Questions - OCR Further Pure Core 1

OCR Further Pure Core 1 2019 June Q1

1 In this question you must show detailed reasoning.
The quadratic equation $x ^ { 2 } - 2 x + 5 = 0$ has roots $\alpha$ and $\beta$.

Write down the values of $\alpha + \beta$ and $\alpha \beta$.
Hence find a quadratic equation with roots $\alpha + \frac { 1 } { \beta }$ and $\beta + \frac { 1 } { \alpha }$.

OCR Further Pure Core 1 2019 June Q2

2 Indicate by shading on an Argand diagram the region
$\{ z : | z | \leqslant | z - 4 | \} \cap \{ z : | z - 3 - 2 i | \leqslant 2 \}$.

OCR Further Pure Core 1 2019 June Q3

3 In this question you must show detailed reasoning.
You are given that $x = 2 + 5 \mathrm { i }$ is a root of the equation $x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0$.
Solve the equation.

OCR Further Pure Core 1 2019 June Q4

4 Using the formulae for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$, show that $\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045$.

OCR Further Pure Core 1 2019 June Q5

5 The diagram shows part of the curve $y = 5 \cosh x + 3 \sinh x$.
\includegraphics[max width=\textwidth, alt={}, center]{a6d9b3ec-5170-4f06-a8a3-b854efe36f07-3_496_771_315_246}

Solve the equation $5 \cosh x + 3 \sinh x = 4$ giving your solution in exact form.
In this question you must show detailed reasoning. Find $\int _ { - 1 } ^ { 1 } ( 5 \cosh x + 3 \sinh x ) \mathrm { d } x$ giving your answer in the form $a \mathrm { e } + \frac { b } { \mathrm { e } }$ where $a$ and $b$ are integers to be determined.

OCR Further Pure Core 1 2019 June Q6

6 You are given that $y = \tan ^ { - 1 } \sqrt { 2 x }$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Show that $\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 2 } } \frac { \sqrt { x } } { \left( x + 2 x ^ { 2 } \right) } \mathrm { d } x = k \pi$ where $k$ is a number to be determined in exact form.

OCR Further Pure Core 1 2019 June Q7

7 The function sech $x$ is defined by $\operatorname { sech } x = \frac { 1 } { \cosh x }$.

Show that $\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }$.
Using a suitable substitution, find $\int \operatorname { sech } x \mathrm {~d} x$.

OCR Further Pure Core 1 2019 June Q8

8 The equation of a plane is $4 x + 2 y + z = 7$.
The point $A$ has coordinates $( 9,6,1 )$ and the point $B$ is the reflection of $A$ in the plane.
Find the coordinates of the point $B$.

OCR Further Pure Core 1 2019 June Q9

9 In this question you must show detailed reasoning.
You are given the complex number $\omega = \cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi$ and the equation $z ^ { 5 } = 1$.

Show that $\omega$ is a root of the equation.
Write down the other four roots of the equation.
Show that $\omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1$.
Hence show that $\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0$.
Hence determine the value of $\cos \frac { 2 } { 5 } \pi$ in the form $a + b \sqrt { c }$ where $a , b$ and $c$ are rational numbers to be found.

OCR Further Pure Core 1 2019 June Q10

10 You are given the matrix $\mathbf { A }$ where $\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0
0 & a & 2
4 & 5 & 1 \end{array} \right)$.

Find, in terms of $a$, the determinant of $\mathbf { A }$, simplifying your answer.
Hence find the values of $a$ for which $\mathbf { A }$ is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6
a y + 2 z & = 8
4 x + 5 y + z & = 16 \end{aligned}$$
For each of the values of $a$ found in part (b) determine whether the equations have
- a unique solution, which should be found, or
- an infinite set of solutions or
- no solution.

OCR Further Pure Core 1 2019 June Q11

11 A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line. The displacement of the particle may be modelled by the differential equation $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t$
where $x$ is the displacement of the particle below the equilibrium position at time $t$.
When $t = 0$ the particle is stationary and its displacement is 2 .

