Edexcel P2 (Pure Mathematics 2) 2022 January

1. The table below shows corresponding values of \(x\) and \(y\) for

$$y = 2 ^ { 5 - \sqrt { x } }$$ The values of \(y\) are given to 3 decimal places. Using the trapezium rule with all the values of \(y\) in the given table,

1. obtain an estimate for $$\int _ { 5 } ^ { 7 } 2 ^ { 5 - \sqrt { x } } \mathrm {~d} x$$ giving your answer to 2 decimal places.
2. Using your answer to part (a) and making your method clear, estimate
   1. \(\quad \int _ { 5 } ^ { 7 } 2 ^ { 6 - \sqrt { x } } \mathrm {~d} x\)
   2. \(\int _ { 5 } ^ { 7 } \left( 3 + 2 ^ { 5 - \sqrt { x } } \right) \mathrm { d } x\)

2. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in simplest form.
2. Hence find the coordinates of the stationary point of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence determine the nature of the stationary point of \(C\).

3. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ where \(k\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = ( 5 - 3 x ) \left( 2 - \frac { k x } { 4 } \right) ^ { 8 }$$ In the expansion of \(\mathrm { f } ( x )\), the constant term is 3 times the coefficient of \(x\).
(b) Find the value of \(k\).
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4. Using the laws of logarithms, solve $$\log _ { 3 } ( 32 - 12 x ) = 2 \log _ { 3 } ( 1 - x ) + 3$$

5. $$f ( x ) = 3 x ^ { 3 } + A x ^ { 2 } + B x - 10$$ where \(A\) and \(B\) are integers.
Given that

• when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is \(k\)
• when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is \(- 10 k\)
• \(k\) is a constant
  1. show that

$$11 A + 9 B = 83$$ Given also that \(( 3 x - 2 )\) is a factor of \(\mathrm { f } ( x )\),

find the value of \(A\) and the value of \(B\).

Hence find the quadratic expression \(\mathrm { g } ( x )\) such that $$f ( x ) = ( 3 x - 2 ) g ( x )$$

6. \begin{figure}[h] \end{figure} The points \(P ( 23,14 ) , Q ( 15 , - 30 )\) and \(R ( - 7 , - 26 )\) lie on the circle \(C\), as shown in Figure 1.

1. Show that angle \(P Q R = 90 ^ { \circ }\)
2. Hence, or otherwise, find
   1. the centre of \(C\),
   2. the radius of \(C\). Given that the point \(S\) lies on \(C\) such that the distance \(Q S\) is greatest,
3. find an equation of the tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation $$3 \sin \left( 2 x - 15 ^ { \circ } \right) = \cos \left( 2 x - 15 ^ { \circ } \right)$$ giving your answers to one decimal place.
2. Solve, for \(0 < \theta < 2 \pi\), the equation $$4 \sin ^ { 2 } \theta + 8 \cos \theta = 3$$ giving your answers to 3 significant figures.

8. A metal post is repeatedly hit in order to drive it into the ground. Given that

• on the 1st hit, the post is driven 100 mm into the ground
• on the 2nd hit, the post is driven an additional 98 mm into the ground
• on the 3rd hit, the post is driven an additional 96 mm into the ground
• the additional distances the post travels on each subsequent hit form an arithmetic sequence
  1. show that the post is driven an additional 62 mm into the ground with the 20th hit.
  2. Find the total distance that the post has been driven into the ground after 20 hits.

Given that for each subsequent hit after the 20th hit

• the additional distances the post travels form a geometric sequence with common ratio \(r\)
• on the 22 nd hit, the post is driven an additional 60 mm into the ground
• find the value of \(r\), giving your answer to 3 decimal places.

After a total of \(N\) hits, the post will have been driven more than 3 m into the ground.

Find, showing all steps in your working, the smallest possible value of \(N\).

9. \begin{figure}[h] \end{figure} Figure 2 shows

• the curve \(C\) with equation \(y = x - x ^ { 2 }\)
• the line \(l\) with equation \(y = m x\), where \(m\) is a constant and \(0 < m < 1\)

The line and the curve intersect at the origin \(O\) and at the point \(P\).

1. Find, in terms of \(m\), the coordinates of \(P\). The region \(R _ { 1 }\), shown shaded in Figure 2, is bounded by \(C\) and \(l\).
2. Show that the area of \(R _ { 1 }\) is $$\frac { ( 1 - m ) ^ { 3 } } { 6 }$$ The region \(R _ { 2 }\), also shown shaded in Figure 2, is bounded by \(C\), the \(x\)-axis and \(l\). Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
3. find the exact value of \(m\).
   \includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_108_76_2613_1875}
   \includegraphics[max width=\textwidth, alt={}, center]{59c9f675-e7eb-47b9-b233-dfbe1844f792-33_52_83_2722_1850}

10. (i) Prove by counter example that the statement
"if \(p\) is a prime number then \(2 p + 1\) is also a prime number" is not true.
(ii) Use proof by exhaustion to prove that if \(n\) is an integer then $$5 n ^ { 2 } + n + 12$$ is always even.