Questions P1 (1374 questions)

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CAIE P1 2019 November Q11
11 \end{array} \right) \quad \text { and } \quad \overrightarrow { O X } = \left( \begin{array} { r } - 2
- 2
5 \end{array} \right)$$
  1. Find \(\overrightarrow { A X }\) and show that \(A X B\) is a straight line.
    The position vector of a point \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { r } 1
    - 8
    3 \end{array} \right)\).
  2. Show that \(C X\) is perpendicular to \(A X\).
  3. Find the area of triangle \(A B C\).
    \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-18_949_1087_260_529} The diagram shows part of the curve \(y = ( x - 1 ) ^ { - 2 } + 2\), and the lines \(x = 1\) and \(x = 3\). The point \(A\) on the curve has coordinates \(( 2,3 )\). The normal to the curve at \(A\) crosses the line \(x = 1\) at \(B\).
  4. Show that the normal \(A B\) has equation \(y = \frac { 1 } { 2 } x + 2\).
  5. Find, showing all necessary working, the volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 Specimen Q1
1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.
CAIE P1 Specimen Q2
2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).
CAIE P1 Specimen Q3
3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).
CAIE P1 Specimen Q4
4
  1. Show that the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) can be expressed as $$4 \sin ^ { 2 } \theta - 15 \sin \theta - 4 = 0$$
  2. Hence solve the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 Specimen Q5
5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point.
CAIE P1 Specimen Q6
6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.
  1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
  2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
  3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).
CAIE P1 Specimen Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-10_716_899_258_621} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters \(C A D\) and \(B A E\) are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).
  1. Show that the radius of the larger circle is \(r \sqrt { } 2\).
  2. Find the area of the shaded region in terms of \(r\).
CAIE P1 Specimen Q8
8 The first term of a progression is \(4 x\) and the second term is \(x ^ { 2 }\).
  1. For the case where the progression is arithmetic with a common difference of 12 , find the possible values of \(x\) and the corresponding values of the third term.
  2. For the case where the progression is geometric with a sum to infinity of 8 , find the third term.
CAIE P1 Specimen Q9
9
  1. Express \(- x ^ { 2 } + 6 x - 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
    The function \(\mathrm { f } : x \mapsto - x ^ { 2 } + 6 x - 5\) is defined for \(x \geqslant m\), where \(m\) is a constant.
  2. State the smallest value of \(m\) for which f is one-one.
  3. For the case where \(m = 5\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-16_771_636_260_756} The diagram shows a cuboid \(O A B C P Q R S\) with a horizontal base \(O A B C\) in which \(A B = 6 \mathrm {~cm}\) and \(O A = a \mathrm {~cm}\), where \(a\) is a constant. The height \(O P\) of the cuboid is 10 cm . The point \(T\) on \(B R\) is such that \(B T = 8 \mathrm {~cm}\), and \(M\) is the mid-point of \(A T\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O P\) respectively.
CAIE P1 Specimen Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-18_515_853_260_644} The diagram shows part of the curve \(y = ( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\) and a point \(P ( 6,5 )\) lying on the curve. The line \(P Q\) intersects the \(x\)-axis at \(Q ( 8,0 )\).
  1. Show that \(P Q\) is a normal to the curve.
  2. Find, showing all necessary working, the exact volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
    [0pt] [In part (ii) you may find it useful to apply the fact that the volume, \(V\), of a cone of base radius \(r\) and vertical height \(h\), is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
Edexcel P1 2019 January Q1
  1. Find
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { 2 x ^ { 3 } } + 5 \right) d x$$ simplifying your answer.
Edexcel P1 2019 January Q2
  1. Given
$$\frac { 3 ^ { x } } { 3 ^ { 4 y } } = 27 \sqrt { 3 }$$ find \(y\) as a simplified function of \(x\).
Edexcel P1 2019 January Q3
  1. The line \(l _ { 1 }\) has equation \(3 x + 5 y - 7 = 0\)
    1. Find the gradient of \(l _ { 1 }\)
    The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 6 , - 2 )\).
  2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
Edexcel P1 2019 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-08_857_857_251_548} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a line \(l _ { 1 }\) with equation \(2 y = x\) and a curve \(C\) with equation \(y = 2 x - \frac { 1 } { 8 } x ^ { 2 }\) The region \(R\), shown unshaded in Figure 1, is bounded by the line \(l _ { 1 }\), the curve \(C\) and a line \(l _ { 2 }\) Given that \(l _ { 2 }\) is parallel to the \(y\)-axis and passes through the intercept of \(C\) with the positive \(x\)-axis, identify the inequalities that define \(R\).
