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CAIE P1 2003 June Q2
4 marks Moderate -0.8
2 Find all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) which satisfy the equation \(\sin 3 x + 2 \cos 3 x = 0\).
CAIE P1 2003 June Q3
5 marks Easy -1.3
3
  1. Differentiate \(4 x + \frac { 6 } { x ^ { 2 } }\) with respect to \(x\).
  2. Find \(\int \left( 4 x + \frac { 6 } { x ^ { 2 } } \right) \mathrm { d } x\).
CAIE P1 2003 June Q4
5 marks Moderate -0.5
4 In an arithmetic progression, the 1 st term is - 10 , the 15th term is 11 and the last term is 41 . Find the sum of all the terms in the progression.
CAIE P1 2003 June Q5
5 marks Easy -1.2
5 The function f is defined by \(\mathrm { f } : x \mapsto a x + b\), for \(x \in \mathbb { R }\), where \(a\) and \(b\) are constants. It is given that \(f ( 2 ) = 1\) and \(f ( 5 ) = 7\).
  1. Find the values of \(a\) and \(b\).
  2. Solve the equation \(\operatorname { ff } ( x ) = 0\).
CAIE P1 2003 June Q6
5 marks Moderate -0.3
6
  1. Sketch the graph of the curve \(y = 3 \sin x\), for \(- \pi \leqslant x \leqslant \pi\). The straight line \(y = k x\), where \(k\) is a constant, passes through the maximum point of this curve for \(- \pi \leqslant x \leqslant \pi\).
  2. Find the value of \(k\) in terms of \(\pi\).
  3. State the coordinates of the other point, apart from the origin, where the line and the curve intersect.
CAIE P1 2003 June Q7
8 marks Moderate -0.8
7 The line \(L _ { 1 }\) has equation \(2 x + y = 8\). The line \(L _ { 2 }\) passes through the point \(A ( 7,4 )\) and is perpendicular to \(L _ { 1 }\).
  1. Find the equation of \(L _ { 2 }\).
  2. Given that the lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\), find the length of \(A B\).
CAIE P1 2003 June Q8
8 marks Moderate -0.3
8 The points \(A , B , C\) and \(D\) have position vectors \(3 \mathbf { i } + 2 \mathbf { k } , 2 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , 2 \mathbf { j } + 7 \mathbf { k }\) and \(- 2 \mathbf { i } + 10 \mathbf { j } + 7 \mathbf { k }\) respectively.
  1. Use a scalar product to show that \(B A\) and \(B C\) are perpendicular.
  2. Show that \(B C\) and \(A D\) are parallel and find the ratio of the length of \(B C\) to the length of \(A D\).
CAIE P1 2003 June Q9
9 marks Moderate -0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{8214ccb9-0894-4c3c-a8d9-d8f8749fdbe1-3_321_636_267_758} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 8 cm . Angle \(A O B = \theta\) radians.
  1. In the case where \(\theta = 1\), calculate the area of the sector BOC.
  2. Find the value of \(\theta\) for which the perimeter of sector \(A O B\) is one half of the perimeter of sector BOC.
  3. In the case where \(\theta = \frac { 1 } { 3 } \pi\), show that the exact length of the perimeter of triangle \(A B C\) is \(( 24 + 8 \sqrt { } 3 ) \mathrm { cm }\).
CAIE P1 2003 June Q10
10 marks Moderate -0.3
10 The equation of a curve is \(y = \sqrt { } ( 5 x + 4 )\).
  1. Calculate the gradient of the curve at the point where \(x = 1\).
  2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(x\) has the constant value 0.03 units per second. Find the rate of increase of \(y\) at the instant when \(x = 1\).
  3. Find the area enclosed by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).
CAIE P1 2003 June Q11
13 marks Moderate -0.3
11 The equation of a curve is \(y = 8 x - x ^ { 2 }\).
  1. Express \(8 x - x ^ { 2 }\) in the form \(a - ( x + b ) ^ { 2 }\), stating the numerical values of \(a\) and \(b\).
  2. Hence, or otherwise, find the coordinates of the stationary point of the curve.
  3. Find the set of values of \(x\) for which \(y \geqslant - 20\). The function g is defined by \(\mathrm { g } : x \mapsto 8 x - x ^ { 2 }\), for \(x \geqslant 4\).
  4. State the domain and range of \(\mathrm { g } ^ { - 1 }\).
  5. Find an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2004 June Q1
4 marks Easy -1.2
1 A geometric progression has first term 64 and sum to infinity 256. Find
  1. the common ratio,
  2. the sum of the first ten terms.
