Edexcel P2 (Pure Mathematics 2) 2022 October

1. Given that \(a , b\) and \(c\) are integers greater than 0 such that

• \(c = b + 2\)
• \(a + b + c = 10\)

Prove, by exhaustion, that the product of \(a , b\) and \(c\) is always even.
You may use the table below to illustrate your answer. You may not need to use all rows of this table.

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = ( 2 - k x ) ^ { 5 }$$ and \(k\) is a constant.
Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)

1. show that \(k = \frac { 2 } { 5 }\)
2. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$ giving each term in simplest form. Using the solution to part (b) and making your method clear,
3. find the gradient of \(C\) at the point where \(x = 0\)

1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$a _ { n } = \cos ^ { 2 } \left( \frac { \mathrm { n } \pi } { 3 } \right)$$ Find the exact values of

1. 1. \(a _ { 1 }\)
   2. \(a _ { 2 }\)
   3. \(a _ { 3 }\)
2. Hence find the exact value of 50 $$n + \cos ^ { 2 } \frac { n \pi } { 3 }$$ You must make your method clear.

1. The weight of a baby mammal is monitored over a 16 -month period.

The weight of the mammal, \(w \mathrm {~kg}\), is given by $$w = \log _ { a } ( t + 5 ) - \log _ { a } 4 \quad 2 \leqslant t \leqslant 18$$ where \(t\) is the age of the mammal in months and \(a\) is a constant.
Given that the weight of the mammal was 10 kg when \(t = 3\)

1. show that \(a = 1.072\) correct to 3 decimal places. Using \(a = 1.072\)
2. find an equation for \(t\) in terms of \(w\)
3. find the value of \(t\) when \(w = 15\), giving your answer to 3 significant figures.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$ can be written as $$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) $$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$

1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).

A table of values of \(x\) and \(y\) for \(y = \mathrm { f } ( x )\) is shown below, with the \(y\) values rounded to 4 decimal places where appropriate.

1. Use the trapezium rule with all the values of \(y\) in the table to find an approximation for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$ giving your answer to 3 decimal places. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_629_592_1105_402} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-16_540_456_1194_1192} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The region \(R\), shown shaded in Figure 1, is bounded by
   • the curve \(C _ { 1 }\)
   • the curve \(C _ { 2 }\) with equation \(y = 2 - \frac { 1 } { 4 } x ^ { 2 }\)
   • the line with equation \(x = 2\)
   • the \(y\)-axis
   The region \(R\) forms part of the design for a logo shown in Figure 2.
   The design consists of the shaded region \(R\) inside a rectangle of width 2 and height 3 Using calculus and the answer to part (a),
2. calculate an estimate for the percentage of the logo which is shaded.

1. The curve \(C\) has equation

$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$

1. Write the equation of \(C\) in the form $$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$ where \(a , b , c\) and \(d\) are fully simplified constants. The curve \(C\) has three turning points.
   Using calculus,
2. show that the \(x\) coordinates of the three turning points satisfy the equation $$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$ Given that the \(x\) coordinate of one of the turning points is 1
3. find, using algebra, the exact \(x\) coordinates of the other two turning points.
   (Solutions based entirely on calculator technology are not acceptable.)

1. A geometric sequence has first term \(a\) and common ratio \(r\)

Given that \(S _ { \infty } = 3 a\)

1. show that \(r = \frac { 2 } { 3 }\) Given also that $$u _ { 2 } - u _ { 4 } = 16$$ where \(u _ { k }\) is the \(k ^ { \text {th } }\) term of this sequence,
2. find the value of \(S _ { 10 }\) giving your answer to one decimal place.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
• the circle \(C _ { 2 }\) with centre \(T\)

The point \(T\) is the minimum turning point of \(C _ { 1 }\)
Using Figure 3 and calculus,

1. find the coordinates of \(T\) The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
2. Find an equation of the circle \(C _ { 2 }\) The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
3. Show that an equation of \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$ The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
4. Find the exact area of \(R\).

1. Given \(a = \log _ { 2 } 3\)
   1. write, in simplest form, in terms of \(a\),
      (a) \(\log _ { 2 } 9\)
      (b) \(\log _ { 2 } \left( \frac { \sqrt { 3 } } { 16 } \right)\)
   2. Solve
   $$3 ^ { x } \times 2 ^ { x + 4 } = 6$$ giving your answer, in simplest form, in terms of \(a\).