Questions - OCR - A-Level Maths

OCR C2 Q9

9

1. Write down the exact values of $\cos \frac { 1 } { 6 } \pi$ and $\tan \frac { 1 } { 3 } \pi$ (where the angles are in radians). Hence verify that $x = \frac { 1 } { 6 } \pi$ is a solution of the equation $$2 \cos x = \tan 2 x$$
2. Sketch, on a single diagram, the graphs of $y = 2 \cos x$ and $y = \tan 2 x$, for $x$ (radians) such that $0 \leqslant x \leqslant \pi$. Hence state, in terms of $\pi$, the other values of $x$ between 0 and $\pi$ satisfying the equation $$2 \cos x = \tan 2 x$$
1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve $y = \tan x$, the $x$-axis, and the lines $x = 0.1$ and $x = 0.4$. (Values of $x$ are in radians.)
2. State with a reason whether this approximation is an underestimate or an overestimate. 1 The 20th term of an arithmetic progression is 10 and the 50th term is 70.
3. Find the first term and the common difference.
4. Show that the sum of the first 29 terms is zero. 2 Triangle $A B C$ has $A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}$ and angle $B = 80 ^ { \circ }$. Calculate
5. the area of the triangle,
6. the length of $C A$,
7. the size of angle $C$. 3
8. Find the first three terms of the expansion, in ascending powers of $x$, of $( 1 - 2 x ) ^ { 12 }$.
9. Hence find the coefficient of $x ^ { 2 }$ in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 }$$ 4
  \includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-05_657_803_1283_671} The diagram shows a sector $O A B$ of a circle with centre $O$. The angle $A O B$ is 1.8 radians. The points $C$ and $D$ lie on $O A$ and $O B$ respectively. It is given that $O A = O B = 20 \mathrm {~cm}$ and $O C = O D = 15 \mathrm {~cm}$. The shaded region is bounded by the arcs $A B$ and $C D$ and by the lines $C A$ and $D B$.
10. Find the perimeter of the shaded region.
11. Find the area of the shaded region. 5 In a geometric progression, the first term is 5 and the second term is 4.8.
12. Show that the sum to infinity is 125 .
13. The sum of the first $n$ terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of $n$. 6
Find $\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x$.
1. Find the value, in terms of $a$, of $\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x$, where $a$ is a constant greater than 1 .
2. Deduce the value of $\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x$. 7
3. Express each of the following in terms of $\log _ { 10 } x$ and $\log _ { 10 } y$.
$\log _ { 10 } \left( \frac { x } { y } \right)$
$\log _ { 10 } \left( 10 x ^ { 2 } y \right)$
(ii) Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of $y$ correct to 3 decimal places. 8 The cubic polynomial $2 x ^ { 3 } + k x ^ { 2 } - x + 6$ is denoted by $\mathrm { f } ( x )$. It is given that ( $x + 1$ ) is a factor of $\mathrm { f } ( x )$.
(i) Show that $k = - 5$, and factorise $\mathrm { f } ( x )$ completely.
(ii) Find $\int _ { - 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$.
(iii) Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve $y = \mathrm { f } ( x )$ and the $x$-axis for $- 1 \leqslant x \leqslant 2$. 9 (i) Sketch, on a single diagram showing values of $x$ from $- 180 ^ { \circ }$ to $+ 180 ^ { \circ }$, the graphs of $y = \tan x$ and $y = 4 \cos x$. The equation $$\tan x = 4 \cos x$$ has two roots in the interval $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$. These are denoted by $\alpha$ and $\beta$, where $\alpha < \beta$.
(ii) Show $\alpha$ and $\beta$ on your sketch, and express $\beta$ in terms of $\alpha$.
(iii) Show that the equation $\tan x = 4 \cos x$ may be written as $$4 \sin ^ { 2 } x + \sin x - 4 = 0$$ Hence find the value of $\beta - \alpha$, correct to the nearest degree. 1 Find the binomial expansion of $( 3 x - 2 ) ^ { 4 }$. 2 A sequence of terms $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { 1 } = 2 \quad \text { and } \quad u _ { n + 1 } = 1 - u _ { n } \text { for } n \geqslant 1 .$$ (i) Write down the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
(ii) Find $\sum _ { n = 1 } ^ { 100 } u _ { n }$. 3 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }$, and the curve passes through the point (4,5). Find the equation of the curve. 4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-08_636_670_1123_740} The diagram shows the curve $y = 4 - x ^ { 2 }$ and the line $y = x + 2$.
(i) Find the $x$-coordinates of the points of intersection of the curve and the line.
(ii) Use integration to find the area of the shaded region bounded by the line and the curve. 5 Solve each of the following equations, for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
(i) $2 \sin ^ { 2 } x = 1 + \cos x$.
(ii) $\sin 2 x = - \cos 2 x$. 6 (i) John aims to pay a certain amount of money each month into a pension fund. He plans to pay $\pounds 100$ in the first month, and then to increase the amount paid by $\pounds 5$ each month, i.e. paying $\pounds 105$ in the second month, $\pounds 110$ in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
how much he will pay in the final month,
how much he will pay altogether over the whole period.
(ii) Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with $\pounds 100$ in the first month. Her monthly payments will form a geometric progression, and she will pay $\pounds 1500$ in the final month. Calculate how much Rachel will pay altogether over the whole period. 7
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-09_488_1027_995_559} The diagram shows a triangle $A B C$, and a sector $A C D$ of a circle with centre $A$. It is given that $A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}$, angle $A B C = 0.8$ radians and angle $D A C = 1.7$ radians. The shaded segment is bounded by the line $D C$ and the arc $D C$.
(i) Show that the length of $A C$ is 7.90 cm , correct to 3 significant figures.
(ii) Find the area of the shaded segment.
(iii) Find the perimeter of the shaded segment. 8 The cubic polynomial $2 x ^ { 3 } + a x ^ { 2 } + b x - 10$ is denoted by $\mathrm { f } ( x )$. It is given that, when $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 12 . It is also given that $( x + 1 )$ is a factor of $\mathrm { f } ( x )$.
(i) Find the values of $a$ and $b$.
(ii) Divide $\mathrm { f } ( x )$ by $( x + 2 )$ to find the quotient and the remainder.
(i) Sketch the curve $y = \left( \frac { 1 } { 2 } \right) ^ { x }$, and state the coordinates of any point where the curve crosses an axis.
(ii) Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve $y = \left( \frac { 1 } { 2 } \right) ^ { x }$, the axes, and the line $x = 2$.
(iii) The point $P$ on the curve $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ has $y$-coordinate equal to $\frac { 1 } { 6 }$. Prove that the $x$-coordinate of $P$ may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$ 1 In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms. 2 The diagram shows a sector $O A B$ of a circle, centre $O$ and radius 8 cm . The angle $A O B$ is $46 ^ { \circ }$.
(i) Express $46 ^ { \circ }$ in radians, correct to 3 significant figures.
(ii) Find the length of the arc $A B$.
(iii) Find the area of the sector $O A B$. 3 (i) Find $\int ( 4 x - 5 ) \mathrm { d } x$.
(ii) The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x - 5$. The curve passes through the point (3, 7). Find the equation of the curve. 4 In a triangle $A B C , A B = 5 \sqrt { 2 } \mathrm {~cm} , B C = 8 \mathrm {~cm}$ and angle $B = 60 ^ { \circ }$.
(i) Find the exact area of the triangle, giving your answer as simply as possible.
(ii) Find the length of $A C$, correct to 3 significant figures. 5
1. Express $\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x$ as a single logarithm.
2. Hence solve the equation $\log _ { 3 } ( 4 x + 7 ) - \log _ { 3 } x = 2$.
Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int _ { 3 } ^ { 9 } \log _ { 10 } x d x$$ giving your answer correct to 3 significant figures.
1. Find and simplify the first four terms in the expansion of $( 1 + 4 x ) ^ { 7 }$ in ascending powers of $x$.
2. In the expansion of $$( 3 + a x ) ( 1 + 4 x ) ^ { 7 } ,$$ the coefficient of $x ^ { 2 }$ is 1001 . Find the value of $a$.
3. (a) Sketch the graph of $y = 2 \cos x$ for values of $x$ such that $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$, indicating the coordinates of any points where the curve meets the axes.
Solve the equation $2 \cos x = 0.8$, giving all values of $x$ between $0 ^ { \circ }$ and $360 ^ { \circ }$.
(ii) Solve the equation $2 \cos x = \sin x$, giving all values of $x$ between $- 180 ^ { \circ }$ and $180 ^ { \circ }$. 8 The polynomial $\mathrm { f } ( x )$ is defined by $\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 7 x + 33$.
(i) Find the remainder when $\mathrm { f } ( x )$ is divided by ( $x + 2$ ).
(ii) Show that $( x - 3 )$ is a factor of $\mathrm { f } ( x )$.
(iii) Solve the equation $\mathrm { f } ( x ) = 0$, giving each root in an exact form as simply as possible. On its first trip between Malby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses $2 \%$ more coal than the previous trip.
(i) Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures.
(ii) There are 39 tonnes of coal available. An engineer wishes to calculate $N$, the total number of trips possible. Show that $N$ satisfies the inequality $$1.02 ^ { N } \leqslant 1.52 .$$ (iii) Hence, by using logarithms, find the greatest number of trips possible. \section*{Jan 2007}