Find the particular solution of the differential equation.
Write down an approximate equation for the displacement when $t$ is large.

OCR Further Pure Core 1 2022 June Q2

2 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { r r } 2 & - 2
1 & 3 \end{array} \right)$.

Calculate $\operatorname { det } \mathbf { A }$.
Write down $\mathbf { A } ^ { - 1 }$.
Hence solve the equation $\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }$.
Write down the matrix $\mathbf { B }$ such that $\mathbf { A B } = 4 \mathbf { I }$. Matrices $\mathbf { C }$ and $\mathbf { D }$ are given by $\mathbf { C } = \left( \begin{array} { l } 2
0
1 \end{array} \right)$ and $\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)$ where $p$ is a constant.
Find, in terms of $p$,
- the matrix CD
- the matrix DC.
It is observed that $\mathbf { C D } \neq \mathbf { D C }$.
The result that $\mathbf { C D } \neq \mathbf { D C }$ is a counter example to the claim that matrix multiplication has a particular property. Name this property.

OCR Further Pure Core 1 2022 June Q3

3 In this question you must show detailed reasoning.

Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$. The loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ are given by $| z | = | z - 2 \mathrm { i } |$ and $| z - 2 | = \sqrt { 5 }$ respectively.
1. Sketch on a single Argand diagram the loci $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$, showing any intercepts with the imaginary axis.
2. Indicate, by shading on your Argand diagram, the region $$\{ z : | z | \leqslant | z - 2 i | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \} .$$
1. Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.
2. State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.
On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.

OCR Further Pure Core 1 2022 June Q4

4 Determine the acute angle between the line $\mathbf { r } = \left( \begin{array} { c } - \sqrt { 3 }
1
3 \end{array} \right) + \lambda \left( \begin{array} { c } 1
2 \sqrt { 3 }
- \sqrt { 3 } \end{array} \right)$ and the $y$-axis.

OCR Further Pure Core 1 2022 June Q5

5 The diagram below shows the curve $C$ with polar equation $r = 3 ( 1 - \sin 2 \theta )$ for $0 \leqslant \theta \leqslant 2 \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-5_728_963_303_239}

Show that a cartesian equation of $C$ is $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 9 ( x - y ) ^ { 4 }$.
Show that the line with equation $\mathrm { y } = \mathrm { x }$ is a line of symmetry of $C$.
In this question you must show detailed reasoning. Find the exact area of each of the loops of $C$.

OCR Further Pure Core 1 2022 June Q6

6 Let $\mathrm { y } = \mathrm { x } \cosh \mathrm { x }$.
Prove by induction that, for all integers $n \geqslant 1 , \frac { d ^ { 2 n - 1 } y } { d x ^ { 2 n - 1 } } = x \sinh x + ( 2 n - 1 ) \cosh x$.

OCR Further Pure Core 1 2022 June Q7

7

Determine the values of $A , B , C$ and $D$ such that $\frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \equiv \frac { A } { x } + \frac { B } { x ^ { 2 } } + \frac { C x + D } { x ^ { 2 } + 9 }$.
In this question you must show detailed reasoning. Hence determine the exact value of $\int _ { 3 } ^ { \infty } \frac { x ^ { 2 } + 18 } { x ^ { 2 } \left( x ^ { 2 } + 9 \right) } \mathrm { d } x$.

