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Edexcel P1 2023 October Q3
6 marks Moderate -0.3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Write \(\frac { 8 - \sqrt { 15 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(a \sqrt { 3 } + b \sqrt { 5 }\) where \(a\) and \(b\) are integers to be found.
  2. Hence, or otherwise, solve $$( x + 5 \sqrt { 3 } ) \sqrt { 5 } = 40 - 2 x \sqrt { 3 }$$ giving your answer in simplest form.
Edexcel P1 2023 October Q4
7 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-08_687_775_248_646} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \frac { 1 } { x + 2 }\)
  1. State the equation of the asymptote of \(C\) that is parallel to the \(y\)-axis.
  2. Factorise fully \(x ^ { 3 } + 4 x ^ { 2 } + 4 x\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.
  3. On Diagram 1, add a sketch of the curve with equation $$y = x ^ { 3 } + 4 x ^ { 2 } + 4 x$$ On your sketch, state clearly the coordinates of each point where this curve cuts or meets the coordinate axes.
  4. Hence state the number of real solutions of the equation $$( x + 2 ) \left( x ^ { 3 } + 4 x ^ { 2 } + 4 x \right) = 1$$ giving a reason for your answer.
    \includegraphics[max width=\textwidth, alt={}]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-09_800_1700_1053_185}
    Only use the copy of Diagram 1 if you need to redraw your answer to part (c).
Edexcel P1 2023 October Q5
7 marks Standard +0.3
5. Figure 2 Diagram NOT accurately drawn Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that
  • \(B D = x \mathrm {~m}\)
  • \(C D = ( 1 + x ) \mathrm { m }\)
  • \(B C = 5 \mathrm {~m}\)
  • angle \(B C D = \theta ^ { \circ }\)
    1. show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)
Given also that
Edexcel P1 2023 October Q6
6 marks Standard +0.3
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.} The equation $$4 ( p - 2 x ) = \frac { 12 + 15 p } { x + p } \quad x \neq - p$$ where \(p\) is a constant, has two distinct real roots.
  1. Show that $$3 p ^ { 2 } - 10 p - 8 > 0$$
  2. Hence, using algebra, find the range of possible values of \(p\)
Edexcel P1 2023 October Q7
10 marks Moderate -0.3
  1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)
Given that
  • \(f ^ { \prime } ( x ) = \frac { 4 x ^ { 2 } + 10 - 7 x ^ { \frac { 1 } { 2 } } } { 4 x ^ { \frac { 1 } { 2 } } }\)
  • the point \(P ( 4 , - 1 )\) lies on \(C\)
      1. find the value of the gradient of \(C\) at \(P\)
      2. Hence find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
    2. Find \(\mathrm { f } ( x )\).
Edexcel P1 2023 October Q8
7 marks Standard +0.8
  1. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.} The curve \(C _ { 1 }\) has equation $$x y = \frac { 15 } { 2 } - 5 x \quad x \neq 0$$ The curve \(C _ { 2 }\) has equation $$y = x ^ { 3 } - \frac { 7 } { 2 } x - 5$$
  1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet when $$2 x ^ { 4 } - 7 x ^ { 2 } - 15 = 0$$ Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at points \(P\) and \(Q\)
  2. find, using algebra, the exact distance \(P Q\)
Edexcel P1 2023 October Q9
7 marks Standard +0.3
9. Diagram NOT accurately drawn \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-24_581_1491_340_296} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the plan view of the area being used for a ball-throwing competition.
Competitors must stand within the circle \(C\) and throw a ball as far as possible into the target area, \(P Q R S\), shown shaded in Figure 3. Given that
  • circle \(C\) has centre \(O\)
  • \(P\) and \(S\) are points on \(C\)
  • \(O P Q R S O\) is a sector of a circle with centre \(O\)
  • the length of arc \(P S\) is 0.72 m
  • the size of angle \(P O S\) is 0.6 radians
    1. show that \(O P = 1.2 \mathrm {~m}\)
Given also that
$$5 x ^ { 2 } + 12 x - 1500 = 0$$
  • Hence calculate the total perimeter of the target area, \(P Q R S\), giving your answer to the nearest metre.