OCR C2 Q10

7 marks

10
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-13_696_931_269_607} The diagram shows the graph of $y = 1 - 3 x ^ { - \frac { 1 } { 2 } }$.

Verify that the curve intersects the $x$-axis at $( 9,0 )$.
The shaded region is enclosed by the curve, the $x$-axis and the line $x = a$ (where $a > 9$ ). Given that the area of the shaded region is 4 square units, find the value of $a$. June 2007 1 A geometric progression $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 \text {. }$$
Write down the values of $u _ { 2 } , u _ { 3 }$ and $u _ { 4 }$.
Find $\sum _ { n = 1 } ^ { 20 } u _ { n }$. 2 Expand $\left( x + \frac { 2 } { x } \right) ^ { 4 }$ completely, simplifying the terms. 3 Use logarithms to solve the equation $3 ^ { 2 x + 1 } = 5 ^ { 200 }$, giving the value of $x$ correct to 3 significant figures. 4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-14_543_855_1155_646} The diagram shows the curve $\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }$.
Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve $\mathrm { y } = \sqrt { 4 \mathrm { x } + 1 }$, the x -axis, and the lines $\mathrm { x } = 1$ and $\mathrm { x } = 3$. Give your answer correct to 3 significant figures.
State with a reason whether this approximation is an under-estimate or an over-estimate. 5
Show that the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1$$ can be expressed in the form $$3 \sin ^ { 2 } \theta + \sin \theta - 2 = 0 .$$
Hence solve the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1 ,$$ giving all values of $\theta$ between $0 ^ { \circ }$ and $360 ^ { \circ }$. 6 (a) (i) Find $\int x \left( x ^ { 2 } - 4 \right) d x$
Hence evaluate $\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x$.
(b) Find $\int \frac { 6 } { x ^ { 3 } } d x$ 7 (a) In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
(b) In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio. 8
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-15_305_744_1043_703} The diagram shows a triangle ABC , where angle BAC is 0.9 radians. BAD is a sector of the circle with centre $A$ and radius $A B$.
The area of the sector $B A D$ is $16.2 \mathrm {~cm} ^ { 2 }$. Show that the length of $A B$ is 6 cm .
The area of triangle $A B C$ is twice the area of sector $B A D$. Find the length of $A C$.
Find the perimeter of the region $B C D$. 9 The polynomial $f ( x )$ is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$
(a) Show that $( x + 1 )$ is a factor of $f ( x )$.
(b) Hence find the exact roots of the equation $f ( x ) = 0$.
(a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form $\mathrm { f } ( \mathrm { x } ) = 0$.
(b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root. \section*{Jan 2008} 1 The diagram shows a sector $A O B$ of a circle with centre $O$ and radius 11 cm . The angle $A O B$ is 0.7 radians. Find the area of the segment shaded in the diagram. 2 Use the trapezium rule, with 3 strips each of width 2, to estimate the value of $$\int _ { 1 } ^ { 7 } \sqrt { x ^ { 2 } + 3 } \mathrm {~d} x$$ 3 Express each of the following as a single logarithm:
$\log _ { a } 2 + \log _ { a } 3$,
$2 \log _ { 10 } x - 3 \log _ { 10 } y$. 4
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-16_515_713_1567_715} In the diagram, angle $B D C = 50 ^ { \circ }$ and angle $B C D = 62 ^ { \circ }$. It is given that $A B = 10 \mathrm {~cm} , A D = 20 \mathrm {~cm}$ and $B C = 16 \mathrm {~cm}$.
Find the length of $B D$.
Find angle $B A D$. 5 The gradient of a curve is given by $\frac { \mathrm { d } y } { \mathrm {~d} x } = 12 \sqrt { x }$. The curve passes through the point (4,50). Find the equation of the curve. 6 A sequence of terms $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { n } = 2 n + 5 , \quad \text { for } n \geqslant 1$$
Write down the values of $u _ { 1 } , u _ { 2 }$ and $u _ { 3 }$.
State what type of sequence it is.
Given that $\sum _ { n = 1 } ^ { N } u _ { n } = 2200$, find the value of $N$. 7
\includegraphics[max width=\textwidth, alt={}, center]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-17_588_569_854_788} The diagram shows part of the curve $y = x ^ { 2 } - 3 x$ and the line $x = 5$.
Explain why $\int _ { 0 } ^ { 5 } \left( x ^ { 2 } - 3 x \right) \mathrm { d } x$ does not give the total area of the regions shaded in the diagram.
Use integration to find the exact total area of the shaded regions. 8 The first term of a geometric progression is 10 and the common ratio is 0.8.
Find the fourth term.
Find the sum of the first 20 terms, giving your answer correct to 3 significant figures.
The sum of the first $N$ terms is denoted by $S _ { N }$, and the sum to infinity is denoted by $S _ { \infty }$. Show that the inequality $S _ { \infty } - S _ { N } < 0.01$ can be written as $$0.8 ^ { N } < 0.0002$$ and use logarithms to find the smallest possible value of $N$. 9
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_770_274_731} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Fig. 1 shows the curve $y = 2 \sin x$ for values of $x$ such that $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$. State the coordinates of the maximum and minimum points on this part of the curve.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fbe396af-4d5b-4c3f-b528-b2e0783c7bc4-18_378_771_954_730} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 shows the curve $y = 2 \sin x$ and the line $y = k$. The smallest positive solution of the equation $2 \sin x = k$ is denoted by $\alpha$. State, in terms of $\alpha$, and in the range $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$,
(a) another solution of the equation $2 \sin x = k$,
(b) one solution of the equation $2 \sin x = - k$.
Find the $x$-coordinates of the points where the curve $y = 2 \sin x$ intersects the curve $y = 2 - 3 \cos ^ { 2 } x$, for values of $x$ such that $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$. 10
Find the binomial expansion of $( 2 x + 5 ) ^ { 4 }$, simplifying the terms.
Hence show that $( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 }$ can be written as $$320 x ^ { 3 } + k x$$ where the value of the constant $k$ is to be stated.
Verify that $x = 2$ is a root of the equation $$( 2 x + 5 ) ^ { 4 } - ( 2 x - 5 ) ^ { 4 } = 3680 x - 800$$ and find the other possible values of $x$.

OCR C2 Q1

Giving your answers in terms of $\pi$, solve the equation

$$3 \tan ^ { 2 } \theta - 1 = 0 ,$$ for $\theta$ in the interval $- \pi \leq \theta \leq \pi$.

OCR C2 Q2

2. Given that $p = \log _ { 2 } 3$ and $q = \log _ { 2 } 5$, find expressions in terms of $p$ and $q$ for

$\quad \log _ { 2 } 45$,
$\log _ { 2 } 0.3$

OCR C2 Q3

3. For the binomial expansion in ascending powers of $x$ of $\left( 1 + \frac { 1 } { 4 } x \right) ^ { n }$, where $n$ is an integer and $n \geq 2$,

find and simplify the first three terms,
find the value of $n$ for which the coefficient of $x$ is equal to the coefficient of $x ^ { 2 }$.

OCR C2 Q4

4.
\includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-1_572_803_1336_461} The diagram shows the curves with equations $y = 7 - 2 x - 3 x ^ { 2 }$ and $y = \frac { 2 } { x }$.
The two curves intersect at the points $P , Q$ and $R$.

Show that the $x$-coordinates of $P , Q$ and $R$ satisfy the equation $$3 x ^ { 3 } + 2 x ^ { 2 } - 7 x + 2 = 0$$ Given that $P$ has coordinates $( - 2 , - 1 )$,
find the coordinates of $Q$ and $R$.

OCR C2 Q5

5. The curve $y = \mathrm { f } ( x )$ passes through the point $P ( - 1,3 )$ and is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { x ^ { 3 } } , \quad x \neq 0$$

Find $\mathrm { f } ( x )$.
Show that the area of the finite region bounded by the curve $y = \mathrm { f } ( x )$, the $x$-axis and the lines $x = 1$ and $x = 4$ is $4 \frac { 1 } { 2 }$.

OCR C2 Q6

6.
\includegraphics[max width=\textwidth, alt={}, center]{30d4e6e5-8235-44b0-ad8e-c4c0b313677f-2_577_970_799_360} The diagram shows triangle $A B C$ in which $A C = 14 \mathrm {~cm} , B C = 8 \mathrm {~cm}$ and $\angle A B C = 1.7$ radians.

Find the size of $\angle A C B$ in radians. The point $D$ lies on $A C$ such that $B D$ is an arc of a circle, centre $C$.
Find the perimeter of the shaded region bounded by the arc $B D$ and the straight lines $A B$ and $A D$.

OCR C2 Q7

7. (a) Given that $y = 3 ^ { x }$, find expressions in terms of $y$ for

$3 ^ { x + 1 }$,
$3 ^ { 2 x - 1 }$.
(b) Hence, or otherwise, solve the equation $$3 ^ { x + 1 } - 3 ^ { 2 x - 1 } = 6$$

OCR C2 Q8

(i) Given that

$$\int _ { 1 } ^ { 3 } \left( x ^ { 2 } - 2 x + k \right) d x = 8 \frac { 2 } { 3 }$$ find the value of the constant $k$.
(ii) Evaluate $$\int _ { 2 } ^ { \infty } \frac { 6 } { x ^ { \frac { 5 } { 2 } } } \mathrm {~d} x$$ giving your answer in its simplest form.

