Questions Further Pure Core

OCR MEI Further Pure Core Specimen Q2

6 marks Standard +0.8

On an Argand diagram draw the locus of points which satisfy $\arg(z - 4i) = \frac{\pi}{4}$. [2]
Give, in complex form, the equation of the circle which has centre at $6 + 4i$ and touches the locus in part (i). [4]

OCR MEI Further Pure Core Specimen Q3

6 marks Standard +0.3

Transformation M is represented by matrix $\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$.

On the diagram in the Printed Answer Booklet draw the image of the unit square under M. [2]
1. Show that there is a constant $k$ such that $\mathbf{M} \begin{pmatrix} x \\ kx \end{pmatrix} = 5 \begin{pmatrix} x \\ kx \end{pmatrix}$ for all $x$. [2]
2. Hence find the equation of an invariant line under M. [1]
3. Draw the invariant line from part (ii) (B) on your diagram for part (i). [1]

OCR MEI Further Pure Core Specimen Q4

5 marks Standard +0.3

You are given that $z = 1 + 2i$ is a root of the equation $z^3 - 5z^2 + qz - 15 = 0$, where $q \in \mathbb{R}$. Find • the other roots, • the value of $q$. [5]

OCR MEI Further Pure Core Specimen Q5

7 marks Standard +0.8

Express $\frac{2}{(r+1)(r+3)}$ in partial fractions. [2]
Hence find $\sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}$, expressing your answer as a single fraction. [5]

OCR MEI Further Pure Core Specimen Q6

6 marks Standard +0.8

A curve is in the first quadrant. It has parametric equations $x = \cosh t + \sinh t$, $y = \cosh t - \sinh t$ where $t \in \mathbb{R}$. Show that the cartesian equation of the curve is $xy = 1$. [2]

Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the $x$-axis, point B lies on the $y$-axis and OAPB is a rectangle. \includegraphics{figure_6}

Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer. [4]

OCR MEI Further Pure Core Specimen Q7

11 marks Standard +0.3

Use the Maclaurin series for $\ln(1 + x)$ up to the term in $x^3$ to obtain an approximation to $\ln 1.5$. [2]
1. Find the error in the approximation in part (i). [1]
2. Explain why the Maclaurin series in part (i), with $x = 2$, should not be used to find an approximation to $\ln 3$. [1]
Find a cubic approximation to $\ln\left(\frac{1+x}{1-x}\right)$. [2]
1. Use the approximation in part (iii) to find approximations to • $\ln 1.5$ and • $\ln 3$. [3]
2. Comment on your answers to part (iv) (A). [2]

OCR MEI Further Pure Core Specimen Q8

5 marks Standard +0.3

Find the cartesian equation of the plane which contains the three points $(1, 0, -1)$, $(2, 2, 1)$ and $(1, 1, 2)$. [5]

OCR MEI Further Pure Core Specimen Q9

7 marks Challenging +1.3

A curve has polar equation $r = a \sin 3\theta$ for $-\frac{1}{4}\pi \leq \theta \leq \frac{1}{4}\pi$, where $a$ is a positive constant.

Sketch the curve. [2]
In this question you must show detailed reasoning. Find, in terms of $a$ and $\pi$, the area enclosed by one of the loops of the curve. [5]

OCR MEI Further Pure Core Specimen Q10

9 marks Standard +0.8

Obtain the solution to the differential equation $$x \frac{dy}{dx} + 3y = \frac{1}{x}, \text{ where } x > 0,$$ given that $y = 1$ when $x = 1$. [7]
Deduce that $y$ decreases as $x$ increases. [2]

OCR MEI Further Pure Core Specimen Q11

9 marks Standard +0.8

It is conjectured that $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + ... + \frac{n-1}{n!} = a - \frac{b}{n!},$$ where $a$ and $b$ are constants, and $n$ is an integer such that $n \geq 2$. By considering particular cases, show that if the conjecture is correct then $a = b = 1$. [2]
Use induction to prove that $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + ... + \frac{n-1}{n!} = 1 - \frac{1}{n!} \text{ for } n \geq 2.$$ [7]

OCR MEI Further Pure Core Specimen Q12

13 marks Standard +0.3

In this question you must show detailed reasoning.