CAIE P1 2004 June Q2
4 marks Moderate -0.5
2 Evaluate \(\int _ { 0 } ^ { 1 } \sqrt { } ( 3 x + 1 ) \mathrm { d } x\).
CAIE P1 2004 June Q3
5 marks Standard +0.3
3
  1. Show that the equation \(\sin ^ { 2 } \theta + 3 \sin \theta \cos \theta = 4 \cos ^ { 2 } \theta\) can be written as a quadratic equation in \(\tan \theta\).
  2. Hence, or otherwise, solve the equation in part (i) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2004 June Q4
6 marks Moderate -0.8
4 Find the coefficient of \(x ^ { 3 }\) in the expansion of
  1. \(( 1 + 2 x ) ^ { 6 }\),
  2. \(( 1 - 3 x ) ( 1 + 2 x ) ^ { 6 }\).
CAIE P1 2004 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-2_501_682_1302_735} In the diagram, \(O C D\) is an isosceles triangle with \(O C = O D = 10 \mathrm {~cm}\) and angle \(C O D = 0.8\) radians. The points \(A\) and \(B\), on \(O C\) and \(O D\) respectively, are joined by an arc of a circle with centre \(O\) and radius 6 cm . Find
  1. the area of the shaded region,
  2. the perimeter of the shaded region.
CAIE P1 2004 June Q6
8 marks Moderate -0.3
6 The curve \(y = 9 - \frac { 6 } { x }\) and the line \(y + x = 8\) intersect at two points. Find
  1. the coordinates of the two points,
  2. the equation of the perpendicular bisector of the line joining the two points.
CAIE P1 2004 June Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-3_646_841_593_651} The diagram shows part of the graph of \(y = \frac { 18 } { x }\) and the normal to the curve at \(P ( 6,3 )\). This normal meets the \(x\)-axis at \(R\). The point \(Q\) on the \(x\)-axis and the point \(S\) on the curve are such that \(P Q\) and \(S R\) are parallel to the \(y\)-axis.
  1. Find the equation of the normal at \(P\) and show that \(R\) is the point ( \(4 \frac { 1 } { 2 } , 0\) ).
  2. Show that the volume of the solid obtained when the shaded region \(P Q R S\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(18 \pi\).
CAIE P1 2004 June Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817} The diagram shows a glass window consisting of a rectangle of height \(h \mathrm {~m}\) and width \(2 r \mathrm {~m}\) and a semicircle of radius \(r \mathrm {~m}\). The perimeter of the window is 8 m .
  1. Express \(h\) in terms of \(r\).
  2. Show that the area of the window, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$ Given that \(r\) can vary,
  3. find the value of \(r\) for which \(A\) has a stationary value,
  4. determine whether this stationary value is a maximum or a minimum.
CAIE P1 2004 June Q9
10 marks Moderate -0.8
9 Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ - 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } - 1 \\ 0 \\ q \end{array} \right) ,$$ where \(p\) and \(q\) are constants. Find
  1. the unit vector in the direction of \(\overrightarrow { A B }\),
  2. the value of \(p\) for which angle \(A O C = 90 ^ { \circ }\),
  3. the values of \(q\) for which the length of \(\overrightarrow { A D }\) is 7 units.
CAIE P1 2004 June Q10
12 marks Moderate -0.8
10 The functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } : x \mapsto x ^ { 2 } - 2 x , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto 2 x + 3 , & x \in \mathbb { R } . \end{array}$$
  1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 15\).
  2. Find the range of f and state, with a reason, whether f has an inverse.
  3. Show that the equation \(\operatorname { gf } ( x ) = 0\) has no real solutions.
  4. Sketch, in a single diagram, the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
CAIE P1 2005 June Q1
4 marks Easy -1.3
1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 2 } - 5\). Given that the point \(( 3,8 )\) lies on the curve, find the equation of the curve.
CAIE P1 2005 June Q2
4 marks Moderate -0.5
2 Find the gradient of the curve \(y = \frac { 12 } { x ^ { 2 } - 4 x }\) at the point where \(x = 3\).
CAIE P1 2005 June Q3
4 marks Moderate -0.8
3
  1. Show that the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\) can be expressed as \(\tan \theta = 3\).
  2. Hence solve the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2005 June Q4
5 marks Moderate -0.3
4
  1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
  2. Find the value of \(k\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + k x ) ( 2 - x ) ^ { 6 }\).
CAIE P1 2005 June Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-2_591_1061_1098_541} The diagram shows a rhombus \(A B C D\). The points \(B\) and \(D\) have coordinates \(( 2,10 )\) and \(( 6,2 )\) respectively, and \(A\) lies on the \(x\)-axis. The mid-point of \(B D\) is \(M\). Find, by calculation, the coordinates of each of \(M , A\) and \(C\).