OCR C2 Q9

9. The second and fifth terms of a geometric series are - 48 and 6 respectively.

Find the first term and the common ratio of the series.
Find the sum to infinity of the series.
Show that the difference between the sum of the first $n$ terms of the series and its sum to infinity is given by $2 ^ { 6 - n }$.

OCR C3 Q1

1 The function f is defined for all real values of $x$ by $$f ( x ) = 10 - ( x + 3 ) ^ { 2 }$$

State the range of f .
Find the value of $\mathrm { ff } ( - 1 )$.

OCR C3 Q3

3 The mass, $m$ grams, of a substance at time $t$ years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t }$$

Find the value of $t$ for which the mass is 25 grams.
Find the rate at which the mass is decreasing when $t = 55$.

OCR C3 Q4

4

\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-02_586_793_1274_717} The diagram shows the curve $y = \frac { 2 } { \sqrt { } x }$. The region $R$, shaded in the diagram, is bounded by the curve and by the lines $x = 1 , x = 5$ and $y = 0$. The region $R$ is rotated completely about the $x$-axis. Find the exact volume of the solid formed.
Use Simpson's rule, with 4 strips, to find an approximate value for $$\int _ { 1 } ^ { 5 } \sqrt { } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer correct to 3 decimal places.

OCR C3 Q5

5

Express $3 \sin \theta + 2 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
Hence solve the equation $3 \sin \theta + 2 \cos \theta = \frac { 7 } { 2 }$, giving all solutions for which $0 ^ { \circ } < \theta < 360 ^ { \circ }$. \section*{June 2005}

OCR C3 Q8

8
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-03_579_901_959_623} The diagram shows part of each of the curves $y = e ^ { \frac { 1 } { 5 } x }$ and $y = \sqrt [ 3 ] { } ( 3 x + 8 )$. The curves meet, as shown in the diagram, at the point $P$. The region $R$, shaded in the diagram, is bounded by the two curves and by the $y$-axis.

Show by calculation that the $x$-coordinate of $P$ lies between 5.2 and 5.3.
Show that the $x$-coordinate of $P$ satisfies the equation $x = \frac { 5 } { 3 } \ln ( 3 x + 8 )$.
Use an iterative formula, based on the equation in part (ii), to find the $x$-coordinate of $P$ correct to 2 decimal places.
Use integration, and your answer to part (iii), to find an approximate value of the area of the region $R$.

OCR C3 Q9

17 marks

9
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-04_629_647_262_749} The function f is defined by $\mathrm { f } ( x ) = \sqrt { } ( m x + 7 ) - 4$, where $x \geqslant - \frac { 7 } { m }$ and $m$ is a positive constant. The diagram shows the curve $y = \mathrm { f } ( x )$.

