Questions - OCR - A-Level Maths

OCR Further Pure Core 1 2020 November Q10

13 marks Standard +0.8

10 A particle of mass 0.5 kg is initially at point $O$. It moves from rest along the $x$-axis under the influence of two forces $F _ { 1 } \mathrm {~N}$ and $F _ { 2 } \mathrm {~N}$ which act parallel to the $x$-axis. At time $t$ seconds the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$. $F _ { 1 }$ is acting in the direction of motion of the particle and $F _ { 2 }$ is resisting motion.
In an initial model

$F _ { 1 }$ is proportional to $t$ with constant of proportionality $\lambda > 0$,
$F _ { 2 }$ is proportional to $v$ with constant of proportionality $\mu > 0$.
1. Show that the motion of the particle can be modelled by the following differential equation.

$$\frac { 1 } { 2 } \frac { d v } { d t } = \lambda t - \mu v$$

Solve the differential equation in part (a), giving the particular solution for $v$ in terms of $t$, $\lambda$ and $\mu$. You are now given that $\lambda = 2$ and $\mu = 1$.

Find a formula for an approximation for $v$ in terms of $t$ when $t$ is large. In a refined model

OCR Further Pure Core 1 2020 November Q11

8 marks Standard +0.8

11 A curve has cartesian equation $x ^ { 3 } + y ^ { 3 } = 2 x y$. $C$ is the portion of the curve for which $x \geqslant 0$ and $y \geqslant 0$. The equation of $C$ in polar form is given by $r = \mathrm { f } ( \theta )$ for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.

Find $f ( \theta )$.
Find an expression for $\mathrm { f } \left( \frac { 1 } { 2 } \pi - \theta \right)$, giving your answer in terms of $\sin \theta$ and $\cos \theta$.
Hence find the line of symmetry of $C$.
Find the value of $r$ when $\theta = \frac { 1 } { 4 } \pi$.
By finding values of $\theta$ when $r = 0$, show that $C$ has a loop.

OCR Further Pure Core 1 2020 November Q12

6 marks Challenging +1.8

12 Show that $\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } d x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }$ where $a , b$ and $c$ are integers to be determined.

OCR Further Pure Core 1 Specimen Q1

2 marks Easy -1.8

1 Show that $\frac { 5 } { 2 - 4 \mathrm { i } } = \frac { 1 } { 2 } + \mathrm { i }$.

OCR Further Pure Core 1 Specimen Q2

5 marks Standard +0.3

2 In this question you must show detailed reasoning. The equation $\mathrm { f } ( x ) = 0$, where $\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169$, has a root $x = 2 + 3 \mathrm { i }$.

Express $\mathrm { f } ( x )$ as a product of two quadratic factors.
Hence write down all the roots of the equation $\mathrm { f } ( x ) = 0$.

OCR Further Pure Core 1 Specimen Q3

6 marks Challenging +1.2

3 In this question you must show detailed reasoning. The diagram below shows the curve $r = 2 \cos 4 \theta$ for $- k \pi \leq \theta \leq k \pi$ where $k$ is a constant to be determined. Calculate the exact area enclosed by the curve.

OCR Further Pure Core 1 Specimen Q4

3 marks Standard +0.3

4 Draw the region in an Argand diagram for which $| z | \leq 2$ and $| z | > | z - 3 i |$.

OCR Further Pure Core 1 Specimen Q5

5 marks Standard +0.8

5

Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }$.
Find $\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x$.

OCR Further Pure Core 1 Specimen Q6

5 marks Standard +0.8

6 The equation $x ^ { 3 } + 2 x ^ { 2 } + x + 3 = 0$ has roots $\alpha , \beta$ and $\gamma$.
The equation $x ^ { 3 } + p x ^ { 2 } + q x + r = 0$ has roots $\alpha \beta , \beta \gamma$ and $\gamma \alpha$.
Find the values of $p , q$ and $r$.

OCR Further Pure Core 1 Specimen Q7

7 marks Challenging +1.2

7 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\frac { x - 3 } { 1 } = \frac { y - 5 } { 2 } = \frac { z + 2 } { - 3 }$ and $\frac { x - 4 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z - 7 } { 4 }$.

Find the shortest distance between $l _ { 1 }$ and $l _ { 2 }$.
Find a cartesian equation of the plane which contains $l _ { 1 }$ and is parallel to $l _ { 2 }$.