Given that $y = \arctan x$, show that $\frac{dy}{dx} = \frac{1}{1+x^2}$. [3]

Fig. 12 shows the curve $y = \frac{1}{1+x^2}$. \includegraphics{figure_12}

Find, in exact form, the mean value of the function $f(x) = \frac{1}{1+x^2}$ for $-1 \leq x \leq 1$. [3]
The region bounded by the curve, the $x$-axis, and the lines $x = 1$ and $x = -1$ is rotated through $2\pi$ radians about the $x$-axis. Find, in exact form, the volume of the solid of revolution generated. [7]

OCR MEI Further Pure Core Specimen Q13

13 marks Challenging +1.2

Matrix M is given by $\mathbf{M} = \begin{pmatrix} k & 1 & -5 \\ 2 & 3 & -3 \\ -1 & 2 & 2 \end{pmatrix}$, where $k$ is a constant.

Show that $\det \mathbf{M} = 12(k - 3)$. [2]
Find a solution of the following simultaneous equations for which $x \neq z$. $$4x^2 + y^2 - 5z^2 = 6$$ $$2x^2 + 3y^2 - 3z^2 = 6$$ $$-x^2 + 2y^2 + 2z^2 = -6$$ [3]
1. Verify that the point $(2, 0, 1)$ lies on each of the following three planes. $$3x + y - 5z = 1$$ $$2x + 3y - 3z = 1$$ $$-x + 2y + 2z = 0$$ [1]
2. Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer. [4]
Find the values of $k$ for which the transformation represented by M has a volume scale factor of 6. [3]

OCR MEI Further Pure Core Specimen Q14

18 marks Challenging +1.2

Starting with the result $$e^{i\theta} = \cos \theta + i \sin \theta,$$ show that
1. $(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$ [2]
2. $\cos \theta = \frac{1}{2}(e^{i\theta} + e^{-i\theta})$. [2]
Using the result in part (i) (A), obtain the values of the constants $a$, $b$, $c$ and $d$ in the identity $$\cos 6\theta = a \cos^6 \theta + b \cos^4 \theta + c \cos^2 \theta + d.$$ [6]
Using the result in part (i) (B), obtain the values of the constants $P$, $Q$, $R$ and $S$ in the identity $$\cos^6 \theta = P \cos 6\theta + Q \cos 4\theta + R \cos 2\theta + S.$$ [5]
Show that $\cos \frac{\pi}{12} = \left(\frac{26 + 15\sqrt{3}}{64}\right)^{\frac{1}{4}}$. [3]

OCR MEI Further Pure Core Specimen Q15

8 marks Challenging +1.8

In this question you must show detailed reasoning. Show that $$\int_0^{\frac{\pi}{3}} \operatorname{arcsinh} 2x \, dx = \frac{2}{3} \ln 3 - \frac{1}{3}.$$ [8]

OCR MEI Further Pure Core Specimen Q16

18 marks Challenging +1.2

A small object is attached to a spring and performs oscillations in a vertical line. The displacement of the object at time $t$ seconds is denoted by $x$ cm. Preliminary observations suggest that the object performs simple harmonic motion (SHM) with a period of 2 seconds about the point at which $x = 0$.

1. Write down a differential equation to model this motion. [3]
2. Give the general solution of the differential equation in part (i) (A). [1]

Subsequent observations indicate that the object's motion would be better modelled by the differential equation $$\frac{d^2x}{dt^2} + 2k \frac{dx}{dt} + (k^2 + 9)x = 0 \qquad (*)$$ where $k$ is a positive constant.

1. Obtain the general solution of (*). [3]
2. State two ways in which the motion given by this model differs from that in part (i). [2]

The amplitude of the object's motion is observed to reduce with a scale factor of 0.98 from one oscillation to the next.

Find the value of $k$. [3]

At the start of the object's motion, $x = 0$ and the velocity is 12 cm s$^{-1}$ in the positive $x$ direction.

Find an equation for $x$ as a function of $t$. [4]
Without doing any further calculations, explain why, according to this model, the greatest distance of the object from its starting point in the subsequent motion will be slightly less than 4 cm. [2]

Questions Further Pure Core (115 questions)

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