Edexcel P1 2019 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-10_677_1036_260_456} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a plot of part of the curve with equation \(y = \cos 2 x\) with \(x\) being measured in radians. The point \(P\), shown on Figure 2, is a minimum point on the curve.
  1. State the coordinates of \(P\). A copy of Figure 2, called Diagram 1, is shown at the top of the next page.
  2. Sketch, on Diagram 1, the curve with equation \(y = \sin x\)
  3. Hence, or otherwise, deduce the number of solutions of the equation
    1. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 20 \pi\)
    2. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 21 \pi\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-11_693_1050_301_447} \captionsetup{labelformat=empty} \caption{
      Diagram 1}\}
      \end{figure} \textbackslash section*\{Diagram 1
Edexcel P1 2019 January Q6
  1. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Given $$\mathrm { f } ( x ) = 2 x ^ { \frac { 5 } { 2 } } - 40 x + 8 \quad x > 0$$
  1. solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\)
  2. solve the equation \(\mathrm { f } ^ { \prime \prime } ( x ) = 5\)
Edexcel P1 2019 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-14_327_595_251_676} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Not to scale Figure 3 shows the design for a structure used to support a roof. The structure consists of four wooden beams, \(A B , B D , B C\) and \(A D\). Given \(A B = 6.5 \mathrm {~m} , B C = B D = 4.7 \mathrm {~m}\) and angle \(B A C = 35 ^ { \circ }\)
  1. find, to one decimal place, the size of angle \(A C B\),
  2. find, to the nearest metre, the total length of wood required to make this structure.
Edexcel P1 2019 January Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-16_647_970_306_488} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is shown in Figure 4. The curve \(C\)
  • has a single turning point, a maximum at ( 4,9 )
  • crosses the coordinate axes at only two places, \(( - 3,0 )\) and \(( 0,6 )\)
  • has a single asymptote with equation \(y = 4\)
    as shown in Figure 4.
    1. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( - x )\).
    2. State the coordinates of the turning point on the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)\).
Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,
  • state the possible values for \(k\). The curve \(C\) is transformed to a new curve that passes through the origin.
    1. Given that the new curve has equation \(y = \mathrm { f } ( x ) - a\), state the value of the constant \(a\).
    2. Write down an equation for another single transformation of \(C\) that also passes through the origin.
    •
    Edexcel P1 2019 January Q9
    1. The equation
    $$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
    Find the range of possible values of \(c\).
    Edexcel P1 2019 January Q10
    1. A sector \(A O B\), of a circle centre \(O\), has radius \(r \mathrm {~cm}\) and angle \(\theta\) radians.
    Given that the area of the sector is \(6 \mathrm {~cm} ^ { 2 }\) and that the perimeter of the sector is 10 cm ,
    1. show that $$3 \theta ^ { 2 } - 13 \theta + 12 = 0$$
    2. Hence find possible values of \(r\) and \(\theta\).
      □
      \includegraphics[max width=\textwidth, alt={}, center]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-21_131_19_2627_1882}
    Edexcel P1 2019 January Q11
    11. (a) On Diagram 1 sketch the graphs of
    1. \(y = x ( 3 - x )\)
    2. \(y = x ( x - 2 ) ( 5 - x )\)
      showing clearly the coordinates of the points where the curves cross the coordinate axes.
      (b) Show that the \(x\) coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 13 \right) = 0\) The point \(P\) lies on both curves. Given that \(P\) lies in the first quadrant,
      (c) find, using algebra and showing your working, the exact coordinates of \(P\).
      \includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
      \section*{Diagram 1}
    Edexcel P1 2019 January Q12
    12. The curve with equation \(y = \mathrm { f } ( x ) , x > 0\), passes through the point \(P ( 4 , - 2 )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x \sqrt { x } - 10 x ^ { - \frac { 1 } { 2 } }$$
    1. find the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
    2. Find \(\mathrm { f } ( x )\).
    Edexcel P1 2020 January Q1
    1. Find, in simplest form,
    $$\int \left( \frac { 8 x ^ { 3 } } { 3 } - \frac { 1 } { 2 \sqrt { x } } - 5 \right) \mathrm { d } x$$
    Edexcel P1 2020 January Q2
    2. Given \(y = 3 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.
    1. \(3 ^ { 3 x }\)
    2. \(\frac { 1 } { 3 ^ { x - 2 } }\)
    3. \(\frac { 81 } { 9 ^ { 2 - 3 x } }\)