    •
    Edexcel P1 2023 October Q10
    6 marks Moderate -0.8
    10. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-28_538_652_255_708} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 3 \cos \left( \frac { x } { n } \right) ^ { \circ } \quad x \geqslant 0$$ where \(n\) is a constant.
    The curve \(C _ { 1 }\) cuts the positive \(x\)-axis for the first time at point \(P ( 270,0 )\), as shown in Figure 4.
      1. State the value of \(n\)
      2. State the period of \(C _ { 1 }\) The point \(Q\), shown in Figure 4, is a minimum point of \(C _ { 1 }\)
    2. State the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 2 \sin x ^ { \circ } + k\), where \(k\) is a constant.
      The point \(R \left( a , \frac { 12 } { 5 } \right)\) and the point \(S \left( - a , - \frac { 3 } { 5 } \right)\), both lie on \(C _ { 2 }\) Given that \(a\) is a constant less than 90
    3. find the value of \(k\).
    Edexcel P1 2023 October Q11
    10 marks Easy -1.2
    11. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c0b4165d-b8bb-419c-b75a-d6c0c2431510-30_595_869_255_568} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} Figure 5 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$
    1. Write \(2 x ^ { 2 } - 12 x + 14\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. Given that \(C\) has a minimum at the point \(P\)
    2. state the coordinates of \(P\) The line \(l\) intersects \(C\) at \(( - 1,28 )\) and at \(P\) as shown in Figure 5.
    3. Find the equation of \(l\) giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. The finite region \(R\), shown shaded in Figure 5, is bounded by the \(x\)-axis, \(l\), the \(y\)-axis, and \(C\).
    4. Use inequalities to define the region \(R\).
    Edexcel C12 2014 January Q1
    4 marks Easy -1.2
    1. Find the first 3 terms in ascending powers of \(x\) of
    $$\left( 2 - \frac { x } { 2 } \right) ^ { 6 }$$ giving each term in its simplest form.
    Edexcel C12 2014 January Q2
    7 marks Easy -1.2
    2. $$\mathrm { f } ( x ) = \frac { 8 } { x ^ { 2 } } - 4 \sqrt { x } + 3 x - 1 , \quad x > 0$$ Giving your answers in their simplest form, find
    1. \(\mathrm { f } ^ { \prime } ( x )\)
    2. \(\int \mathrm { f } ( x ) \mathrm { d } x\)
    Edexcel C12 2014 January Q3
    7 marks Moderate -0.8
    3. $$f ( x ) = 10 x ^ { 3 } + 27 x ^ { 2 } - 13 x - 12$$
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
      1. \(x - 2\)
      2. \(x + 3\)
    2. Hence factorise \(\mathrm { f } ( x )\) completely.
    Edexcel C12 2014 January Q4
    7 marks Moderate -0.8
    4. Answer this question without the use of a calculator and show all your working.
    1. Show that $$\frac { 4 } { 2 \sqrt { 2 } - \sqrt { 6 } } = 2 \sqrt { 2 } ( 2 + \sqrt { 3 } )$$
    2. Show that $$\sqrt { 27 } + \sqrt { 21 } \times \sqrt { 7 } - \frac { 6 } { \sqrt { 3 } } = 8 \sqrt { 3 }$$
    Edexcel C12 2014 January Q5
    7 marks Standard +0.8
    5. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3 \\ u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of
    1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
    2. \(u _ { 61 }\)
    3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)
    Edexcel C12 2014 January Q6
    6 marks Standard +0.3
    6. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a b = 25 \\ \log _ { 4 } a - \log _ { 4 } b = 3 \end{gathered}$$ Show each step of your working, giving exact values for \(a\) and \(b\).