A sequence of transformations maps the curve $y = \sqrt { } x$ to the curve $y = \mathrm { f } ( x )$. Give details of these transformations.
Explain how you can tell that f is a one-one function and find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
It is given that the curves $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ do not meet. Explain how it can be deduced that neither curve meets the line $y = x$, and hence determine the set of possible values of $m$. [5] Jan 2006
1 Show that $\int _ { 2 } ^ { 8 } \frac { 3 } { x } \mathrm {~d} x = \ln 64$. 2 Solve, for $0 ^ { \circ } < \theta < 360 ^ { \circ }$, the equation $\sec ^ { 2 } \theta = 4 \tan \theta - 2$. 3 (a) Differentiate $x ^ { 2 } ( x + 1 ) ^ { 6 }$ with respect to $x$.
(b) Find the gradient of the curve $y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 3 }$ at the point where $x = 1$. 4
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-05_531_737_884_705} The function f is defined by $\mathrm { f } ( x ) = 2 - \sqrt { x }$ for $x \geqslant 0$. The graph of $y = \mathrm { f } ( x )$ is shown above.
State the range of f .
Find the value of $\mathrm { ff } ( 4 )$.
Given that the equation $| \mathrm { f } ( x ) | = k$ has two distinct roots, determine the possible values of the constant $k$. 5
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-05_490_750_1966_701} The diagram shows the curves $y = ( 1 - 2 x ) ^ { 5 }$ and $y = \mathrm { e } ^ { 2 x - 1 } - 1$. The curves meet at the point $\left( \frac { 1 } { 2 } , 0 \right)$. Find the exact area of the region (shaded in the diagram) bounded by the $y$-axis and by part of each curve. 6 (a)
$t$ 0 10 20
$X$ 275 440
The quantity $X$ is increasing exponentially with respect to time $t$. The table above shows values of $X$ for different values of $t$. Find the value of $X$ when $t = 20$.
(b) The quantity $Y$ is decreasing exponentially with respect to time $t$ where $$Y = 80 \mathrm { e } ^ { - 0.02 t } .$$
Find the value of $t$ for which $Y = 20$, giving your answer correct to 2 significant figures.
Find by differentiation the rate at which $Y$ is decreasing when $t = 30$, giving your answer correct to 2 significant figures. 7
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-06_461_737_1123_705} The diagram shows the curve with equation $y = \cos ^ { - 1 } x$.
Sketch the curve with equation $y = 3 \cos ^ { - 1 } ( x - 1 )$, showing the coordinates of the points where the curve meets the axes.
By drawing an appropriate straight line on your sketch in part (i), show that the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$ has exactly one root.
Show by calculation that the root of the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$ lies between 1.8 and 1.9.
The sequence defined by $$x _ { 1 } = 2 , \quad x _ { n + 1 } = 1 + \cos \left( \frac { 1 } { 3 } x _ { n } \right)$$ converges to a number $\alpha$. Find the value of $\alpha$ correct to 2 decimal places and explain why $\alpha$ is the root of the equation $3 \cos ^ { - 1 } ( x - 1 ) = x$. \section*{[Questions 8 and 9 are printed overleaf.]} 8
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-07_790_748_264_699} The diagram shows part of the curve $y = \ln \left( 5 - x ^ { 2 } \right)$ which meets the $x$-axis at the point $P$ with coordinates ( 2,0 ). The tangent to the curve at $P$ meets the $y$-axis at the point $Q$. The region $A$ is bounded by the curve and the lines $x = 0$ and $y = 0$. The region $B$ is bounded by the curve and the lines $P Q$ and $x = 0$.
Find the equation of the tangent to the curve at $P$.
Use Simpson's Rule with four strips to find an approximation to the area of the region $A$, giving your answer correct to 3 significant figures.
Deduce an approximation to the area of the region $B$. 9
By first writing $\sin 3 \theta$ as $\sin ( 2 \theta + \theta )$, show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
Determine the greatest possible value of $$9 \sin \left( \frac { 10 } { 3 } \alpha \right) - 12 \sin ^ { 3 } \left( \frac { 10 } { 3 } \alpha \right)$$ and find the smallest positive value of $\alpha$ (in degrees) for which that greatest value occurs.
Solve, for $0 ^ { \circ } < \beta < 90 ^ { \circ }$, the equation $3 \sin 6 \beta \operatorname { cosec } 2 \beta = 4$. \section*{June 2006} 1 Find the equation of the tangent to the curve $y = \sqrt { 4 x + 1 }$ at the point ( 2,3 ). 2 Solve the inequality $| 2 x - 3 | < | x + 1 |$. 3 The equation $2 x ^ { 3 } + 4 x - 35 = 0$ has one real root.
Show by calculation that this real root lies between 2 and 3 .
Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { 17.5 - 2 x _ { n } }$$ with a suitable starting value, to find the real root of the equation $2 x ^ { 3 } + 4 x - 35 = 0$ correct to 2 decimal places. You should show the result of each iteration. 4 It is given that $y = 5 ^ { x - 1 }$.
Show that $x = 1 + \frac { \ln y } { \ln 5 }$.
Find an expression for $\frac { \mathrm { d } x } { \mathrm {~d} y }$ in terms of $y$.
Hence find the exact value of the gradient of the curve $y = 5 ^ { x - 1 }$ at the point (3, 25). 5
Write down the identity expressing $\sin 2 \theta$ in terms of $\sin \theta$ and $\cos \theta$.
Given that $\sin \alpha = \frac { 1 } { 4 }$ and $\alpha$ is acute, show that $\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }$.
Solve, for $0 ^ { \circ } < \beta < 90 ^ { \circ }$, the equation $5 \sin 2 \beta \sec \beta = 3$. \section*{June 2006} 6
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-09_570_591_264_776} The diagram shows the graph of $y = \mathrm { f } ( x )$, where $$\mathrm { f } ( x ) = 2 - x ^ { 2 } , \quad x \leqslant 0 .$$
Evaluate $\mathrm { ff } ( - 3 )$.
Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