OCR Further Pure Core 1 2022 June Q8

8 A biologist is studying the effect of pesticides on crops. On a certain farm pesticide is regularly applied to a particular crop which grows in soil. Over time, pesticide is transferred between the crop and the soil at a rate which depends on the amount of pesticide in both the crop and the soil. The amount of pesticide in the crop after $t$ days is $x$ grams. The amount of pesticide in the soil after $t$ days is $y$ grams. Initially, when $t = 0$, there is no pesticide in either the crop or the soil. At first it is assumed that no pesticide is lost from the system. The biologist further assumes that pesticide is added to the crop at a constant rate of $k$ grams per day, where $k > 6$. After collecting some initial data, the biologist suggests that for $t \geqslant 0$, this situation can be modelled by the following pair of first order linear differential equations.
$\frac { d x } { d t } = - 2 x + 78 y + k$
$\frac { d y } { d t } = 2 x - 78 y$

1. Show that $\frac { d ^ { 2 } x } { d t ^ { 2 } } + 80 \frac { d x } { d t } = 78 \mathrm { k }$.
2. Determine the particular solution for $x$ in terms of $k$ and $t$. If more than 250 grams of pesticide is found in the crop, then it will fail food safety standards.
3. The crop is tested 50 days after the pesticide is first added to it. Explain why, according to this model, the crop will fail food safety standards as a result of this test. Further data collection suggests that some pesticide decays in the soil and so is lost from the system. The model is refined in light of this data. The particular solution for $x$ for this refined model is
  $\mathrm { x } = \mathrm { k } \left( 20 - \mathrm { e } ^ { - 41 \mathrm { t } } \left( 20 \cosh ( \sqrt { 1677 } \mathrm { t } ) + \frac { 819 } { \sqrt { 1677 } } \sinh ( \sqrt { 1677 } \mathrm { t } ) \right) \right.$.
Given now that $k < 12$, determine whether the crop will fail food safety standards in the long run according to this refined model. In the refined model, it is still assumed that pesticide is added to the crop at a constant rate.
Suggest a reason why it might be more realistic to model the addition of pesticide as not being at a constant rate.

OCR Further Pure Core 1 2022 June Q9

9 The cube roots of unity are represented on the Argand diagram below by the points $A , B$ and $C$.
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-8_760_800_303_244} The points $L , M$ and $N$ are the midpoints of the line segments $A B , B C$ and $C A$ respectively. Determine a degree 6 polynomial equation with integer coefficients whose roots are the complex numbers represented by the points $A , B , C , L , M$ and $N$. \section*{END OF QUESTION PAPER}

OCR Further Pure Core 1 2023 June Q1

1 In this question you must show detailed reasoning.
Determine the value of $\sum _ { r = 1 } ^ { 50 } r ^ { 2 } ( 16 - r )$.

OCR Further Pure Core 1 2023 June Q2

2 In this question you must show detailed reasoning.
The equation $z ^ { 4 } + 4 z ^ { 3 } + 9 z ^ { 2 } + 10 z + 6 = 0$ has roots $\alpha , \beta , \gamma$ and $\delta$.

Show that a quartic equation whose roots are $\alpha + 1 , \beta + 1 , \gamma + 1$ and $\delta + 1$ is $w ^ { 4 } + 3 w ^ { 2 } + 2 = 0$.
Hence determine the exact roots of the equation $z ^ { 4 } + 4 z ^ { 3 } + 9 z ^ { 2 } + 10 z + 6 = 0$.

OCR Further Pure Core 1 2023 June Q3

3

Show that $\frac { - 3 + \sqrt { 3 } \mathrm { i } } { 2 } = \sqrt { 3 } \mathrm { e } ^ { \frac { 5 } { 6 } \pi \mathrm { i } }$.
Hence determine the exact roots of the equation $z ^ { 5 } = \frac { 9 ( - 3 + \sqrt { 3 } \mathrm { i } ) } { 2 }$, giving the roots in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ where $r > 0$ and $0 \leqslant \theta < 2 \pi$.

