1 } { r + 1 } \equiv \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }$.
Hence find an expression, in terms of $n$, for $\sum _ { r = 2 } ^ { n } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }$.
Hence find the value of $\sum _ { r = 4 } ^ { \infty } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }$.

OCR FP1 2015 June Q9

9 The matrix $\mathbf { D }$ is given by $\mathbf { D } = \left( \begin{array} { l l l } 1 & 3 & 4
2 & a & 3
0 & 1 & a \end{array} \right)$.

Find the values of $a$ for which $\mathbf { D }$ is singular.
Three simultaneous equations are shown below. $$\begin{array} { r } x + 3 y + 4 z = 3
2 x + a y + 3 z = 2
y + a z = 0 \end{array}$$ For each of the following values of $a$, determine whether or not there is a unique solution. If a unique solution does not exist, determine whether the equations are consistent or inconsistent.
(a) $a = 3$
(b) $a = 1$

OCR FP1 2015 June Q10

10 The cubic equation $x ^ { 3 } + 4 x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.

Use the substitution $x = \sqrt { u }$ to obtain a cubic equation in $u$.
Find the value of $\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \alpha \beta \gamma$.

OCR FP1 2016 June Q1

1 Find $\sum _ { r = 1 } ^ { n } ( 3 r + 1 ) ( r - 1 )$, giving your answer in a fully factorised form.

OCR FP1 2016 June Q2

2 The complex number $z$ has modulus $2 \sqrt { 3 }$ and argument $- \frac { 1 } { 3 } \pi$. Giving your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are exact real numbers, and showing clearly how you obtain them, find

$z$,
$\frac { 1 } { \left( z ^ { * } - 5 \mathrm { i } \right) ^ { 2 } }$.

OCR FP1 2016 June Q3

3 The quadratic equation $k x ^ { 2 } + x + k = 0$ has roots $\alpha$ and $\beta$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Find the value of $\left( \alpha + \frac { 1 } { \alpha } \right) \left( \beta + \frac { 1 } { \beta } \right)$ in terms of $k$.

OCR FP1 2016 June Q4

4 The matrices $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$ are given by $\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { l l l } b & 0 & 5 \end{array} \right)$ and $\mathbf { C } = \left( \begin{array} { r } 6
4
- 1 \end{array} \right)$. Find

$5 \mathbf { A } - 3 \mathbf { B }$,
BC,
CA .

OCR FP1 2016 June Q5

5 The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { 1 } = 5 \text { and } u _ { n + 1 } = 3 u _ { n } + 2 \text { for } n \geqslant 1 \text {. }$$ Prove by induction that $u _ { n } = 2 \times 3 ^ { n } - 1$.

OCR FP1 2016 June Q6

6 In an Argand diagram the points $A$ and $B$ represent the complex numbers $5 + 4 \mathrm { i }$ and $1 + 2 \mathrm { i }$ respectively.

Given that $A$ and $B$ are the ends of a diameter of a circle $C$, find the equation of $C$ in complex number form. The perpendicular bisector of $A B$ is denoted by $l$.
Sketch $C$ and $l$ on a single Argand diagram.
Find the complex numbers represented by the points of intersection of $C$ and $l$.

OCR FP1 2016 June Q7

7 The matrix $\left( \begin{array} { l l } 1 & 3
0 & 1 \end{array} \right)$ represents a transformation P .

Describe fully the transformation P . The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1
- 1 & 0 \end{array} \right)$.
Given that $\mathbf { M }$ represents transformation Q followed by transformation P , find the matrix that represents the transformation Q and describe fully the transformation Q .

OCR FP1 2016 June Q9

9

The matrix $\mathbf { X }$ is given by $\mathbf { X } = \left( \begin{array} { r r r } a & 3 & - 2
0 & a & 5
1 & 2 & 1 \end{array} \right)$. Show that the determinant of $\mathbf { X }$ is $a ^ { 2 } - 8 a + 15$.
Explain briefly why the equations $$\begin{array} { r } 3 x + 3 y - 2 z = 1
3 y + 5 z = 5
x + 2 y + z = 2 \end{array}$$ do not have a unique solution and determine whether these equations are consistent or inconsistent.
Use an algebraic method to find the square roots of the complex number $9 + 40 \mathrm { i }$.
Show that $9 + 40 \mathrm { i }$ is a root of the quadratic equation $z ^ { 2 } - 18 z + 1681 = 0$.
By using the substitution $z = \frac { 1 } { u ^ { 2 } }$, find the roots of the equation $1681 u ^ { 4 } - 18 u ^ { 2 } + 1 = 0$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.

OCR FP1 2011 January Q7

Write down the matrix, $\mathbf { A }$, that represents a shear with $x$-axis invariant in which the image of the point $( 1,1 )$ is $( 4,1 )$.
The matrix $\mathbf { B }$ is given by $\mathbf { B } = \left( \begin{array} { c c } \sqrt { 3 } & 0
0 & \sqrt { 3 } \end{array} \right)$. Describe fully the geometrical transformation represented by $\mathbf { B }$.
The matrix $\mathbf { C }$ is given by $\mathbf { C } = \left( \begin{array} { l l } 2 & 6
0 & 2 \end{array} \right)$.
(a) Draw a diagram showing the unit square and its image under the transformation represented by $\mathbf { C }$.
(b) Write down the determinant of $\mathbf { C }$ and explain briefly how this value relates to the transformation represented by $\mathbf { C }$. 8 The quadratic equation $2 x ^ { 2 } - x + 3 = 0$ has roots $\alpha$ and $\beta$, and the quadratic equation $x ^ { 2 } - p x + q = 0$ has roots $\alpha + \frac { 1 } { \alpha }$ and $\beta + \frac { 1 } { \beta }$.
Show that $p = \frac { 5 } { 6 }$.
Find the value of $q$. 9 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r r } a & - a & 1
3 & a & 1
4 & 2 & 1 \end{array} \right)$.
Find, in terms of $a$, the determinant of $\mathbf { M }$.
Hence find the values of $a$ for which $\mathbf { M } ^ { - 1 }$ does not exist.
Determine whether the simultaneous equations $$\begin{aligned} & 6 x - 6 y + z = 3 k
& 3 x + 6 y + z = 0
& 4 x + 2 y + z = k \end{aligned}$$ where $k$ is a non-zero constant, have a unique solution, no solution or an infinite number of solutions, justifying your answer.
Show that $\frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$.
Hence find an expression, in terms of $n$, for $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$
Show that $\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { ( n + 1 ) ( n + 2 ) }$.

Questions — OCR FP1 (201 questions)

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