Sketch the graph of $y = \mathrm { f } ^ { - 1 } ( x )$. Indicate the coordinates of the points where the graph meets the axes. 7 (a) Find the exact value of $\int _ { 1 } ^ { 2 } \frac { 2 } { ( 4 x - 1 ) ^ { 2 } } \mathrm {~d} x$.
(b)
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-09_570_761_1676_731} The diagram shows part of the curve $y = \frac { 1 } { x }$. The point $P$ has coordinates $\left( a , \frac { 1 } { a } \right)$ and the point $Q$ has coordinates $\left( 2 a , \frac { 1 } { 2 a } \right)$, where $a$ is a positive constant. The point $R$ is such that $P R$ is parallel to the $x$-axis and $Q R$ is parallel to the $y$-axis. The region shaded in the diagram is bounded by the curve and by the lines $P R$ and $Q R$. Show that the area of this shaded region is $\ln \left( \frac { 1 } { 2 } \mathrm { e } \right)$.
Express $5 \cos x + 12 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
Hence give details of a pair of transformations which transforms the curve $y = \cos x$ to the curve $y = 5 \cos x + 12 \sin x$.
Solve, for $0 ^ { \circ } < x < 360 ^ { \circ }$, the equation $5 \cos x + 12 \sin x = 2$, giving your answers correct to the nearest $0.1 ^ { \circ }$. 9
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-10_565_725_671_712} The diagram shows the curve with equation $y = 2 \ln ( x - 1 )$. The point $P$ has coordinates ( $0 , p$ ). The region $R$, shaded in the diagram, is bounded by the curve and the lines $x = 0 , y = 0$ and $y = p$. The units on the axes are centimetres. The region $R$ is rotated completely about the $\boldsymbol { y }$-axis to form a solid.
Show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the solid is given by $$V = \pi \left( \mathrm { e } ^ { p } + 4 \mathrm { e } ^ { \frac { 1 } { 2 } p } + p - 5 \right) .$$
It is given that the point $P$ is moving in the positive direction along the $y$-axis at a constant rate of $0.2 \mathrm {~cm} \mathrm {~min} ^ { - 1 }$. Find the rate at which the volume of the solid is increasing at the instant when $p = 4$, giving your answer correct to 2 significant figures. 1 Find the equation of the tangent to the curve $y = \frac { 2 x + 1 } { 3 x - 1 }$ at the point $\left( 1 , \frac { 3 } { 2 } \right)$, giving your answer in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers. 2 It is given that $\theta$ is the acute angle such that $\sin \theta = \frac { 12 } { 13 }$. Find the exact value of
$\cot \theta$,
$\cos 2 \theta$. 3 (a) It is given that $a$ and $b$ are positive constants. By sketching graphs of $$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$ on the same diagram, show that the equation $$x ^ { 5 } + b x - a = 0$$ has exactly one real root.
(b) Use the iterative formula $x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }$, with a suitable starting value, to find the real root of the equation $x ^ { 5 } + 2 x - 53 = 0$. Show the result of each iteration, and give the root correct to 3 decimal places. 4
Given that $x = ( 4 t + 9 ) ^ { \frac { 1 } { 2 } }$ and $y = 6 \mathrm { e } ^ { \frac { 1 } { 2 } x + 1 }$, find expressions for $\frac { \mathrm { d } x } { \mathrm {~d} t }$ and $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Hence find the value of $\frac { \mathrm { d } y } { \mathrm {~d} t }$ when $t = 4$, giving your answer correct to 3 significant figures. 5
Express $4 \cos \theta - \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
Hence solve the equation $4 \cos \theta - \sin \theta = 2$, giving all solutions for which $- 180 ^ { \circ } < \theta < 180 ^ { \circ }$. \section*{Jan 2007} 6
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-12_485_960_262_589} The diagram shows the curve with equation $y = \frac { 1 } { \sqrt { 3 x + 2 } }$. The shaded region is bounded by the curve and the lines $x = 0 , x = 2$ and $y = 0$.
Find the exact area of the shaded region.
The shaded region is rotated completely about the $x$-axis. Find the exact volume of the solid formed, simplifying your answer. 7 The curve $y = \ln x$ is transformed to the curve $y = \ln \left( \frac { 1 } { 2 } x - a \right)$ by means of a translation followed by a stretch. It is given that $a$ is a positive constant.
Give full details of the translation and stretch involved.
Sketch the graph of $y = \ln \left( \frac { 1 } { 2 } x - a \right)$.
Sketch, on another diagram, the graph of $y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|$.
State, in terms of $a$, the set of values of $x$ for which $\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)$. 8
\includegraphics[max width=\textwidth, alt={}, center]{ceca0210-939e-4797-8ee1-8bf663534fcd-13_528_1435_267_354} The diagram shows the curve with equation $y = x ^ { 8 } \mathrm { e } ^ { - x ^ { 2 } }$. The curve has maximum points at $P$ and $Q$. The shaded region $A$ is bounded by the curve, the line $y = 0$ and the line through $Q$ parallel to the $y$-axis. The shaded region $B$ is bounded by the curve and the line $P Q$.
Show by differentiation that the $x$-coordinate of $Q$ is 2 .
Use Simpson's rule with 4 strips to find an approximation to the area of region $A$. Give your answer correct to 3 decimal places.
Deduce an approximation to the area of region $B$. 9 Functions $f$ and $g$ are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = 2 \sin x & \text { for } - \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi ,
\mathrm {~g} ( x ) = 4 - 2 x ^ { 2 } & \text { for } x \in \mathbb { R } . \end{array}$$
State the range of f and the range of g .
Show that $\operatorname { gf } ( 0.5 ) = 2.16$, correct to 3 significant figures, and explain why $\mathrm { fg } ( 0.5 )$ is not defined.
Find the set of values of $x$ for which $\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )$ is not defined.

