Questions - OCR Further Pure Core 2

By using partial fractions show that $\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } + 3 r + 2 } = \frac { 1 } { 2 } - \frac { 1 } { n + 2 }$.
Hence determine the value of $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } + 3 r + 2 }$.

OCR Further Pure Core 2 2019 June Q2

A plane $\Pi$ has the equation $\mathbf { r } \cdot \left( \begin{array} { r } 3
6
- 2 \end{array} \right) = 15 . C$ is the point $( 4 , - 5,1 )$.
Find the shortest distance between $\Pi$ and $C$.
Lines $l _ { 1 }$ and $l _ { 2 }$ have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 4

OCR Further Pure Core 2 2019 June Q4

4
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

OCR Further Pure Core 2 2019 June Q5

5
2
4 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
1 \end{array} \right) \end{aligned}$$ Find, in exact form, the distance between $l _ { 1 }$ and $l _ { 2 }$.

OCR Further Pure Core 2 2019 June Q7

7 In an Argand diagram the points representing the numbers $2 + 3 \mathrm { i }$ and $1 - \mathrm { i }$ are two adjacent vertices of a square, $S$.

Find the area of $S$.
Find all the possible pairs of numbers represented by the other two vertices of $S$.

OCR Further Pure Core 2 2019 June Q8

8 In this question you must show detailed reasoning.

By writing $\sin \theta$ in terms of $\mathrm { e } ^ { \mathrm { i } \theta }$ and $\mathrm { e } ^ { - \mathrm { i } \theta }$ show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
Hence show that $\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }$.

OCR Further Pure Core 2 2019 June Q10

Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln \left( \frac { 1 } { 2 } + \cos x \right)$.
By considering the root of the equation $\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0$ deduce that $\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }$. \section*{END OF QUESTION PAPER}

OCR Further Pure Core 2 2022 June Q1

Find a vector which is perpendicular to both $3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }$ and $\mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$. The equations of two lines are $\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$ and $\mathbf { r } = \mathbf { i } + 11 \mathbf { j } - 4 \mathbf { k } + \mu ( - \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$.
Show that the lines intersect, stating the point of intersection.

OCR Further Pure Core 2 2022 June Q2

2 Two polar curves, $C _ { 1 }$ and $C _ { 2 }$, are defined by $C _ { 1 } : r = 2 \theta$ and $C _ { 2 } : r = \theta + 1$ where $0 \leqslant \theta \leqslant 2 \pi$. $C _ { 1 }$ intersects the initial line at two points, the pole and the point $A$.

Write down the polar coordinates of $A$.
Determine the polar coordinates of the point of intersection of $C _ { 1 }$ and $C _ { 2 }$. The diagram below shows a sketch of $C _ { 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{007f07ee-cb29-4a97-93d9-2328079c4aea-2_681_1353_1318_244}
On the copy of this sketch in the Printed Answer Booklet, sketch $C _ { 2 }$.

OCR Further Pure Core 2 2022 June Q4

4 In this question you must show detailed reasoning.
Determine the smallest value of $n$ for which $\frac { 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } } { 1 + 2 + \ldots + n } > 341$.

OCR Further Pure Core 2 2022 June Q5

By using the exponential definitions of $\sinh x$ and $\cosh x$, prove the identity $\cosh 2 x \equiv \cosh ^ { 2 } x + \sinh ^ { 2 } x$.
Hence find an expression for $\cosh 2 x$ in terms of $\cosh x$.
Determine the solutions of the equation $5 \cosh 2 x = 16 \cosh x + 21$, giving your answers in exact logarithmic form.

OCR Further Pure Core 2 2022 June Q6

6 A particle, $P$, positioned at the origin, $O$, is projected with a certain velocity along the $x$-axis. $P$ is then acted on by a single force which varies in such a way that $P$ moves backwards and forwards along the $x$-axis. When the time after projection is $t$ seconds, the displacement of $P$ from the origin is $x \mathrm {~m}$ and its velocity is $v \mathrm {~ms} ^ { - 1 }$. The motion of $P$ is modelled using the differential equation $\ddot { x } + \omega ^ { 2 } x = 0$, where $\omega$ rads $^ { - 1 }$ is a positive constant.

Write down the general solution of this differential equation.
$D$ is the point where $x = d$ for some positive constant, $d$. When $P$ reaches $D$ it comes to instantaneous rest.
Using the answer to part (a), determine expressions, in terms of $\omega$, $d$ and $t$ only, for the following quantities
- $X$
- $v$
- Hence show that, according to the model, $v ^ { 2 } = \omega ^ { 2 } \left( d ^ { 2 } - x ^ { 2 } \right)$.
The quantity $z$ is defined by $z = \frac { 1 } { v }$.
Using part (c), determine an expression for $\mathrm { Z } _ { \mathrm { m } }$, the mean value of z with respect to the displacement, as $P$ moves directly from $O$ to $D$. One measure of the validity of the model is consideration of the value of $\mathrm { z } _ { \mathrm { m } }$. If $\mathrm { z } _ { \mathrm { m } }$ exceeds 8 then the model is considered to be valid. The value of $d$ is measured as 0.25 to 2 significant figures. The value of $\omega$ is measured as $0.75 \pm 0.02$.
Determine what can be inferred about the validity of the model from the given information.
Find, according to the model, the least possible value of the velocity with which $P$ was initially projected. Give your answer to $\mathbf { 2 }$ significant figures.

