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AQA FP2 2015 June Q4

7 marks Standard +0.8

4 The expression $\mathrm { f } ( n )$ is given by $\mathrm { f } ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }$.

Show that $\mathrm { f } ( k + 1 ) - 16 \mathrm { f } ( k )$ can be expressed in the form $A \times 3 ^ { 3 k }$, where $A$ is an integer.
Prove by induction that $\mathrm { f } ( n )$ is a multiple of 11 for all integers $n \geqslant 1$.

AQA FP2 2015 June Q5

9 marks Standard +0.3

5 The locus of points, $L$, satisfies the equation $$| z - 2 + 4 \mathrm { i } | = | z |$$

Sketch $L$ on the Argand diagram below.
The locus $L$ cuts the real axis at $A$ and the imaginary axis at $B$.
1. Show that the complex number represented by $C$, the midpoint of $A B$, is $$\frac { 5 } { 2 } - \frac { 5 } { 4 } \mathrm { i }$$
2. The point $O$ is the origin of the Argand diagram. Find the equation of the circle that passes through the points $O , A$ and $B$, giving your answer in the form $| z - \alpha | = k$.
  [0pt] [2 marks] \section*{(a)}
  \includegraphics[max width=\textwidth, alt={}]{bc3aaed2-4aef-4aec-b657-098b1e581e55-10_1173_1242_1217_463}

AQA FP2 2015 June Q6

8 marks Challenging +1.2

Given that $y = ( x - 2 ) \sqrt { 5 + 4 x - x ^ { 2 } } + 9 \sin ^ { - 1 } \left( \frac { x - 2 } { 3 } \right)$, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 5 + 4 x - x ^ { 2 } }$$ where $k$ is an integer.
Hence show that $$\int _ { 2 } ^ { \frac { 7 } { 2 } } \sqrt { 5 + 4 x - x ^ { 2 } } \mathrm {~d} x = p \sqrt { 3 } + q \pi$$ where $p$ and $q$ are rational numbers.
[0pt] [3 marks]

AQA FP2 2015 June Q7

17 marks Standard +0.8

7 The cubic equation $27 z ^ { 3 } + k z ^ { 2 } + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.

Write down the values of $\alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.
1. In the case where $\beta = \gamma$, find the roots of the equation.
2. Find the value of $k$ in this case.
1. In the case where $\alpha = 1 - \mathrm { i }$, find $\alpha ^ { 2 }$ and $\alpha ^ { 3 }$.
2. Hence find the value of $k$ in this case.
In the case where $k = - 12$, find a cubic equation with integer coefficients which has roots $\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1$ and $\frac { 1 } { \gamma } + 1$.
[0pt] [5 marks]

AQA FP2 2015 June Q8

9 marks Challenging +1.2

8 The complex number $\omega$ is given by $\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }$.

1. Verify that $\omega$ is a root of the equation $z ^ { 5 } = 1$.
2. Write down the three other non-real roots of $z ^ { 5 } = 1$, in terms of $\omega$.
1. Show that $1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0$.
2. Hence show that $\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0$.
Hence show that $\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }$.

OCR FP2 Q1

6 marks Standard +0.3

Write down and simplify the first three non-zero terms of the Maclaurin series for $\ln ( 1 + 3 x )$.
Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.

Use the copy of the sketch on the insert to show how an iteration of the form $x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)$, with starting value $x _ { 1 }$ such that $0 < x _ { 1 } < \alpha$ as shown, converges to the root $x = \alpha$.
State what happens in the iteration in the following two cases.
1. $x _ { 1 }$ is chosen such that $\alpha < x _ { 1 } < \beta$.
2. $x _ { 1 }$ is chosen such that $x _ { 1 } > \beta$. \section*{Jan 2006} 4
  1. \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
  2. (a) $\_\_\_\_$
  (b) $\_\_\_\_$ \section*{Jan 2006}

OCR FP2 Q5

8 marks Challenging +1.2

Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
Show that $y$ cannot take values between - 3 and 1 .

OCR FP2 Q6

8 marks Standard +0.8

It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - x } x ^ { n } \mathrm {~d} x$$ Prove that, for $n \geqslant 1$, $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
Evaluate $I _ { 3 }$, giving the answer in terms of e.

OCR FP2 Q7

9 marks Challenging +1.2

7 \includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-04_673_1285_1176_429} The diagram shows the curve with equation $y = \sqrt { x }$. A set of $N$ rectangles of unit width is drawn, starting at $x = 1$ and ending at $x = N + 1$, where $N$ is an integer (see diagram).

