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OCR MEI C2 Q2

5 marks Easy -1.2

2 The $n$th term of a sequence, $u _ { n }$, is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$

Write down the values of $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$. State what type of sequence this is.
Find $\sum _ { n = 1 } ^ { 30 } u _ { n }$.

Jim's first payment is $\pounds 500$, and he plans to pay $\pounds 10$ less each month, so that his second payment is $\pounds 490$, his third is $\pounds 480$, and so on.
(A) Calculate his 12th payment.
(B) He plans to make 24 payments altogether. Show that he pays $\pounds 9240$ in total.
Mary's first payment is $\pounds 460$ and she plans to pay $2 \%$ less each month than the previous month, so that her second payment is $\pounds 450.80$, her third is $\pounds 441.784$, and so on.
(A) Calculate her 12th payment.
(B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
(C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
(D) How much would Mary's first payment need to be if she wishes to pay $2 \%$ less each month as before, but to pay the same in total as Jim, $\pounds 9240$, over the 24 months?

OCR MEI C2 Q6

2 marks Easy -1.2

6 You are given that $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$. Give your answers as fractions.

OCR MEI C2 Q7

4 marks Easy -1.3

Evaluate $\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }$.
Express the series $2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7$ in the form $\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )$ where $\mathrm { f } ( r )$ and $a$ are to be determined.

OCR MEI C2 Q8

3 marks Moderate -0.8

Find $\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)$.
State whether the sequence with $k$ th term $k ^ { 2 } - 1$ is convergent or divergent, giving a reason for your answer.

OCR MEI C2 Q9

4 marks Easy -1.2

Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
Find $\sum _ { k = 1 } ^ { 3 } k ( k + 1 )$.

OCR MEI C2 Q10

3 marks Easy -1.2

10 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.

$3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots$
$3,7,11,15 , \ldots$
$3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots$

OCR MEI C2 Q11

2 marks Easy -1.2

11 Find $\sum _ { r = 3 } ^ { 6 } r ( r + 2 )$.

OCR MEI C2 Q12

5 marks Moderate -0.8

12 Calculate $\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }$. 12 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.

Find the 55th term of this sequence, showing your method.
Find the sum of the first 55 terms of the sequence.

OCR C3 Q1

5 marks Moderate -0.3

1. $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$ Find an equation for the tangent to the curve $y = \mathrm { f } ( x )$ at the point where $x = - 2$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

OCR C3 Q2

7 marks Standard +0.3

2. \includegraphics[max width=\textwidth, alt={}, center]{b124d427-1f9b-4770-95bb-ed79bae5b4fb-1_460_805_587_486} The diagram shows the curve with equation $y = \frac { 1 } { 2 } \ln 3 x$.

Express the equation of the curve in the form $x = \mathrm { f } ( y )$. The shaded region is bounded by the curve, the coordinate axes and the line $y = 1$.
Find, in terms of $\pi$ and e, the volume of the solid formed when the shaded region is rotated through four right angles about the $y$-axis.

OCR C3 Q3

8 marks Standard +0.3

Use the identity for $\sin ( A + B )$ to show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$
Hence find, in terms of $\pi$, the solutions of the equation $$\sin 3 x - \sin x = 0$$ for $x$ in the interval $0 \leq x < 2 \pi$.

OCR C3 Q4

9 marks Moderate -0.3

4. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R }$$ where $a$ is a positive constant.

Showing the coordinates of any points where the graph meets the axes, sketch the graph of $y = | \mathrm { f } ( x ) |$. The function $g$ is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
Find $\mathrm { fg } ( \mathrm { a } )$ in terms of $a$.
Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

OCR C3 Q5

9 marks Standard +0.8

Find, as natural logarithms, the solutions of the equation $$\mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } + 15 = 0$$
Use proof by contradiction to prove that $\log _ { 2 } 3$ is irrational.

OCR C3 Q6

11 marks Standard +0.3

6. $\quad f ( x ) = 2 x ^ { 2 } + 3 \ln ( 2 - x ) , \quad x \in \mathbb { R } , \quad x < 2$.

Show that the equation $\mathrm { f } ( x ) = 0$ can be written in the form $$x = 2 - \mathrm { e } ^ { k x ^ { 2 } }$$ where $k$ is a constant to be found. The root, $\alpha$, of the equation $\mathrm { f } ( x ) = 0$ is 1.9 correct to 1 decimal place.
Use the iterative formula $$x _ { n + 1 } = 2 - \mathrm { e } ^ { k x _ { n } ^ { 2 } }$$ with $x _ { 0 } = 1.9$ and your value of $k$, to find $\alpha$ correct to 3 decimal places.
You should show the result of each iteration.
Solve the equation $\mathrm { f } ^ { \prime } ( x ) = 0$.

OCR C3 Q7

11 marks Standard +0.3

Use the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$
Prove that, for $\sin x \neq 0$, $$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$
Find the values of $x$ in the interval $0 \leq x \leq 360 ^ { \circ }$ for which $$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$ giving your answers to 1 decimal place where appropriate.

OCR C3 Q8

12 marks Standard +0.3

8. $f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1$.

Express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.
State the range of f .
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
Describe fully two transformations that would map the graph of $y = \mathrm { f } ^ { - 1 } ( x )$ onto the graph of $y = \sqrt { x } , x \geq 0$.
Find an equation for the normal to the curve $y = \mathrm { f } ^ { - 1 } ( x )$ at the point where $x = 8$.

OCR MEI C2 Q1

2 marks Easy -1.2

1 Find $\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }$.

OCR MEI C2 Q2

5 marks Moderate -0.8

2 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 , \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$

Find the third term of this sequence and state what type of sequence it is.
Show that the series $u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$ converges and find its sum to infinity.