Questions — CAIE P1 (1202 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 \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE P1 2021 March Q10
  1. For the case where angle \(B A C = \frac { 1 } { 6 } \pi\) radians, find \(k\) correct to 4 significant figures.
  2. For the general case in which angle \(B A C = \theta\) radians, where \(0 < \theta < \frac { 1 } { 2 } \pi\), it is given that \(\frac { \theta } { \sin \theta } > 1\). Find the set of possible values of \(k\).
CAIE P1 2022 March Q6
  1. Find, by calculation, the coordinates of \(A\) and \(B\).
  2. Find an equation of the circle which has its centre at \(C\) and for which the line with equation \(y = 3 x - 20\) is a tangent to the circle.
CAIE P1 2022 March Q10
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
CAIE P1 2024 March Q10
  1. Find the equation of the tangent to the circle at the point \(( - 6,9 )\).
  2. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  3. Find the value of \(\theta\) correct to 4 significant figures.
  4. Find the perimeter and area of the segment shaded in the diagram.
CAIE P1 2020 November Q10
  1. Find \(C D\) in terms of \(r\) and \(\sin \theta\).
    It is now given that \(r = 4\) and \(\theta = \frac { 1 } { 6 } \pi\).
  2. Find the perimeter of sector \(C A B\) in terms of \(\pi\).
  3. Find the area of the shaded region in terms of \(\pi\) and \(\sqrt { 3 }\).
CAIE P1 2021 November Q6
  1. Find the perimeter of the plate, giving your answer in terms of \(\pi\).
  2. Find the area of the plate, giving your answer in terms of \(\pi\) and \(\sqrt { 3 }\).
CAIE P1 2022 November Q10
  1. Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
  2. Find the difference in area of the two triangles \(A O B\) and \(A P B\), giving your answer correct to 2 decimal places.
  3. Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.
CAIE P1 2022 November Q10
  1. Find the coordinates of \(A\).
  2. Find the volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in the form \(\frac { \pi } { a } ( b \sqrt { c } - d )\), where \(a , b , c\) and \(d\) are integers.
  3. Find an exact expression for the perimeter of the shaded region.
CAIE P1 2017 June Q4
  1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 5\) and \(\theta = \frac { 1 } { 6 } \pi\), find the area of the shaded region.
CAIE P1 2018 June Q5
  1. Express each of the vectors \(\overrightarrow { D A }\) and \(\overrightarrow { C A }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(C A D\).
CAIE P1 2017 March Q4
  1. Show that angle \(C B D = \frac { 9 } { 14 } \pi\) radians.
  2. Find the perimeter of the shaded region.
CAIE P1 2005 November Q5
  1. Express \(h\) in terms of \(r\) and hence show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = 12 \pi r ^ { 2 } - 2 \pi r ^ { 3 }$$
  2. Given that \(r\) varies, find the stationary value of \(V\).
CAIE P1 2014 November Q5
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). The equation of a curve is \(y = x ^ { 3 } + a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants.
  3. In the case where the curve has no stationary point, show that \(a ^ { 2 } < 3 b\).
  4. In the case where \(a = - 6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\).
    \includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-3_634_711_952_717} The diagram shows a pyramid \(O A B C X\). The horizontal square base \(O A B C\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(X D = 10\) units. The mid-point of \(O X\) is \(M\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
  5. Express the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { A C }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  6. Use a scalar product to find angle \(M A C\).
    (a) The sum, \(S _ { n }\), of the first \(n\) terms of an arithmetic progression is given by \(S _ { n } = 32 n - n ^ { 2 }\). Find the first term and the common difference.
    (b) A geometric progression in which all the terms are positive has sum to infinity 20 . The sum of the first two terms is 12.8 . Find the first term of the progression.
CAIE P1 2015 November Q10
  1. For the case where \(a = 2\), find the unit vector in the direction of \(\overrightarrow { P M }\).
  2. For the case where angle \(A T P = \cos ^ { - 1 } \left( \frac { 2 } { 7 } \right)\), find the value of \(a\).