Questions - A-Level Maths

Find the probability generating function of $Z$, expressing your answer as a polynomial.
Use the probability generating function of $Z$ to find $\mathrm{E}(Z)$ and $\operatorname{Var}(Z)$.

OCR MEI FP2 2011 June Q1

18 marks Standard +0.8

A curve has polar equation $r = a ( 1 - \sin \theta )$, where $a > 0$ and $0 \leqslant \theta < 2 \pi$.
1. Sketch the curve.
2. Find, in an exact form, the area of the region enclosed by the curve.
1. Find, in an exact form, the value of the integral $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x$.
2. Find, in an exact form, the value of the integral $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 + 4 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$.

OCR MEI FP2 2011 June Q2

18 marks Challenging +1.2

Use de Moivre's theorem to find expressions for $\sin 5 \theta$ and $\cos 5 \theta$ in terms of $\sin \theta$ and $\cos \theta$.
Hence show that, if $t = \tan \theta$, then $$\tan 5 \theta = \frac { t \left( t ^ { 4 } - 10 t ^ { 2 } + 5 \right) } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$
1. Find the 5th roots of $- 4 \sqrt { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$. These 5th roots are represented in the Argand diagram, in order of increasing $\theta$, by the points A , $\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$.
2. Draw the Argand diagram, making clear which point is which. The mid-point of AB is the point P which represents the complex number $w$.
3. Find, in exact form, the modulus and argument of $w$.
4. $w$ is an $n$th root of a real number $a$, where $n$ is a positive integer. State the least possible value of $n$ and find the corresponding value of $a$.

OCR MEI FP2 2011 June Q3

18 marks Challenging +1.2

Find the value of $k$ for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 1 & k \\ 5 & 4 & 6 \\ 3 & 2 & 4 \end{array} \right)$$ does not have an inverse.
Assuming that $k$ does not take this value, find the inverse of $\mathbf { M }$ in terms of $k$.
In the case $k = 3$, evaluate $$\mathbf { M } \left( \begin{array} { r } - 3 \\ 3 \\ 1 \end{array} \right)$$
State the significance of what you have found in part (ii).
Find the value of $t$ for which the system of equations $$\begin{array} { r } x - y + 3 z = t \\ 5 x + 4 y + 6 z = 1 \\ 3 x + 2 y + 4 z = 0 \end{array}$$ has solutions. Find the general solution in this case and describe the solution geometrically.

OCR MEI FP2 2011 June Q4

18 marks Challenging +1.2

Given that $\cosh y = x$, show that $y = \pm \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)$ and that $\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)$.
Find $\int _ { \frac { 4 } { 5 } } ^ { 1 } \frac { 1 } { \sqrt { 25 x ^ { 2 } - 16 } } \mathrm {~d} x$, expressing your answer in an exact logarithmic form.
Solve the equation $$5 \cosh x - \cosh 2 x = 3$$ giving your answers in an exact logarithmic form.

OCR MEI FP2 2011 June Q5

18 marks Standard +0.8

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of $m$ and $n$.

On separate diagrams, sketch the curve in each of the following cases.
(A) $m = 1 , n = 1$,
(B) $m = 2 , n = 2$,
(C) $m = 2 , n = 4$,
(D) $m = 4 , n = 2$.
What feature does the curve have when $m = n$ ? What is the effect on the curve of interchanging $m$ and $n$ when $m \neq n$ ?
Describe how the $x$-coordinate of the maximum on the curve varies as $m$ and $n$ vary. Use calculus to determine the $x$-coordinate of the maximum.
Find the condition on $m$ for the gradient to be zero when $x = 0$. State a corresponding result for the gradient to be zero when $x = 1$.
Use your calculator to investigate the shape of the curve for large values of $m$ and $n$. Hence conjecture what happens to the value of the integral $\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x$ as $m$ and $n$ tend to infinity.
Use your calculator to investigate the shape of the curve for small values of $m$ and $n$. Hence conjecture what happens to the shape of the curve as $m$ and $n$ tend to zero. }{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 1 GE.
OCR is part of the

OCR C1 2009 January Q1

3 marks Easy -1.2

1 Express $\sqrt { 45 } + \frac { 20 } { \sqrt { 5 } }$ in the form $k \sqrt { 5 }$, where $k$ is an integer.

OCR C1 2009 January Q2

4 marks Easy -1.3

2 Simplify

$( \sqrt [ 3 ] { x } ) ^ { 6 }$,
$\frac { 3 y ^ { 4 } \times ( 10 y ) ^ { 3 } } { 2 y ^ { 5 } }$.