OCR Further Pure Core 1 Specimen Q8

8 marks Standard +0.3

8

Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3 \\ 2 x + 4 y + 5 z = & 9 \\ 7 x + 11 y + 12 z = & 20 \end{array}$$
Determine the values of $p$ and $k$ for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r l } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$

OCR Further Pure Core 1 Specimen Q9

5 marks Standard +0.8

9 Prove by induction that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } \frac { 5 - 4 r } { 5 ^ { r } } = \frac { n } { 5 ^ { n } }$$

OCR Further Pure Core 1 Specimen Q10

10 marks Standard +0.3

10 The Argand diagram below shows the origin $O$ and pentagon $A B C D E$, where $A , B , C , D$ and $E$ are the points that represent the complex numbers $a , b , c , d$ and $e$, and where $a$ is a positive real number. You are given that these five complex numbers are the roots of the equation $z ^ { 5 } - a ^ { 5 } = 0$. \includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

Justify each of the following statements.
1. $A , B , C , D$ and $E$ lie on a circle with centre $O$.
2. $A B C D E$ is a regular pentagon.
3. $b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c$
4. $b ^ { * } = e$
5. $a + b + c + d + e = 0$
6. The midpoints of sides $A B , B C , C D , D E$ and $E A$ represent the complex numbers $p , q , r , s$ and $t$. Determine a polynomial equation, with real coefficients, that has roots $p , q , r , s$ and $t$.

OCR Further Pure Core 1 Specimen Q11

19 marks Challenging +1.2

11 A company is required to weigh any goods before exporting them overseas. When a crate is placed on a set of weighing scales, the mass displayed takes time to settle down to its final value. The company wishes to model the mass, $m \mathrm {~kg}$, which is displayed $t$ seconds after a crate X is placed on the scales.
For the displayed mass it is assumed that the rate of change of the quantity $\left( 0.5 \frac { \mathrm {~d} m } { \mathrm {~d} t } + m \right)$ with respect to time is proportional to $( 80 - m )$.

Show that $\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 2 \mathrm {~km} = 160 \mathrm { k }$, where $k$ is a real constant. It is given that the complementary function for the differential equation in part (i) is $\mathrm { e } ^ { \lambda t } ( A \cos 2 t + B \sin 2 t )$, where $A$ and $B$ are arbitrary constants.
Show that $k = \frac { 5 } { 2 }$ and state the value of the constant $\lambda$. When X is initially placed on the scales the displayed mass is zero and the rate of increase of the displayed mass is $160 \mathrm {~kg} \mathrm {~s} ^ { - 1 }$.
Find $m$ in terms of $t$.
Describe the long term behaviour of $m$.
With reference to your answer to part (iv), comment on a limitation of the model.
(a) Find the value of $m$ that corresponds to the stationary point on the curve $m = \mathrm { f } ( t )$ with the smallest positive value of $t$.
(b) Interpret this value of $m$ in the context of the model.
Adapt the differential equation $\frac { \mathrm { d } ^ { 2 } m } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} m } { \mathrm {~d} t } + 5 m = 400$ to model the mass displayed $t$ seconds after a crate Y , of mass 100 kg , is placed on the scales. \section*{END OF QUESTION PAPER} \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the

OCR Further Pure Core 2 2019 June Q1

7 marks Standard +0.3

1 In this question you must show detailed reasoning.

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

8 marks Standard +0.3

2

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
3
1 \end{array} \right)

\end{aligned}

OCR Further Pure Core 2 2019 June Q4

5 marks Standard +0.3

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

OCR Further Pure Core 2 2019 June Q5

11 marks Standard +0.8

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

OCR Further Pure Core 2 2019 June Q7

7 marks Standard +0.8

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 marks Challenging +1.2

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

7 marks Challenging +1.3

10

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

6 marks Moderate -0.3

1

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

5 marks Standard +0.3

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 marks Standard +0.8

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

7 marks Standard +0.3

5

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.

Questions — OCR (4907 questions)

Browse by module

Browse by board

OCR Further Pure Core 1 2020 November Q10

OCR Further Pure Core 1 2020 November Q11

OCR Further Pure Core 1 2020 November Q12

OCR Further Pure Core 1 Specimen Q1

OCR Further Pure Core 1 Specimen Q2

OCR Further Pure Core 1 Specimen Q3

OCR Further Pure Core 1 Specimen Q4

OCR Further Pure Core 1 Specimen Q5

OCR Further Pure Core 1 Specimen Q6

OCR Further Pure Core 1 Specimen Q7

OCR Further Pure Core 1 Specimen Q8

OCR Further Pure Core 1 Specimen Q9

OCR Further Pure Core 1 Specimen Q10

OCR Further Pure Core 1 Specimen Q11

OCR Further Pure Core 2 2019 June Q1

OCR Further Pure Core 2 2019 June Q2

OCR Further Pure Core 2 2019 June Q4

OCR Further Pure Core 2 2019 June Q5

OCR Further Pure Core 2 2019 June Q7

OCR Further Pure Core 2 2019 June Q8

OCR Further Pure Core 2 2019 June Q10

OCR Further Pure Core 2 2022 June Q1

OCR Further Pure Core 2 2022 June Q2

OCR Further Pure Core 2 2022 June Q4

OCR Further Pure Core 2 2022 June Q5