OCR C3 2010 June Q1

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

$y = x ^ { 3 } \mathrm { e } ^ { 2 x }$,
$y = \ln \left( 3 + 2 x ^ { 2 } \right)$,
$y = \frac { x } { 2 x + 1 }$.

OCR C3 2010 June Q2

2 The transformations R, S and T are defined as follows.
R : reflection in the $x$-axis
S : stretch in the $x$-direction with scale factor 3
$\mathrm { T } : \quad$ translation in the positive $x$-direction by 4 units

The curve $y = \ln x$ is transformed by R followed by T . Find the equation of the resulting curve.
Find, in terms of S and T, a sequence of transformations that transforms the curve $y = x ^ { 3 }$ to the curve $y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }$. You should make clear the order of the transformations.

OCR C3 2010 June Q3

3

Express the equation $\operatorname { cosec } \theta ( 3 \cos 2 \theta + 7 ) + 11 = 0$ in the form $a \sin ^ { 2 } \theta + b \sin \theta + c = 0$, where $a , b$ and $c$ are constants.
Hence solve, for $- 180 ^ { \circ } < \theta < 180 ^ { \circ }$, the equation $\operatorname { cosec } \theta ( 3 \cos 2 \theta + 7 ) + 11 = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{cd1bde44-ab7e-45e6-ac22-346145eba3a0-2_648_951_1530_598} The diagram shows part of the curve $y = \frac { k } { x }$, where $k$ is a positive constant. The points $A$ and $B$ on the curve have $x$-coordinates 2 and 6 respectively. Lines through $A$ and $B$ parallel to the axes as shown meet at the point $C$. The region $R$ is bounded by the curve and the lines $x = 2 , x = 6$ and $y = 0$. The region $S$ is bounded by the curve and the lines $A C$ and $B C$. It is given that the area of the region $R$ is $\ln 81$.
Show that $k = 4$.
Find the exact volume of the solid produced when the region $S$ is rotated completely about the $x$-axis.
Solve the inequality $| 2 x + 1 | \leqslant | x - 3 |$.
Given that $x$ satisfies the inequality $| 2 x + 1 | \leqslant | x - 3 |$, find the greatest possible value of $| x + 2 |$.
Show by calculation that the equation $$\tan ^ { 2 } x - x - 2 = 0$$ where $x$ is measured in radians, has a root between 1.0 and 1.1.
Use the iteration formula $x _ { n + 1 } = \tan ^ { - 1 } \sqrt { 2 + x _ { n } }$ with a suitable starting value to find this root correct to 5 decimal places. You should show the outcome of each step of the process.
Deduce a root of the equation $$\sec ^ { 2 } 2 x - 2 x - 3 = 0$$
\includegraphics[max width=\textwidth, alt={}]{cd1bde44-ab7e-45e6-ac22-346145eba3a0-3_771_1087_1128_529}
The diagram shows the curve with equation $y = ( 3 x - 1 ) ^ { 4 }$. The point $P$ on the curve has coordinates $( 1,16 )$ and the tangent to the curve at $P$ meets the $x$-axis at the point $Q$. The shaded region is bounded by $P Q$, the $x$-axis and that part of the curve for which $\frac { 1 } { 3 } \leqslant x \leqslant 1$. Find the exact area of this shaded region.
Express $3 \cos x + 3 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
The expression $\mathrm { T } ( x )$ is defined by $\mathrm { T } ( x ) = \frac { 8 } { 3 \cos x + 3 \sin x }$.
(a) Determine a value of $x$ for which $\mathrm { T } ( x )$ is not defined.
(b) Find the smallest positive value of $x$ satisfying $\mathrm { T } ( 3 x ) = \frac { 8 } { 9 } \sqrt { 6 }$, giving your answer in an exact form. \section*{[Question 9 is printed overleaf.]}

OCR C3 2010 June Q9

9 The functions f and g are defined for all real values of $x$ by $$\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x \quad \text { and } \quad \mathrm { g } ( x ) = a x + b$$ where $a$ and $b$ are non-zero constants.

Find the range of f .
Explain why the function $f$ has no inverse.
Given that $\mathrm { g } ^ { - 1 } ( x ) = \mathrm { g } ( x )$ for all values of $x$, show that $a = - 1$.
Given further that $\operatorname { gf } ( x ) < 5$ for all values of $x$, find the set of possible values of $b$.

