Questions — CAIE P1 (1228 questions)

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CAIE P1 2019 November Q6
7 marks Standard +0.3
6 A line has equation \(y = 3 k x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the line and curve meet at two distinct points.
  2. For each of two particular values of \(k\), the line is a tangent to the curve. Show that these two tangents meet on the \(x\)-axis.
CAIE P1 2019 November Q7
7 marks Moderate -0.3
7
  1. Show that the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) can be expressed as \(3 x ^ { 2 } - 4 x + 1 = 0\), where \(x = \cos ^ { 2 } \theta\).
  2. Hence solve the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2019 November Q8
8 marks Moderate -0.3
8 A function f is defined for \(x > \frac { 1 } { 2 }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 ( 2 x - 1 ) ^ { \frac { 1 } { 2 } } - 6\).
  1. Find the set of values of \(x\) for which f is decreasing.
  2. It is now given that \(\mathrm { f } ( 1 ) = - 3\). Find \(\mathrm { f } ( x )\).
CAIE P1 2019 November Q9
8 marks Standard +0.3
9 The first, second and third terms of a geometric progression are \(3 k , 5 k - 6\) and \(6 k - 4\), respectively.
  1. Show that \(k\) satisfies the equation \(7 k ^ { 2 } - 48 k + 36 = 0\).
  2. Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of \(k\).
  3. One of these ratios gives a progression which is convergent. Find the sum to infinity.
CAIE P1 2019 November Q10
9 marks Standard +0.3
10 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(X\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 8 \\ - 4 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 10 \\ 2 \\ 11 \end{array} \right) \quad \text { and } \quad \overrightarrow { O X } = \left( \begin{array} { r } - 2 \\ - 2 \\ 5 \end{array} \right)$$
  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. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
CAIE P1 Specimen Q1
3 marks Standard +0.8
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
3 marks Easy -1.2
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
4 marks Standard +0.3
3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).
CAIE P1 Specimen Q4
6 marks Moderate -0.3
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
8 marks Moderate -0.8
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
7 marks Moderate -0.8
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 marks Standard +0.3
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 marks Moderate -0.3
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
8 marks Moderate -0.8
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
12 marks Standard +0.3
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
8 marks Challenging +1.2
  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
8 marks Standard +0.3
  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
8 marks Standard +0.3
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
CAIE P1 2024 March Q10
12 marks Standard +0.3
  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
10 marks Moderate -0.3
[diagram]
The diagram shows a sector \(CAB\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D\), \(E\), and \(F\), where \(COD\) is a straight line and angle \(ACD\) is \(\theta\) radians.
  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
7 marks Standard +0.3
  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
8 marks Standard +0.3
  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
10 marks Standard +0.3
  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
7 marks Moderate -0.8
  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
6 marks Moderate -0.5
  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 \(C A D\).