Edexcel Paper 1 (Paper 1) 2021 October

1. $$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$ Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
You must make your method clear.

1. Given that

$$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$

1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
   • meets the \(y\)-axis at the point \(P\)
   • has a minimum turning point at the point \(Q\)
   • Write down
     1. the coordinates of \(P\)
     2. the coordinates of \(Q\)

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

$$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where \(k\) is an integer.
Given that \(u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0\)

1. show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
2. Find the value of \(k\), giving a reason for your answer.
3. Find the value of \(u _ { 3 }\)

1. The curve with equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = x ^ { 2 } + \ln \left( 2 x ^ { 2 } - 4 x + 5 \right)$$ has a single turning point at \(x = \alpha\)

1. Show that \(\alpha\) is a solution of the equation $$2 x ^ { 3 } - 4 x ^ { 2 } + 7 x - 2 = 0$$ The iterative formula $$x _ { n + 1 } = \frac { 1 } { 7 } \left( 2 + 4 x _ { n } ^ { 2 } - 2 x _ { n } ^ { 3 } \right)$$ is used to find an approximate value for \(\alpha\).
   Starting with \(x _ { 1 } = 0.3\)
2. calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(x _ { 4 }\) Using a suitable interval and a suitable function that should be stated,
3. show that \(\alpha\) is 0.341 to 3 decimal places.

1. In this question you should show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} A company made a profit of \(\pounds 20000\) in its first year of trading, Year 1
A model for future trading predicts that the yearly profit will increase by \(8 \%\) each year, so that the yearly profits will form a geometric sequence. According to the model,

1. show that the profit for Year 3 will be \(\pounds 23328\)
2. find the first year when the yearly profit will exceed £65000
3. find the total profit for the first 20 years of trading, giving your answer to the nearest £1000

6. Figure 1 Figure 1 shows a sketch of triangle \(A B C\).
Given that

• \(\overrightarrow { A B } = - 3 \mathbf { i } - 4 \mathbf { j } - 5 \mathbf { k }\)
• \(\overrightarrow { B C } = \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
  1. find \(\overrightarrow { A C }\)
  2. show that \(\cos A B C = \frac { 9 } { 10 }\)

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
      Given that \(l\) is a tangent to \(C\),
2. find the possible values of \(k\), giving your answers as simplified surds.

1. A scientist is studying the growth of two different populations of bacteria.

The number of bacteria, \(N\), in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where \(A\) and \(k\) are positive constants and \(t\) is the time in hours from the start of the study.
Given that

• there were 1000 bacteria in this population at the start of the study
• it took exactly 5 hours from the start of the study for this population to double
  1. find a complete equation for the model.
  2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, \(M\), in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where \(k\) has the value found in part (a) and \(t\) is the time in hours from the start of the study.
Given that \(T\) hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of \(T\).

9. $$f ( x ) = \frac { 50 x ^ { 2 } + 38 x + 9 } { ( 5 x + 2 ) ^ { 2 } ( 1 - 2 x ) } \quad x \neq - \frac { 2 } { 5 } \quad x \neq \frac { 1 } { 2 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$\frac { A } { 5 x + 2 } + \frac { B } { ( 5 x + 2 ) ^ { 2 } } + \frac { C } { 1 - 2 x }$$ where \(A\), \(B\) and \(C\) are constants

1. 1. find the value of \(B\) and the value of \(C\)
   2. show that \(A = 0\)
2. 1. Use binomial expansions to show that, in ascending powers of \(x\) $$f ( x ) = p + q x + r x ^ { 2 } + \ldots$$ where \(p , q\) and \(r\) are simplified fractions to be found.
   2. Find the range of values of \(x\) for which this expansion is valid.

1. In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Given that \(1 + \cos 2 \theta + \sin 2 \theta \neq 0\) prove that $$\frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta } \equiv \tan \theta$$
2. Hence solve, for \(0 < x < 180 ^ { \circ }\) $$\frac { 1 - \cos 4 x + \sin 4 x } { 1 + \cos 4 x + \sin 4 x } = 3 \sin 2 x$$ giving your answers to one decimal place where appropriate.

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = ( \ln x ) ^ { 2 } \quad x > 0$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 2\), the \(x\)-axis and the line with equation \(x = 4\) The table below shows corresponding values of \(x\) and \(y\), with the values of \(y\) given to 4 decimal places.

1. Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(R\), giving your answer to 3 significant figures.
2. Use algebraic integration to find the exact area of \(R\), giving your answer in the form $$y = a ( \ln 2 ) ^ { 2 } + b \ln 2 + c$$ where \(a\), \(b\) and \(c\) are integers to be found.

12. \begin{figure}[h] \end{figure} Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground. The vertical height, \(H\) metres, of the ball above the ground has been plotted against the horizontal distance travelled, \(x\) metres, measured from where the ball was hit. The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground
• reaches its maximum vertical height after travelling a horizontal distance of 90 m
• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that \(H\) is modelled as a quadratic function in \(x\)

1. find \(H\) in terms of \(x\)
2. Hence find, according to the model,
   1. the maximum vertical height of the ball above the ground,
   2. the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.
3. The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

1. A curve \(C\) has parametric equations

$$x = \frac { t ^ { 2 } + 5 } { t ^ { 2 } + 1 } \quad y = \frac { 4 t } { t ^ { 2 } + 1 } \quad t \in \mathbb { R }$$ Show that all points on \(C\) satisfy $$( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$$

1. Given that

$$y = \frac { x - 4 } { 2 + \sqrt { x } } \quad x > 0$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \mathrm {~A} \sqrt { \mathrm { x } } } \quad x > 0$$ where \(A\) is a constant to be found.

1. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\)

$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.