OCR C4 Q3

3 The line $L _ { 1 }$ passes through the points $( 2 , - 3,1 )$ and $( - 1 , - 2 , - 4 )$. The line $L _ { 2 }$ passes through the point $( 3,2 , - 9 )$ and is parallel to the vector $4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }$.

Find an equation for $L _ { 1 }$ in the form $\mathbf { r } = \mathbf { a } + t \mathbf { b }$.
Prove that $L _ { 1 }$ and $L _ { 2 }$ are skew.

OCR C4 Q4

4

Show that the substitution $x = \tan \theta$ transforms $\int \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$ to $\int \cos ^ { 2 } \theta \mathrm {~d} \theta$.
Hence find the exact value of $\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x$.
$5 A B C D$ is a parallelogram. The position vectors of $A , B$ and $C$ are given respectively by $$\mathbf { a } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { c } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } .$$
Find the position vector of $D$.
Determine, to the nearest degree, the angle $A B C$. 6 The equation of a curve is $x y ^ { 2 } = 2 x + 3 y$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - y ^ { 2 } } { 2 x y - 3 }$.
Show that the curve has no tangents which are parallel to the $y$-axis. 7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent at the point $P \left( 4 , - \frac { 1 } { 2 } \right)$ is $$x - 16 y = 12$$
Find the value of the parameter at the point where the tangent at $P$ meets the curve again. June 2005
8
Given that $\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { 1 + x } + \frac { B } { 2 + x } + \frac { C } { ( 2 + x ) ^ { 2 } }$, find $A , B$ and $C$.
Hence or otherwise expand $\frac { 3 x + 4 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
State the set of values of $x$ for which the expansion in part (ii) is valid. 9 Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of $20 ^ { \circ } \mathrm { C }$. At time $t$ minutes later, the temperature of the liquid is $\theta ^ { \circ } \mathrm { C }$.
Explain how the information above leads to the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$ where $k$ is a positive constant.
The liquid is initially at a temperature of $100 ^ { \circ } \mathrm { C }$. It takes 5 minutes for the liquid to cool from $100 ^ { \circ } \mathrm { C }$ to $68 ^ { \circ } \mathrm { C }$. Show that $$\theta = 20 + 80 \mathrm { e } ^ { - \left( \frac { 1 } { 5 } \ln \frac { 5 } { 3 } \right) t }$$
Calculate how much longer it takes for the liquid to cool by a further $32 ^ { \circ } \mathrm { C }$. 1 Simplify $\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }$. 2 Given that $\sin y = x y + x ^ { 2 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. 3
Find the quotient and the remainder when $3 x ^ { 3 } - 2 x ^ { 2 } + x + 7$ is divided by $x ^ { 2 } - 2 x + 5$.
Hence, or otherwise, determine the values of the constants $a$ and $b$ such that, when $3 x ^ { 3 } - 2 x ^ { 2 } + a x + b$ is divided by $x ^ { 2 } - 2 x + 5$, there is no remainder. 4
Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.
Hence find $\int x \tan ^ { 2 } x \mathrm {~d} x$.

OCR C4 Q7

7 A curve is given parametrically by the equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { t }$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent at the point $P \left( 4 , - \frac { 1 } { 2 } \right)$ is $$x - 16 y = 12$$
Find the value of the parameter at the point where the tangent at $P$ meets the curve again. June 2005

OCR C4 Q9

9 Newton's law of cooling states that the rate at which the temperature of an object is falling at any instant is proportional to the difference between the temperature of the object and the temperature of its surroundings at that instant. A container of hot liquid is placed in a room which has a constant temperature of $20 ^ { \circ } \mathrm { C }$. At time $t$ minutes later, the temperature of the liquid is $\theta ^ { \circ } \mathrm { C }$.