OCR Further Pure Core 1 2023 June Q4

4 The transformations $T _ { A }$ and $T _ { B }$ are represented by the matrices $\mathbf { A }$ and $\mathbf { B }$ respectively, where $\mathbf { A } = \left( \begin{array} { r r } 0 & - 1
1 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)$

Describe geometrically the single transformation consisting of $T _ { A }$ followed by $T _ { B }$.
By considering the transformation $\mathrm { T } _ { \mathrm { A } }$, determine the matrix $\mathrm { A } ^ { 423 }$. The transformation $\mathrm { T } _ { \mathrm { C } }$ is represented by the matrix $\mathbf { C }$, where $\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0
0 & \frac { 1 } { 3 } \end{array} \right)$. The region $R$ is defined by the set of points $( x , y )$ satisfying the inequality $x ^ { 2 } + y ^ { 2 } \leqslant 36$. The region $R ^ { \prime }$ is defined as the image of $R$ under $\mathrm { T } _ { \mathrm { C } }$.
1. Find the exact area of the region $R ^ { \prime }$.
2. Sketch the region $R ^ { \prime }$, specifying all the points where the boundary of $R ^ { \prime }$ intersects the coordinate axes.

OCR Further Pure Core 1 2023 June Q5

5

Find the general solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + 5 y = 0$.
Hence find the general solution of the differential equation $\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + 5 y = x ( 4 - 5 x )$.

OCR Further Pure Core 1 2023 June Q7

7 An engineer is modelling the motion of a particle $P$ of mass 0.5 kg in a wind tunnel.
$P$ is modelled as travelling in a straight line. The point $O$ is a fixed point within the wind tunnel. The displacement of $P$ from $O$ at time $t$ seconds is $x$ metres, for $t \geqslant 0$. You are given that $x \geqslant 0$ for all $t \geqslant 0$ and that $P$ does not reach the end of the wind tunnel.
If $t \geqslant 0$, then $P$ is subject to three forces which are modelled in the following way.

The first force has a magnitude of $5 ( t + 1 ) \cosh t \mathrm {~N}$ and acts in the positive $x$-direction.
The second force has a magnitude of $0.5 x \mathrm {~N}$ and acts towards $O$.
The third force has a magnitude of $\left| \frac { d x } { d t } \right| \mathrm { N }$ and acts in the direction of motion of the particle.
1. The engineer applies the equation " $F = m a$ " to the model of the motion of $P$ and derives the following differential equation.
  $5 ( t + 1 ) \operatorname { cosht } - 0.5 x + \frac { d x } { d t } = 0.5 \frac { d ^ { 2 } x } { d t ^ { 2 } }$
  1. Explain the sign of the $\frac { \mathrm { dx } } { \mathrm { dt } }$ term in the engineer's differential equation.

When $t = 0$ the displacement of $P$ is 6 m , and it is travelling towards $O$ with a speed of $5 \mathrm {~ms} ^ { - 1 }$.
(ii) Without attempting to solve the differential equation, find the acceleration of $P$ when $t = 0$. Let the particular solution to the differential equation in part (a) be a function f such that $\mathrm { x } = \mathrm { f } ( \mathrm { t } )$ for $t \geqslant 0$. The particular solution to the differential equation can be expressed as a Maclaurin series.

Show that the Maclaurin series for $\mathrm { f } ( t )$ up to and including the term in $t$ is $6 - 5 t$.
Use your answer to part (a)(ii) to show that the term in $t ^ { 2 }$ in the Maclaurin series for $\mathrm { f } ( t )$ is $- 3 t ^ { 2 }$.
By differentiating the differential equation in part (a) with respect to $t$, show that the term in $t ^ { 3 }$ in the Maclaurin series for $\mathrm { f } ( t )$ is $0.5 t ^ { 3 }$. You are given that the complete Maclaurin series for the function f is valid for all values of $t \geqslant 0$.
After 0.25 seconds $P$ has travelled 1.43 m towards the origin.

By using the Maclaurin series for $\mathrm { f } ( t )$ up to and including the term in $t ^ { 3 }$, evaluate the suitability of the model for determining the displacement of $P$ from $O$ when $t = 0.25$.
Explain why it might not be sensible to use the Maclaurin series for $\mathrm { f } ( t )$ up to and including the term in $t ^ { 3 }$ to evaluate the suitability of the model for determining the displacement of $P$ from $O$ when $t = 10$.

Questions — OCR Further Pure Core 1 (134 questions)

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