OCR C1 2009 January Q3

5 marks Standard +0.3

3 Solve the equation $3 x ^ { \frac { 2 } { 3 } } + x ^ { \frac { 1 } { 3 } } - 2 = 0$.

OCR C1 2009 January Q4

6 marks Moderate -0.8

Sketch the curve $y = \frac { 1 } { x ^ { 2 } }$.
The curve $y = \frac { 1 } { x ^ { 2 } }$ is translated by 3 units in the negative $x$-direction. State the equation of the curve after it has been translated.
The curve $y = \frac { 1 } { x ^ { 2 } }$ is stretched parallel to the $y$-axis with scale factor 4 and, as a result, the point $P ( 1,1 )$ is transformed to the point $Q$. State the coordinates of $Q$.

OCR C1 2009 January Q5

9 marks Easy -1.3

5 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in each of the following cases:

$y = 10 x ^ { - 5 }$,
$y = \sqrt [ 4 ] { x }$,
$y = x ( x + 3 ) ( 1 - 5 x )$.

OCR C1 2009 January Q6

8 marks Moderate -0.8

Express $5 x ^ { 2 } + 20 x - 8$ in the form $p ( x + q ) ^ { 2 } + r$.
State the equation of the line of symmetry of the curve $y = 5 x ^ { 2 } + 20 x - 8$.
Calculate the discriminant of $5 x ^ { 2 } + 20 x - 8$.
State the number of real roots of the equation $5 x ^ { 2 } + 20 x - 8 = 0$.

OCR C1 2009 January Q7

8 marks Moderate -0.8

7 The line with equation $3 x + 4 y - 10 = 0$ passes through point $A ( 2,1 )$ and point $B ( 10 , k )$.

Find the value of $k$.
Calculate the length of $A B$. A circle has equation $( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25$.
Write down the coordinates of the centre and the radius of the circle.
Verify that $A B$ is a diameter of the circle.

OCR C1 2009 January Q8

10 marks Moderate -0.3

Solve the equation $5 - 8 x - x ^ { 2 } = 0$, giving your answers in simplified surd form.
Solve the inequality $5 - 8 x - x ^ { 2 } \leqslant 0$.
Sketch the curve $y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )$, giving the coordinates of the points where the curve crosses the coordinate axes.

OCR C1 2009 January Q9

7 marks Moderate -0.3

9 The curve $y = x ^ { 3 } + p x ^ { 2 } + 2$ has a stationary point when $x = 4$. Find the value of the constant $p$ and determine whether the stationary point is a maximum or minimum point.

OCR C1 2009 January Q10

12 marks Standard +0.3

10 A curve has equation $y = x ^ { 2 } + x$.

Find the gradient of the curve at the point for which $x = 2$.
Find the equation of the normal to the curve at the point for which $x = 2$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
Find the values of $k$ for which the line $y = k x - 4$ is a tangent to the curve.

OCR C1 2010 January Q1

3 marks Easy -1.2

1 Express $x ^ { 2 } - 12 x + 1$ in the form $( x - p ) ^ { 2 } + q$.

OCR C1 2010 January Q2

4 marks Easy -1.2

2 \includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-2_330_681_390_731} The graph of $y = \mathrm { f } ( x )$ for $- 2 \leqslant x \leqslant 4$ is shown above.

Sketch the graph of $y = 2 \mathrm { f } ( x )$ for $- 2 \leqslant x \leqslant 4$ on the axes provided.
Describe the transformation which transforms the graph of $y = \mathrm { f } ( x )$ to the graph of $y = \mathrm { f } ( x - 1 )$.