    Edexcel C12 2014 January Q7
    5 marks Moderate -0.3
    7.
    1. Show that $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ may be expressed in the form $$12 \cos ^ { 2 } x + \cos x - 1 = 0$$
    2. Hence, using trigonometry, find all the solutions in the interval \(0 \leqslant x \leqslant 360 ^ { \circ }\) of $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ Give each solution, in degrees, to 1 decimal place. \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-15_106_97_2615_1784}
    Edexcel C12 2014 January Q8
    7 marks Moderate -0.3
    8. Find the range of values of \(k\) for which the quadratic equation $$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$ has no real roots.
    Edexcel C12 2014 January Q9
    7 marks Moderate -0.3
    9. In the first month after opening, a mobile phone shop sold 300 phones. A model for future sales assumes that the number of phones sold will increase by \(5 \%\) per month, so that \(300 \times 1.05\) will be sold in the second month, \(300 \times 1.05 ^ { 2 }\) in the third month, and so on. Using this model, calculate
    1. the number of phones sold in the 24th month,
    2. the total number of phones sold over the whole 24 months. This model predicts that, in the \(N\) th month, the number of phones sold in that month exceeds 3000 for the first time.
    3. Find the value of \(N\).
    Edexcel C12 2014 January Q10
    9 marks Moderate -0.8
    10. The curve \(C\) has equation \(y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi\)
    1. In the space below, sketch the curve \(C\).
    2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
    3. Solve, for \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\), $$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.
    Edexcel C12 2014 January Q11
    8 marks Moderate -0.8
    11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.
    1. Show that \(p = 9\)
    2. Find the value of the 20th term of this series.
    3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}
    Edexcel C12 2014 January Q12
    11 marks Standard +0.3
    12.
    [diagram]
    Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\). \(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find
    1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
    2. the perimeter of the platform to 3 significant figures,
    3. the area of the platform to 3 significant figures.
    Edexcel C12 2014 January Q13
    14 marks Moderate -0.3
    13. The curve \(C\) has equation $$y = \frac { ( x - 3 ) ( 3 x - 25 ) } { x } , \quad x > 0$$
    1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a fully simplified form.
    2. Hence find the coordinates of the turning point on the curve \(C\).
    3. Determine whether this turning point is a minimum or maximum, justifying your answer. The point \(P\), with \(x\) coordinate \(2 \frac { 1 } { 2 }\), lies on the curve \(C\).
    4. Find the equation of the normal at \(P\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-35_90_72_2631_1873}
    Edexcel C12 2014 January Q14
    12 marks Standard +0.3
    14. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-36_652_791_223_548} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.
    1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
    2. Use integration to calculate an exact value for the area of \(R\).
    Edexcel C12 2014 January Q15
    14 marks Moderate -0.8
    15. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e878227b-d625-4ef2-ac49-a9dc05c5321a-40_883_824_212_568} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Diagram NOT drawn to scale The points \(X\) and \(Y\) have coordinates \(( 0,3 )\) and \(( 6,11 )\) respectively. \(X Y\) is a chord of a circle \(C\) with centre \(Z\), as shown in Figure 3.
    1. Find the gradient of \(X Y\). The point \(M\) is the midpoint of \(X Y\).
    2. Find an equation for the line which passes through \(Z\) and \(M\). Given that the \(y\) coordinate of \(Z\) is 10 ,
    3. find the \(x\) coordinate of \(Z\),
    4. find the equation of the circle \(C\), giving your answer in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$$ where \(a\), \(b\) and \(c\) are constants.
    Edexcel C12 2015 January Q1
    3 marks Easy -1.8
    Simplify the following expressions fully.
    1. \(\left( x ^ { 6 } \right) ^ { \frac { 1 } { 3 } }\)
    2. \(\sqrt { 2 } \left( x ^ { 3 } \right) \div \sqrt { \frac { 32 } { x ^ { 2 } } }\)