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CAIE P1 2015 June Q9
8 marks Standard +0.3
9
  1. The first term of an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
  2. The first term of a geometric progression is \(\sqrt { } 3\) and the second term is \(2 \cos \theta\), where \(0 < \theta < \pi\). Find the set of values of \(\theta\) for which the progression is convergent.
CAIE P1 2015 June Q10
9 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-3_682_1319_1525_413} Points \(A ( 2,9 )\) and \(B ( 3,0 )\) lie on the curve \(y = 9 + 6 x - 3 x ^ { 2 }\), as shown in the diagram. The tangent at \(A\) intersects the \(x\)-axis at \(C\). Showing all necessary working,
  1. find the equation of the tangent \(A C\) and hence find the \(x\)-coordinate of \(C\),
  2. find the area of the shaded region \(A B C\).
    [0pt] [Question 11 is printed on the next page.]
CAIE P1 2015 June Q11
10 marks Standard +0.8
11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.
  1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
  2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
  3. Find angle \(A O B\) in degrees. {www.cie.org.uk} after the live examination series. }
CAIE P1 2016 June Q1
3 marks Moderate -0.5
1 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 3 } { 2 x } \right) ^ { 6 }\).
CAIE P1 2016 June Q2
4 marks Moderate -0.3
2 Solve the equation \(3 \sin ^ { 2 } \theta = 4 \cos \theta - 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2016 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_497_1106_554_515} The diagram shows part of the curve \(x = \frac { 12 } { y ^ { 2 } } - 2\). The shaded region is bounded by the curve, the \(y\)-axis and the lines \(y = 1\) and \(y = 2\). Showing all necessary working, find the volume, in terms of \(\pi\), when this shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.
CAIE P1 2016 June Q4
6 marks Moderate -0.3
4 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 8 ( 3 x + 4 ) ^ { - \frac { 1 } { 2 } }\).
  1. A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of change of the \(y\)-coordinate as \(P\) crosses the \(y\)-axis. The curve intersects the \(y\)-axis where \(y = \frac { 4 } { 3 }\).
  2. Find the equation of the curve.
CAIE P1 2016 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_364_625_1873_762} A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures \(x \mathrm {~m}\) by \(y \mathrm {~m}\) and is fully enclosed by metal fencing. The farmer uses 480 m of fencing.
  1. Show that the total area of land used for the sheep pens, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 384 x - 9.6 x ^ { 2 }$$
  2. Given that \(x\) and \(y\) can vary, find the dimensions of each sheep pen for which the value of \(A\) is a maximum. (There is no need to verify that the value of \(A\) is a maximum.)
CAIE P1 2016 June Q6
7 marks Moderate -0.8
6
  1. Find the values of the constant \(m\) for which the line \(y = m x\) is a tangent to the curve \(y = 2 x ^ { 2 } - 4 x + 8\).
  2. The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants. The solutions of the equation \(\mathrm { f } ( x ) = 0\) are \(x = 1\) and \(x = 9\). Find
    1. the values of \(a\) and \(b\),
    2. the coordinates of the vertex of the curve \(y = \mathrm { f } ( x )\).
CAIE P1 2016 June Q7
7 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-3_408_451_721_845} In the diagram, \(A O B\) is a quarter circle with centre \(O\) and radius \(r\). The point \(C\) lies on the arc \(A B\) and the point \(D\) lies on \(O B\). The line \(C D\) is parallel to \(A O\) and angle \(A O C = \theta\) radians.
  1. Express the perimeter of the shaded region in terms of \(r , \theta\) and \(\pi\).
  2. For the case where \(r = 5 \mathrm {~cm}\) and \(\theta = 0.6\), find the area of the shaded region.
CAIE P1 2016 June Q8
7 marks Standard +0.8
8 A curve has equation \(y = 3 x - \frac { 4 } { x }\) and passes through the points \(A ( 1 , - 1 )\) and \(B ( 4,11 )\). At each of the points \(C\) and \(D\) on the curve, the tangent is parallel to \(A B\). Find the equation of the perpendicular bisector of \(C D\).
CAIE P1 2016 June Q9
9 marks Moderate -0.3
9
  1. The first term of a geometric progression in which all the terms are positive is 50 . The third term is 32 . Find the sum to infinity of the progression.
  2. The first three terms of an arithmetic progression are \(2 \sin x , 3 \cos x\) and ( \(\sin x + 2 \cos x\) ) respectively, where \(x\) is an acute angle.
    1. Show that \(\tan x = \frac { 4 } { 3 }\).
    2. Find the sum of the first twenty terms of the progression.
CAIE P1 2016 June Q10
10 marks Moderate -0.3
10 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 5 \\ - 1 \\ k \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ 6 \\ - 3 \end{array} \right)$$ respectively, where \(k\) is a constant.
