OCR PURE

1 In triangle \(A B C , A B = 20 \mathrm {~cm}\) and angle \(B = 45 ^ { \circ }\).

1. Given that \(A C = 16 \mathrm {~cm}\), find the two possible values for angle \(C\), correct to 1 decimal place.
2. Given instead that the area of the triangle is \(75 \sqrt { 2 } \mathrm {~cm} ^ { 2 }\), find \(B C\).

2

1. The curve \(y = \frac { 2 } { 3 + x }\) is translated by four units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
2. Describe fully the single transformation that transforms the curve \(y = \frac { 2 } { 3 + x }\) to \(y = \frac { 5 } { 3 + x }\).

3 In each of the following cases choose one of the statements $$P \Rightarrow Q \quad P \Leftarrow Q \quad P \Leftrightarrow Q$$ to describe the relationship between \(P\) and \(Q\).

1. \(P : y = 3 x ^ { 5 } - 4 x ^ { 2 } + 12 x\)
   \(Q : \frac { \mathrm { d } y } { \mathrm {~d} x } = 15 x ^ { 4 } - 8 x + 12\)
2. \(\quad P : x ^ { 5 } - 32 = 0\) where \(x\) is real
   \(Q : x = 2\)
3. \(\quad P : \ln y < 0\)
   \(Q : y < 1\)

4

1. Express \(4 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the number of real roots of the equation \(4 x ^ { 2 } - 12 x + 11 = 0\).
3. Explain fully how the value of \(r\) is related to the number of real roots of the equation \(p ( x + q ) ^ { 2 } + r = 0\) where \(p , q\) and \(r\) are real constants and \(p > 0\).

5 In this question you must show detailed reasoning.
The line \(x + 5 y = k\) is a tangent to the curve \(x ^ { 2 } - 4 y = 10\). Find the value of the constant \(k\).

6 A pan of water is heated until it reaches \(100 ^ { \circ } \mathrm { C }\). Once the water reaches \(100 ^ { \circ } \mathrm { C }\), the heat is switched off and the temperature \(T ^ { \circ } \mathrm { C }\) of the water decreases. The temperature of the water is modelled by the equation $$T = 25 + a \mathrm { e } ^ { - k t }$$ where \(t\) denotes the time, in minutes, after the heat is switched off and \(a\) and \(k\) are positive constants.

1. Write down the value of \(a\).
2. Explain what the value of 25 represents in the equation \(T = 25 + a \mathrm { e } ^ { - k t }\). When the heat is switched off, the initial rate of decrease of the temperature of the water is \(15 ^ { \circ } \mathrm { C }\) per minute.
3. Calculate the value of \(k\).
4. Find the time taken for the temperature of the water to drop from \(100 ^ { \circ } \mathrm { C }\) to \(45 ^ { \circ } \mathrm { C }\).
5. A second pan of water is heated, but the heat is turned off when the water is at a temperature of less than \(100 ^ { \circ } \mathrm { C }\). Suggest how the equation for the temperature as the water cools would be modified by this.

7

1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
2. Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.

8 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }\). The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 8\) and \(x = a\), where \(a > 8\).
\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of \(a\).

9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).

10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}

1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.

11
\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-7_127_1147_260_459} A particle \(P\) is moving along a straight line with constant acceleration. Initially the particle is at \(O\). After 9 s , \(P\) is at a point \(A\), where \(O A = 18 \mathrm {~m}\) (see diagram) and the velocity of \(P\) at \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { O A }\).

1. (a) Show that the initial speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
   (b) Find the acceleration of \(P\).
   \(B\) is a point on the line such that \(O B = 10 \mathrm {~m}\), as shown in the diagram.
2. Show that \(P\) is never at point \(B\). A second particle \(Q\) moves along the same straight line, but has variable acceleration. Initially \(Q\) is at \(O\), and the displacement of \(Q\) from \(O\) at time \(t\) seconds is given by $$x = a t ^ { 3 } + b t ^ { 2 } + c t$$ where \(a\), \(b\) and \(c\) are constants.
   It is given that
   • the velocity and acceleration of \(Q\) at the point \(O\) are the same as those of \(P\) at \(O\),
   • \(\quad Q\) reaches the point \(A\) when \(t = 6\).
   • Find the velocity of \(Q\) at \(A\).
   \section*{OCR} \section*{Oxford Cambridge and RSA}