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CAIE P1 2004 June Q1
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
2 Evaluate \(\int _ { 0 } ^ { 1 } \sqrt { } ( 3 x + 1 ) \mathrm { d } x\).
CAIE P1 2004 June Q3
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
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
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
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
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
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
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
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
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
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
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
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
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 \(C\).
CAIE P1 2005 June Q6
6 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.
CAIE P1 2005 June Q7
7 A function f is defined by f : \(x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. Find the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }\), where \(A\) is a constant.
  3. State the largest value of \(A\) for which g has an inverse.
  4. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2005 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669} In the diagram, \(A B C\) is a semicircle, centre \(O\) and radius 9 cm . The line \(B D\) is perpendicular to the diameter \(A C\) and angle \(A O B = 2.4\) radians.
  1. Show that \(B D = 6.08 \mathrm {~cm}\), correct to 3 significant figures.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
CAIE P1 2005 June Q9
9 A curve has equation \(y = \frac { 4 } { \sqrt { } x }\).
  1. The normal to the curve at the point \(( 4,2 )\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(P Q\), correct to 3 significant figures.
  2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
CAIE P1 2005 June Q10
10 The equation of a curve is \(y = x ^ { 2 } - 3 x + 4\).
  1. Show that the whole of the curve lies above the \(x\)-axis.
  2. Find the set of values of \(x\) for which \(x ^ { 2 } - 3 x + 4\) is a decreasing function of \(x\). The equation of a line is \(y + 2 x = k\), where \(k\) is a constant.
  3. In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.
  4. Find the value of \(k\) for which the line is a tangent to the curve.
CAIE P1 2005 June Q11
11 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$
  1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
  2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
  3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\). \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed.
    University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2006 June Q1
1 A curve has equation \(y = \frac { k } { x }\). Given that the gradient of the curve is - 3 when \(x = 2\), find the value of the constant \(k\).
CAIE P1 2006 June Q2
2 Solve the equation $$\sin 2 x + 3 \cos 2 x = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2006 June Q3
3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(
) 5000$. Find
  1. the grant given in 2011,
  2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.
CAIE P1 2006 June Q4
4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).