  1. Find the value of \(k\) in the case where angle \(A O B = 90 ^ { \circ }\).
  2. Find the possible values of \(k\) for which the lengths of \(A B\) and \(O C\) are equal. The point \(D\) is such that \(\overrightarrow { O D }\) is in the same direction as \(\overrightarrow { O A }\) and has magnitude 9 units. The point \(E\) is such that \(\overrightarrow { O E }\) is in the same direction as \(\overrightarrow { O C }\) and has magnitude 14 units.
  3. Find the magnitude of \(\overrightarrow { D E }\) in the form \(\sqrt { } n\) where \(n\) is an integer.
CAIE P1 2016 June Q11
11 marks Moderate -0.3
11 The function f is defined by \(\mathrm { f } : x \mapsto 4 \sin x - 1\) for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. State the range of f .
  2. Find the coordinates of the points at which the curve \(y = \mathrm { f } ( x )\) intersects the coordinate axes.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\), stating both the domain and range of \(\mathrm { f } ^ { - 1 }\). {www.cie.org.uk} after the live examination series. }
CAIE P1 2016 June Q1
3 marks Moderate -0.8
1 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 10 - 3 x , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto \frac { 10 } { 3 - 2 x } , \quad x \in \mathbb { R } , x \neq \frac { 3 } { 2 } \end{aligned}$$ Solve the equation \(\mathrm { ff } ( x ) = \mathrm { gf } ( 2 )\).
CAIE P1 2016 June Q2
4 marks Moderate -0.3
2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { ( 5 - 2 x ) ^ { 2 } }\). Given that the curve passes through ( 2,7 ), find the equation of the curve.
CAIE P1 2016 June Q3
4 marks Moderate -0.8
3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }$$ The point \(C\) is such that \(\overrightarrow { A B } = \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).
CAIE P1 2016 June Q4
6 marks Moderate -0.3
4 Find the term that is independent of \(x\) in the expansion of
  1. \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\),
  2. \(\left( 2 + \frac { 3 } { x ^ { 2 } } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2016 June Q5
5 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-2_663_446_1562_847} In the diagram, triangle \(A B C\) is right-angled at \(C\) and \(M\) is the mid-point of \(B C\). It is given that angle \(A B C = \frac { 1 } { 3 } \pi\) radians and angle \(B A M = \theta\) radians. Denoting the lengths of \(B M\) and \(M C\) by \(x\),
  1. find \(A M\) in terms of \(x\),
  2. show that \(\theta = \frac { 1 } { 6 } \pi - \tan ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)\).
CAIE P1 2016 June Q6
6 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-3_552_734_255_703} The diagram shows a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The line \(P T\) is the tangent to the circle at \(P\) and angle \(P O T = \alpha\) radians. The line \(O T\) meets the circle at \(Q\).
  1. Express the perimeter of the shaded region \(P Q T\) in terms of \(r\) and \(\alpha\).
  2. In the case where \(\alpha = \frac { 1 } { 3 } \pi\) and \(r = 10\), find the area of the shaded region correct to 2 significant figures.
CAIE P1 2016 June Q7
7 marks Standard +0.3
7
  1. Prove the identity \(\frac { 1 + \cos \theta } { 1 - \cos \theta } - \frac { 1 - \cos \theta } { 1 + \cos \theta } \equiv \frac { 4 } { \sin \theta \tan \theta }\).
  2. Hence solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\sin \theta \left( \frac { 1 + \cos \theta } { 1 - \cos \theta } - \frac { 1 - \cos \theta } { 1 + \cos \theta } \right) = 3 .$$
CAIE P1 2016 June Q8
8 marks Moderate -0.8
8 Three points have coordinates \(A ( 0,7 ) , B ( 8,3 )\) and \(C ( 3 k , k )\). Find the value of the constant \(k\) for which
  1. \(C\) lies on the line that passes through \(A\) and \(B\),
  2. \(C\) lies on the perpendicular bisector of \(A B\).
CAIE P1 2016 June Q9
9 marks Moderate -0.3
9 A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.
  1. On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
    1. How many litres will be lost on the 30th day after filling?
    2. The tank becomes empty during the \(n\)th day after filling. Find the value of \(n\).
    3. Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by \(10 \%\) on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.
      [0pt] [Questions 10 and 11 are printed on the next page.]
CAIE P1 2016 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826} The diagram shows the part of the curve \(y = \frac { 8 } { x } + 2 x\) for \(x > 0\), and the minimum point \(M\).
  1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y ^ { 2 } \mathrm {~d} x\).
  2. Find the coordinates of \(M\) and determine the coordinates and nature of the stationary point on the part of the curve for which \(x < 0\).
  3. Find the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2016 June Q11
11 marks Moderate -0.3
11 The function f is defined by \(\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \in \mathbb { R }\).
  1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
  2. Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\). The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
  3. Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  4. State the smallest value of \(k\) for which g has an inverse.
  5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). {www.cie.org.uk} after the live examination series. }