By considering the areas of these rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } < \int _ { 1 } ^ { N + 1 } \sqrt { x } \mathrm {~d} x$$
By considering the areas of another set of rectangles, explain why $$\sqrt { 1 } + \sqrt { 2 } + \sqrt { 3 } + \ldots + \sqrt { N } > \int _ { 0 } ^ { N } \sqrt { x } \mathrm {~d} x$$
Hence find, in terms of $N$, limits between which $\sum _ { r = 1 } ^ { N } \sqrt { r }$ lies. \section*{Jan 2006}

OCR FP2 Q8

13 marks Challenging +1.2

8 The equation of a curve, in polar coordinates, is $$r = 1 + \cos 2 \theta , \quad \text { for } 0 \leqslant \theta < 2 \pi$$

State the greatest value of $r$ and the corresponding values of $\theta$.
Find the equations of the tangents at the pole.
Find the exact area enclosed by the curve and the lines $\theta = 0$ and $\theta = \frac { 1 } { 2 } \pi$.
Find, in simplified form, the cartesian equation of the curve.

OCR FP2 Q9

12 marks Standard +0.3

Using the definitions of $\cosh x$ and $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, prove that $$\sinh 2 x = 2 \sinh x \cosh x$$
Show that the curve with equation $$y = \cosh 2 x - 6 \sinh x$$ has just one stationary point, and find its $x$-coordinate in logarithmic form. Determine the nature of the stationary point.

AQA FP3 Q5

18 marks Standard +0.8

5 The complex number $z$ satisfies the relation $$| z + 4 - 4 i | = 4$$

Sketch, on an Argand diagram, the locus of $z$.
Show that the greatest value of $| z |$ is $4 ( \sqrt { 2 } + 1 )$.
Find the value of $z$ for which $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form $a + \mathrm { i } b$.

AQA FP3 Q6

17 marks Challenging +1.2

6 It is given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$.

1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form $a + \mathrm { i } b$.

AQA FP3 2006 January Q1

12 marks Standard +0.3

Find the roots of the equation $m ^ { 2 } + 2 m + 2 = 0$ in the form $a + i b$.
(2 marks)
1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x$$
2. Hence express $y$ in terms of $x$, given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$ when $x = 0$.

AQA FP3 2006 January Q2

8 marks Standard +0.3

Find $\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x$, where $a > 0$.
Write down the value of $\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }$, where $k$ is a positive constant.
Hence find $\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x$.

AQA FP3 2006 January Q3

8 marks Standard +0.3

Show that $y = x ^ { 3 } - x$ is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
By differentiating $\left( x ^ { 2 } - 1 \right) y = c$ implicitly, where $y$ is a function of $x$ and $c$ is a constant, show that $y = \frac { c } { x ^ { 2 } - 1 }$ is a solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 0$$
Hence find the general solution of $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$

AQA FP3 2006 January Q4

14 marks Standard +0.8

Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } + \ldots$$ to write down the first four terms in the expansion, in ascending powers of $x$, of $\ln ( 1 - x )$.
The function f is defined by $$\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$$ Use Maclaurin's theorem to show that when $\mathrm { f } ( x )$ is expanded in ascending powers of $x$ :
1. the first three terms are $$1 + x + \frac { 1 } { 2 } x ^ { 2 }$$
2. the coefficient of $x ^ { 3 }$ is zero.
Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { \sin x } - 1 + \ln ( 1 - x ) } { x ^ { 2 } \sin x }$$ (4 marks)

AQA FP3 2006 January Q5

17 marks Standard +0.3

The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x \ln x + \frac { y } { x }$$ and $$y ( 1 ) = 1$$
1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.1$, to obtain an approximation to $y ( 1.1 )$.
2. Use the formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with your answer to part (a)(i) to obtain an approximation to $y ( 1.2 )$, giving your answer to three decimal places.
1. Show that $\frac { 1 } { x }$ is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 1 } { x } y = x \ln x$$
2. Solve this differential equation, given that $y = 1$ when $x = 1$.
3. Calculate the value of $y$ when $x = 1.2$, giving your answer to three decimal places.

AQA FP3 2006 January Q6

16 marks Challenging +1.2

A circle $C _ { 1 }$ has cartesian equation $x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36$. Show that the polar equation of $C _ { 1 }$ is $r = 12 \sin \theta$.
A curve $C _ { 2 }$ with polar equation $r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi$ is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651} Calculate the area bounded by $C _ { 2 }$.
The circle $C _ { 1 }$ intersects the curve $C _ { 2 }$ at the points $P$ and $Q$. Find, in surd form, the area of the quadrilateral $O P M Q$, where $M$ is the centre of the circle and $O$ is the pole.
(6 marks)

AQA FP3 2007 January Q1

9 marks Standard +0.3

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln \left( 1 + x ^ { 2 } + y \right)$$ and $$y ( 1 ) = 0.6$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.05$, to obtain an approximation to $y ( 1.05 )$, giving your answer to four decimal places.
Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.05$, to obtain an approximation to $y ( 1.05 )$, giving your answer to four decimal places.

AQA FP3 2007 January Q2

6 marks Standard +0.8

2 A curve has polar equation $r ( 1 - \sin \theta ) = 4$. Find its cartesian equation in the form $y = \mathrm { f } ( x )$.

AQA FP3 2007 January Q3

9 marks Standard +0.3

Show that $x ^ { 2 }$ is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
Solve this differential equation, given that $y = 1$ when $x = 2$.

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