OCR Further Pure Core 2 2022 June Q7

7 You are given that $a$ is a parameter which can take only real values.
The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6
- 3 & 10 - 4 a & 9
7 & 4 & 4 \end{array} \right)$.

Find an expression for the determinant of $\mathbf { A }$ in terms of $a$. You are given the following system of equations in $x , y$ and $z$. $$\begin{array} { r r } 2 x + & 4 y - 6 z =
- 3 x + & ( 10 - 4 a ) y + 9 z =
7 x + & 4 y + 4 z =
7 x + & 11 \end{array}$$ The system can be written in the form $\mathbf { A } \left( \begin{array} { c } \mathrm { x }
\mathrm { y }
\mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6
- 9
11 \end{array} \right)$.
1. In the case where $\mathbf { A }$ is not singular, solve the given system of equations by using $\mathbf { A } ^ { - 1 }$.
2. In the case where $\mathbf { A }$ is singular describe the configuration of the planes whose equations are the three equations of the system. The transformation represented by $\mathbf { A }$ is denoted by T .
  A 3-D object of volume $| 5 a - 20 |$ is transformed by T to a 3-D image.
1. Determine the range of values of $a$ for which the orientation of the image is the reverse of the orientation of the object.
2. Determine the range of values of $a$ for which the volume of the image is less than the volume of the object.

OCR Further Pure Core 2 2022 June Q10

10 The coordinates of the points $A$ and $B$ are ( $3 , - 2 , - 1$ ) and ( $13,10,9$ ) respectively.

The plane $\Pi _ { A }$ contains $A$ and the plane $\Pi _ { B }$ contains $B$.
The planes $\Pi _ { A }$ and $\Pi _ { B }$ are parallel.
The $x$ and $y$ components of any normal to plane $\Pi _ { A }$ are equal.
The shortest distance between $\Pi _ { A }$ and $\Pi _ { B }$ is 2 .

There are two possible solution planes for $\Pi _ { A }$ which satisfy the above conditions.
Determine the acute angle between these two possible solution planes.

OCR Further Pure Core 2 2023 June Q1

The matrix $\mathbf { P }$ is given by $\mathbf { P } = \left( \begin{array} { l l l l } 1 & 0 & - 2 & 2
4 & 2 & - 2 & 3 \end{array} \right)$.
1. Write down the dimensions of $\mathbf { P }$.
2. Write down the transpose of $\mathbf { P }$.
The matrices $\mathbf { Q } , \mathbf { R }$ and $\mathbf { S }$ are given by $\mathbf { Q } = \left( \begin{array} { l l } 1 & 2 \end{array} \right) , \mathbf { R } = \left( \begin{array} { r r } 3 & - 4
2 & 3 \end{array} \right)$ and $\mathbf { S } = \left( \begin{array} { l l } 3 & - 2 \end{array} \right)$. Write down the sum of the two of these matrices which are conformable for addition.
The dimensions of matrix $\mathbf { A }$ are 4 by 5. The matrices $\mathbf { A }$ and $\mathbf { B }$ are conformable for multiplication so that the matrix $\mathbf { C } = \mathbf { B A }$ can be formed. The matrix $\mathbf { C }$ has 6 rows.
1. Write down the number of columns that $\mathbf { C }$ has.
2. Write down the dimensions of $\mathbf { B }$.
3. Explain whether the matrix $\mathbf { A B }$ can be formed.
Find the value of $c$ for which $\left( \begin{array} { r r } - 2 & 3
6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5
10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5
10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3
6 & 10 \end{array} \right)$.

OCR Further Pure Core 2 2023 June Q2

2 In this question you must show detailed reasoning.

Write the complex number $- 24 + 7 \mathrm { i }$ in modulus-argument form.
Solve the simultaneous equations given below, giving your answers in cartesian form. $$\begin{aligned} i z + 3 w & = - 7 i
- 6 z + 5 i w & = 3 + 13 i \end{aligned}$$

OCR Further Pure Core 2 2023 June Q3

Show that $\frac { \mathrm { d } } { \mathrm { d } u } \left( \sinh ^ { - 1 } u \right) = \frac { 1 } { \sqrt { u ^ { 2 } + 1 } }$.
Find the equation of the normal to the graph of $\mathrm { y } = \sinh ^ { - 1 } 2 \mathrm { x }$ at the point where $x = \sqrt { 6 }$. Give your answer in the form $\mathrm { y } = \mathrm { mx } + \mathrm { c }$ where $m$ and $c$ are given in exact, non-hyperbolic form.

