Edexcel Paper 2 (Paper 2) Specimen

1. \begin{figure}[h] \end{figure} Figure 1 shows a circle with centre \(O\). The points \(A\) and \(B\) lie on the circumference of the circle. The area of the major sector, shown shaded in Figure 1, is \(135 \mathrm {~cm} ^ { 2 }\). The reflex angle \(A O B\) is 4.8 radians. Find the exact length, in cm, of the minor arc \(A B\), giving your answer in the form \(a \pi + b\), where \(a\) and \(b\) are integers to be found.
(4)

1. (a) Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that

$$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a).
Adele's working is shown below. Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\)
Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\)
(b) (i) Identify the mistake made by Adele in her working.
(ii) Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\)
(2)

1. A cup of hot tea was placed on a table. At time \(t\) minutes after the cup was placed on the table, the temperature of the tea in the cup, \(\theta ^ { \circ } \mathrm { C }\), is modelled by the equation

$$\theta = 25 + A \mathrm { e } ^ { - 0.03 t }$$ where \(A\) is a constant. The temperature of the tea was \(75 ^ { \circ } \mathrm { C }\) when the cup was placed on the table.

1. Find a complete equation for the model.
2. Use the model to find the time taken for the tea to cool from \(75 ^ { \circ } \mathrm { C }\) to \(60 ^ { \circ } \mathrm { C }\), giving your answer in minutes to one decimal place. Two hours after the cup was placed on the table, the temperature of the tea was measured as \(20.3 ^ { \circ } \mathrm { C }\). Using this information,
3. evaluate the model, explaining your reasoning.

1. (a) Sketch the graph with equation

$$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
(b) Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$ (c) Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.

1. The line \(l\) has equation

$$3 x - 2 y = k$$ where \(k\) is a real constant.
Given that the line \(l\) intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for \(k\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 2.

1. Find the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\).
2. Show that the \(x\) coordinate of \(Q\) satisfies $$x = \frac { 8 } { 1 + \ln x }$$
3. Show that the \(x\) coordinate of \(Q\) lies between 3.5 and 3.6
4. Use the iterative formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$ with \(x _ { 1 } = 3.5\) to
   1. find the value of \(x _ { 5 }\) to 4 decimal places,
   2. find the \(x\) coordinate of \(Q\) accurate to 2 decimal places.

1. A bacterial culture has area \(p \mathrm {~mm} ^ { 2 }\) at time \(t\) hours after the culture was placed onto a circular dish.

A scientist states that at time \(t\) hours, the rate of increase of the area of the culture can be modelled as being proportional to the area of the culture.

1. Show that the scientist's model for \(p\) leads to the equation $$p = a \mathrm { e } ^ { k t }$$ where \(a\) and \(k\) are constants. The scientist measures the values for \(p\) at regular intervals during the first 24 hours after the culture was placed onto the dish. She plots a graph of \(\ln p\) against \(t\) and finds that the points on the graph lie close to a straight line with gradient 0.14 and vertical intercept 3.95
2. Estimate, to 2 significant figures, the value of \(a\) and the value of \(k\).
3. Hence show that the model for \(p\) can be rewritten as $$p = a b ^ { t }$$ stating, to 3 significant figures, the value of the constant \(b\). With reference to this model,
4. 1. interpret the value of the constant \(a\),
   2. interpret the value of the constant \(b\).
5. State a long term limitation of the model for \(p\).

8. \begin{figure}[h] \end{figure} A bowl is modelled as a hemispherical shell as shown in Figure 3.
Initially the bowl is empty and water begins to flow into the bowl. When the depth of the water is \(h \mathrm {~cm}\), the volume of water, \(V \mathrm {~cm} ^ { 3 }\), according to the model is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 75 - h ) , \quad 0 \leqslant h \leqslant 24$$ The flow of water into the bowl is at a constant rate of \(160 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(0 \leqslant h \leqslant 12\)

1. Find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 10\) Given that the flow of water into the bowl is increased to a constant rate of \(300 \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) for \(12 < h \leqslant 24\)
2. find the rate of change of the depth of the water, in \(\mathrm { cms } ^ { - 1 }\), when \(h = 20\)

1. A circle with centre \(A ( 3 , - 1 )\) passes through the point \(P ( - 9,8 )\) and the point \(Q ( 15 , - 10 )\)
   1. Show that \(P Q\) is a diameter of the circle.
   2. Find an equation for the circle.
   A point \(R\) also lies on the circle. Given that the length of the chord \(P R\) is 20 units,
2. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
3. Find the size of angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( t + 2 ) , \quad y = \frac { 1 } { t + 1 } , \quad t > - \frac { 2 } { 3 }$$

1. State the domain of values of \(x\) for the curve \(C\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = \ln 2\), the \(x\)-axis and the line with equation \(x = \ln 4\)
2. Use calculus to show that the area of \(R\) is \(\ln \left( \frac { 3 } { 2 } \right)\).

1. The second, third and fourth terms of an arithmetic sequence are \(2 k , 5 k - 10\) and \(7 k - 14\) respectively, where \(k\) is a constant.

Show that the sum of the first \(n\) terms of the sequence is a square number.

1. A curve \(C\) is given by the equation

$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

13. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. (i) Kayden claims that

$$3 ^ { x } \geqslant 2 ^ { x }$$ Determine whether Kayden's claim is always true, sometimes true or never true, justifying your answer.
(ii) Prove that \(\sqrt { 3 }\) is an irrational number.