Explain how the information above leads to the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$ where $k$ is a positive constant.
The liquid is initially at a temperature of $100 ^ { \circ } \mathrm { C }$. It takes 5 minutes for the liquid to cool from $100 ^ { \circ } \mathrm { C }$ to $68 ^ { \circ } \mathrm { C }$. Show that $$\theta = 20 + 80 \mathrm { e } ^ { - \left( \frac { 1 } { 5 } \ln \frac { 5 } { 3 } \right) t }$$
Calculate how much longer it takes for the liquid to cool by a further $32 ^ { \circ } \mathrm { C }$. 1 Simplify $\frac { x ^ { 3 } - 3 x ^ { 2 } } { x ^ { 2 } - 9 }$. 2 Given that $\sin y = x y + x ^ { 2 }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$ and $y$. 3
Find the quotient and the remainder when $3 x ^ { 3 } - 2 x ^ { 2 } + x + 7$ is divided by $x ^ { 2 } - 2 x + 5$.
Hence, or otherwise, determine the values of the constants $a$ and $b$ such that, when $3 x ^ { 3 } - 2 x ^ { 2 } + a x + b$ is divided by $x ^ { 2 } - 2 x + 5$, there is no remainder. 4
Use integration by parts to find $\int x \sec ^ { 2 } x \mathrm {~d} x$.
Hence find $\int x \tan ^ { 2 } x \mathrm {~d} x$. 5 A curve is given parametrically by the equations $x = t ^ { 2 } , y = 2 t$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$, giving your answer in its simplest form.
Show that the equation of the tangent to the curve at $\left( p ^ { 2 } , 2 p \right)$ is $$p y = x + p ^ { 2 } .$$
Find the coordinates of the point where the tangent at $( 9,6 )$ meets the tangent at $( 25 , - 10 )$. 6
Show that the substitution $x = \sin ^ { 2 } \theta$ transforms $\int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x$ to $\int 2 \sin ^ { 2 } \theta \mathrm {~d} \theta$.
Hence find $\int _ { 0 } ^ { 1 } \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x$. 7 The expression $\frac { 11 + 8 x } { ( 2 - x ) ( 1 + x ) ^ { 2 } }$ is denoted by $\mathrm { f } ( x )$.
Express $\mathrm { f } ( x )$ in the form $\frac { A } { 2 - x } + \frac { B } { 1 + x } + \frac { C } { ( 1 + x ) ^ { 2 } }$, where $A , B$ and $C$ are constants.
Given that $| x | < 1$, find the first 3 terms in the expansion of $\mathrm { f } ( x )$ in ascending powers of $x$. 8
Solve the differential equation $$\frac { d y } { d x } = \frac { 2 - x } { y - 3 }$$ giving the particular solution that satisfies the condition $y = 4$ when $x = 5$.
Show that this particular solution can be expressed in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where the values of the constants $a , b$ and $k$ are to be stated.
Hence sketch the graph of the particular solution, indicating clearly its main features. 9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4
2
- 6 \end{array} \right) + t \left( \begin{array} { r } - 8
1
- 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2
a
- 2 \end{array} \right) + s \left( \begin{array} { r } - 9
2
- 5 \end{array} \right) ,$$ where $a$ is a constant.
Calculate the acute angle between the lines.
Given that these two lines intersect, find $a$ and the point of intersection. \section*{June 2006} 1 Find the gradient of the curve $4 x ^ { 2 } + 2 x y + y ^ { 2 } = 12$ at the point $( 1,2 )$. 2
Expand $( 1 - 3 x ) ^ { - 2 }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$.
Find the coefficient of $x ^ { 2 }$ in the expansion of $\frac { ( 1 + 2 x ) ^ { 2 } } { ( 1 - 3 x ) ^ { 2 } }$ in ascending powers of $x$. 3
Express $\frac { 3 - 2 x } { x ( 3 - x ) }$ in partial fractions.
Show that $\int _ { 1 } ^ { 2 } \frac { 3 - 2 x } { x ( 3 - x ) } \mathrm { d } x = 0$.
What does the result of part (ii) indicate about the graph of $y = \frac { 3 - 2 x } { x ( 3 - x ) }$ between $x = 1$ and $x = 2$ ? 4 The position vectors of three points $A , B$ and $C$ relative to an origin $O$ are given respectively by and $$\begin{aligned} & \overrightarrow { O A } = 7 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k } ,
& \overrightarrow { O B } = 4 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }
& \overrightarrow { O C } = 5 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } . \end{aligned}$$
Find the angle between $A B$ and $A C$.
Find the area of triangle $A B C$. 5 A forest is burning so that, $t$ hours after the start of the fire, the area burnt is $A$ hectares. It is given that, at any instant, the rate at which this area is increasing is proportional to $A ^ { 2 }$.
Write down a differential equation which models this situation.
After 1 hour, 1000 hectares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt. 6
Show that the substitution $u = \mathrm { e } ^ { x } + 1$ transforms $\int \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x$ to $\int \frac { u - 1 } { u } \mathrm {~d} u$.
Hence show that $\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { x } + 1 } \mathrm {~d} x = \mathrm { e } - 1 - \ln \left( \frac { \mathrm { e } + 1 } { 2 } \right)$. \section*{June 2006} 7 Two lines have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } + \lambda ( 3 \mathbf { i } + \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) ,$$ where $a$ is a constant.
Given that the lines are skew, find the value that $a$ cannot take.
Given instead that the lines intersect, find the point of intersection. 8
Show that $\int \cos ^ { 2 } 6 x \mathrm {~d} x = \frac { 1 } { 2 } x + \frac { 1 } { 24 } \sin 12 x + c$.
Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 12 } \pi } x \cos ^ { 2 } 6 x \mathrm {~d} x$. 9 A curve is given parametrically by the equations $$x = 4 \cos t , \quad y = 3 \sin t$$ where $0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$.
Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
Show that the equation of the tangent at the point $P$, where $t = p$, is $$3 x \cos p + 4 y \sin p = 12$$
The tangent at $P$ meets the $x$-axis at $R$ and the $y$-axis at $S$. $O$ is the origin. Show that the area of triangle $O R S$ is $\frac { 12 } { \sin 2 p }$.
Write down the least possible value of the area of triangle $O R S$, and give the corresponding value of $p$. Jan 2007
1 It is given that $$f ( x ) = \frac { x ^ { 2 } + 2 x - 24 } { x ^ { 2 } - 4 x } \quad \text { for } x \neq 0 , x \neq 4$$ Express $\mathrm { f } ( x )$ in its simplest form. 2 Find the exact value of $\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x$. 3 The points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to an origin $O$, where $\mathbf { a } = 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$ and $\mathbf { b } = - 7 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k }$.
Find the length of $A B$.
Use a scalar product to find angle $O A B$. 4 Use the substitution $u = 2 x - 5$ to show that $\int _ { \frac { 5 } { 2 } } ^ { 3 } ( 4 x - 8 ) ( 2 x - 5 ) ^ { 7 } \mathrm {~d} x = \frac { 17 } { 72 }$.
Expand $( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 - 3 \left( x + x ^ { 3 } \right) \right) ^ { - \frac { 1 } { 3 } }$. 6
Express $\frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } }$ in the form $\frac { A } { x - 3 } + \frac { B } { ( x - 3 ) ^ { 2 } }$, where $A$ and $B$ are constants.
Hence find the exact value of $\int _ { 4 } ^ { 10 } \frac { 2 x + 1 } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x$, giving your answer in the form $a + b \ln c$, where $a , b$ and $c$ are integers. 7 The equation of a curve is $2 x ^ { 2 } + x y + y ^ { 2 } = 14$. Show that there are two stationary points on the curve and find their coordinates. 8 The parametric equations of a curve are $x = 2 t ^ { 2 } , y = 4 t$. Two points on the curve are $P \left( 2 p ^ { 2 } , 4 p \right)$ and $Q \left( 2 q ^ { 2 } , 4 q \right)$.
Show that the gradient of the normal to the curve at $P$ is $- p$.
Show that the gradient of the chord joining the points $P$ and $Q$ is $\frac { 2 } { p + q }$.
The chord $P Q$ is the normal to the curve at $P$. Show that $p ^ { 2 } + p q + 2 = 0$.
The normal at the point $R ( 8,8 )$ meets the curve again at $S$. The normal at $S$ meets the curve again at $T$. Find the coordinates of $T$. 9
Find the general solution of the differential equation $$\frac { \sec ^ { 2 } y } { \cos ^ { 2 } ( 2 x ) } \frac { d y } { d x } = 2$$
For the particular solution in which $y = \frac { 1 } { 4 } \pi$ when $x = 0$, find the value of $y$ when $x = \frac { 1 } { 6 } \pi$.

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