OCR Further Pure Core 2 2023 June Q6

6 The equation of the plane $\Pi$ is $\mathbf { r } = \left( \begin{array} { r } - 1
2
1 \end{array} \right) + \lambda \left( \begin{array} { l } 4
4
3 \end{array} \right) + \mu \left( \begin{array} { r } - 2
3
1 \end{array} \right)$.

Find the acute angle between $\Pi$ and the plane with equation $\mathbf { r } . \left( \begin{array} { l } 2
0
3 \end{array} \right) = 4$. The point $A$ has coordinates ( $9 , - 7,20$ ).
The point $F$ is the point of intersection between $\Pi$ and the perpendicular from $A$ to $\Pi$.
Determine the coordinates of $F$.

OCR Further Pure Core 2 2023 June Q7

7 In this question you must show detailed reasoning.

Show that $$\sum _ { r = 1 } ^ { n } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } } = \frac { a } { n + 1 } + b + c \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$$ where $a$, $b$ and $c$ are integers whose values are to be determined. You are given that $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$ exists and is equal to $\frac { 1 } { 6 } \pi ^ { 2 }$.
Show that $\sum _ { r = 1 } ^ { \infty } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } }$ exists and is equal to $( \pi - 1 ) ( \pi + 1 )$.

OCR Further Pure Core 2 2023 June Q8

8 A surge in the current, $I$ units, through an electrical component at a time, $t$ seconds, is to be modelled. The surge starts when $t = 0$ and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, $I$ is modelled by the following differential equation.
$\left( 2 t - t ^ { 2 } \right) \frac { d l } { d t } = \left( 2 t - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - 2 ( t - 1 ) l$

By using an integrating factor show that, according to the model, while the surge is occurring, $I$ is given by $\mathrm { I } = \left( 2 \mathrm { t } - \mathrm { t } ^ { 2 } \right) \left( \sin ^ { - 1 } ( \mathrm { t } - 1 ) + 5 \right)$. The surge lasts until there is again no current through the component.
Determine the length of time that the surge lasts according to the model.
Determine, according to the model, the rate of increase of the current at the start of the surge. Give your answer in an exact form.

OCR Further Pure Core 2 2023 June Q9

9 A function is defined by $y = f ( t )$ where $f ( t ) = \ln ( 1 + a t )$ and $a$ is a constant.

By considering $\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }$ and $\frac { d ^ { 4 } y } { d t ^ { 4 } }$, make a conjecture for a general formula for $\frac { d ^ { n } y } { d t ^ { n } }$ in terms of $n$ and $a$ for any integer $n \geqslant 1$.
Use induction to prove the formula conjectured in part (a).
In the case where $\mathrm { f } ( t ) = \ln ( 1 + 2 t )$, find the rate at which the $6 ^ { \text {th } }$ derivative of $\mathrm { f } ( t )$ is varying when $t = \frac { 3 } { 2 }$.

OCR Further Pure Core 2 2024 June Q1

Use the method of differences to show that $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( \frac { 1 } { \mathrm { r } } - \frac { 1 } { \mathrm { r } + 1 } \right) = 1 - \frac { 1 } { \mathrm { n } + 1 }$.
Hence determine the following sums.
1. $\quad \sum _ { r = 1 } ^ { 99 } \frac { 1 } { r } - \frac { 1 } { r + 1 }$
2. $\quad \sum _ { r = 100 } ^ { \infty } \frac { 1 } { r } - \frac { 1 } { r + 1 }$

OCR Further Pure Core 2 2024 June Q2

2 In this question you must show detailed reasoning.

Solve the equation $x ^ { 2 } - 6 x + 58 = 0$. Give your solutions in the form $a + b$ i where $a$ and $b$ are real numbers.
Determine, in exact form, $\arg ( - 10 + ( 5 \sqrt { 12 } ) \mathrm { i } ) ^ { 5 }$.

OCR Further Pure Core 2 2024 June Q3

3 Matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $\mathbf { A } = \left( \begin{array} { r r } 4 & - 3
- 2 & 2 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { r r } 3 & - 5
0 & 1 \end{array} \right)$.

Find 2A - 4B.
Write down the matrix $\mathbf { C }$ such that $\mathbf { A C } = 2 \mathbf { A }$.
Find the value of $\operatorname { det } \mathbf { A }$.
In this question you must show detailed reasoning. Use $\mathbf { A } ^ { - 1 }$ to solve the equations $4 \mathrm { x } - 3 \mathrm { y } = 7$ and $- 2 \mathrm { x } + 2 \mathrm { y } = 9$.

OCR Further Pure Core 2 2024 June Q4

4 In this question you must show detailed reasoning.
The series $S$ is defined as being the sum of the squares of all positive odd integers from $1 ^ { 2 }$ to $779 ^ { 2 }$. Determine the value of $S$.

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