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CAIE P1 2019 June Q5
5 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-06_355_634_255_753} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The point \(C\) lies on the circumference and angle \(A O C = \theta\) radians. The perimeter of sector \(B O C\) is twice the perimeter of sector \(A O C\). Find the value of \(\theta\) correct to 2 significant figures.
CAIE P1 2019 June Q6
7 marks Moderate -0.8
6 The equation of a curve is \(y = 3 \cos 2 x\) and the equation of a line is \(2 y + \frac { 3 x } { \pi } = 5\).
  1. State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leqslant x \leqslant 2 \pi\).
  2. Sketch, on the same diagram, the graphs of \(y = 3 \cos 2 x\) and \(2 y + \frac { 3 x } { \pi } = 5\) for \(0 \leqslant x \leqslant 2 \pi\).
  3. State the number of solutions of the equation \(6 \cos 2 x = 5 - \frac { 3 x } { \pi }\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P1 2019 June Q7
7 marks Moderate -0.8
7 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 2 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 2 x + 3 } { x - 1 } , \quad x \in \mathbb { R } , x \neq 1 \end{aligned}$$
  1. Obtain expressions for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined. [4]
  2. Solve the equation \(\operatorname { fg } ( x ) = \frac { 7 } { 3 }\).
CAIE P1 2019 June Q8
8 marks Moderate -0.3
8 The position vectors of points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 6 \\ - 2 \\ - 6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ k \\ - 3 \end{array} \right)$$ where \(k\) is a constant.
  1. Find the value of \(k\) for which angle \(A O B\) is \(90 ^ { \circ }\).
  2. Find the values of \(k\) for which the lengths of \(O A\) and \(O B\) are equal.
    The point \(C\) is such that \(\overrightarrow { A C } = 2 \overrightarrow { C B }\).
  3. In the case where \(k = 4\), find the unit vector in the direction of \(\overrightarrow { O C }\).
CAIE P1 2019 June Q9
8 marks Challenging +1.2
9 The curve \(C _ { 1 }\) has equation \(y = x ^ { 2 } - 4 x + 7\). The curve \(C _ { 2 }\) has equation \(y ^ { 2 } = 4 x + k\), where \(k\) is a constant. The tangent to \(C _ { 1 }\) at the point where \(x = 3\) is also the tangent to \(C _ { 2 }\) at the point \(P\). Find the value of \(k\) and the coordinates of \(P\).
CAIE P1 2019 June Q10
10 marks Standard +0.3
10
  1. In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is \(a\).
    1. Show that the common difference of the progression is \(\frac { 1 } { 3 } a\).
    2. Given that the tenth term is 36 more than the fourth term, find the value of \(a\).
  2. The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12 , find the value of the fifth term.
CAIE P1 2019 June Q11
12 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve \(y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }\) and the minimum point \(M\).
  1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
  2. Find the coordinates of \(M\).
    The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.
  3. Find, showing all necessary working, the area of the shaded region.
    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 2016 March Q1
4 marks Moderate -0.8
1
  1. Find the coefficients of \(x ^ { 4 }\) and \(x ^ { 5 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).
  2. It is given that, when \(( 1 + p x ) ( 1 - 2 x ) ^ { 5 }\) is expanded, there is no term in \(x ^ { 5 }\). Find the value of the constant \(p\).
CAIE P1 2016 March Q2
4 marks Easy -1.2
2 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 2 } { x ^ { 3 } }\) passes through \(( - 1,3 )\). Find the equation of the curve.
CAIE P1 2016 March Q3
5 marks Moderate -0.5
3 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.
CAIE P1 2016 March Q4
6 marks Moderate -0.3
4
  1. Solve the equation \(\sin ^ { - 1 } ( 3 x ) = - \frac { 1 } { 3 } \pi\), giving the solution in an exact form.
  2. Solve, by factorising, the equation \(2 \cos \theta \sin \theta - 2 \cos \theta - \sin \theta + 1 = 0\) for \(0 \leqslant \theta \leqslant \pi\).
CAIE P1 2016 March Q5
8 marks Moderate -0.3
5 Two points have coordinates \(A ( 5,7 )\) and \(B ( 9 , - 1 )\).
  1. Find the equation of the perpendicular bisector of \(A B\). The line through \(C ( 1,2 )\) parallel to \(A B\) meets the perpendicular bisector of \(A B\) at the point \(X\).
  2. Find, by calculation, the distance \(B X\).
CAIE P1 2016 March Q6
8 marks Standard +0.3
6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.
  1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
  2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.
CAIE P1 2016 March Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-3_529_698_260_721} The diagram shows a pyramid \(O A B C\) with a horizontal triangular base \(O A B\) and vertical height \(O C\). Angles \(A O B , B O C\) and \(A O C\) are each right angles. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) respectively, with \(O A = 4\) units, \(O B = 2.4\) units and \(O C = 3\) units. The point \(P\) on \(C A\) is such that \(C P = 3\) units.
  1. Show that \(\overrightarrow { C P } = 2.4 \mathbf { i } - 1.8 \mathbf { k }\).
  2. Express \(\overrightarrow { O P }\) and \(\overrightarrow { B P }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  3. Use a scalar product to find angle \(B P C\).
CAIE P1 2016 March Q8
10 marks Standard +0.3
8 The function f is such that \(\mathrm { f } ( x ) = a ^ { 2 } x ^ { 2 } - a x + 3 b\) for \(x \leqslant \frac { 1 } { 2 a }\), where \(a\) and \(b\) are constants.
  1. For the case where \(\mathrm { f } ( - 2 ) = 4 a ^ { 2 } - b + 8\) and \(\mathrm { f } ( - 3 ) = 7 a ^ { 2 } - b + 14\), find the possible values of \(a\) and \(b\).
  2. For the case where \(a = 1\) and \(b = - 1\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and give the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2016 March Q9
10 marks Standard +0.8
9
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
    1. Show that angle \(A O B = 2 \alpha\).
    2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
    [0pt] [6]
CAIE P1 2016 March Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-5_650_1038_260_550} The diagram shows part of the curve \(y = \frac { 1 } { 16 } ( 3 x - 1 ) ^ { 2 }\), which touches the \(x\)-axis at the point \(P\). The point \(Q ( 3,4 )\) lies on the curve and the tangent to the curve at \(Q\) crosses the \(x\)-axis at \(R\).
  1. State the \(x\)-coordinate of \(P\). Showing all necessary working, find by calculation
  2. the \(x\)-coordinate of \(R\),
  3. the area of the shaded region \(P Q R\).
CAIE P1 2017 March Q1
4 marks Moderate -0.5
1 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + 3 k x + k = 0\) has distinct real roots.
CAIE P1 2017 March Q2
4 marks Standard +0.3
2 In the expansion of \(\left( \frac { 1 } { a x } + 2 a x ^ { 2 } \right) ^ { 5 }\), the coefficient of \(x\) is 5 . Find the value of the constant \(a\).
CAIE P1 2017 March Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-04_489_465_258_840} The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is \(h \mathrm {~cm}\) the surface of the water is a square of side \(\frac { 1 } { 2 } h \mathrm {~cm}\).
  1. Express the volume of water in the container in terms of \(h\).
    [0pt] [The volume of a pyramid having a base area \(A\) and vertical height \(h\) is \(\frac { 1 } { 3 } A h\).]
    Water is steadily dripping into the container at a constant rate of \(20 \mathrm {~cm} ^ { 3 }\) per minute.
  2. Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm . \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-06_403_773_258_685} In the diagram, \(A B = A C = 8 \mathrm {~cm}\) and angle \(C A B = \frac { 2 } { 7 } \pi\) radians. The circular \(\operatorname { arc } B C\) has centre \(A\), the circular arc \(C D\) has centre \(B\) and \(A B D\) is a straight line.
CAIE P1 2017 March Q5
7 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-08_526_499_258_824} The diagram shows the graphs of \(y = \tan x\) and \(y = \cos x\) for \(0 \leqslant x \leqslant \pi\). The graphs intersect at points \(A\) and \(B\).
  1. Find by calculation the \(x\)-coordinate of \(A\).
  2. Find by calculation the coordinates of \(B\).
CAIE P1 2017 March Q6
7 marks Moderate -0.3
6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$
  1. Use a scalar product to find angle \(O A B\).
  2. Find the area of triangle \(O A B\).
CAIE P1 2017 March Q7
8 marks Moderate -0.3
7 The function f is defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
    The first, second and third terms of a geometric progression are respectively \(\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )\) and \(k \mathrm { f } ^ { \prime \prime } ( 2 )\).
  2. Find the value of the constant \(k\).
CAIE P1 2017 March Q8
10 marks Moderate -0.8
8 The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x ^ { 2 } + 3 \\ & \mathrm {~g} : x \mapsto 3 x + 2 \end{aligned}$$
  1. Show that \(\operatorname { gf } ( x ) = 6 x ^ { 2 } + 11\) and obtain an unsimplified expression for \(\operatorname { fg } ( x )\).
  2. Find an expression for \(( \mathrm { fg } ) ^ { - 1 } ( x )\) and determine the domain of \(( \mathrm { fg } ) ^ { - 1 }\).
  3. Solve the equation \(\mathrm { gf } ( 2 x ) = \mathrm { fg } ( x )\).
CAIE P1 2017 March Q9
11 marks Standard +0.8
9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).
  1. Find the equation of the tangent to the curve at \(A\).
    The normal to the curve at \(A\) intersects the curve again at \(B\).
  2. Find the coordinates of \(B\).
    The tangents at \(A\) and \(B\) intersect each other at \(C\).
  3. Find the coordinates of \(C\).