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CAIE P1 2011 June Q6
6 The variables \(x , y\) and \(z\) can take only positive values and are such that $$z = 3 x + 2 y \quad \text { and } \quad x y = 600 .$$
  1. Show that \(z = 3 x + \frac { 1200 } { x }\).
  2. Find the stationary value of \(z\) and determine its nature.
CAIE P1 2011 June Q7
7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { ( 1 + 2 x ) ^ { 2 } }\) and the point \(\left( 1 , \frac { 1 } { 2 } \right)\) lies on the curve.
  1. Find the equation of the curve.
  2. Find the set of values of \(x\) for which the gradient of the curve is less than \(\frac { 1 } { 3 }\).
CAIE P1 2011 June Q8
8 A television quiz show takes place every day. On day 1 the prize money is \(
) 1000$. If this is not won the prize money is increased for day 2 . The prize money is increased in a similar way every day until it is won. The television company considered the following two different models for increasing the prize money. Model 1: Increase the prize money by \(
) 1000$ each day.
Model 2: Increase the prize money by \(10 \%\) each day.
On each day that the prize money is not won the television company makes a donation to charity. The amount donated is \(5 \%\) of the value of the prize on that day. After 40 days the prize money has still not been won. Calculate the total amount donated to charity
  1. if Model 1 is used,
  2. if Model 2 is used.
CAIE P1 2011 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-3_387_1175_1781_486} In the diagram, \(O A B\) is an isosceles triangle with \(O A = O B\) and angle \(A O B = 2 \theta\) radians. Arc \(P S T\) has centre \(O\) and radius \(r\), and the line \(A S B\) is a tangent to the \(\operatorname { arc } P S T\) at \(S\).
  1. Find the total area of the shaded regions in terms of \(r\) and \(\theta\).
  2. In the case where \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 6\), find the total perimeter of the shaded regions, leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).
CAIE P1 2011 June Q10
10
  1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
  2. Find the coordinates of \(Q\).
  3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).
CAIE P1 2011 June Q11
11 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2 \end{aligned}$$
  1. Find and simplify expressions for \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
  2. Hence find the value of \(a\) for which \(\mathrm { fg } ( a ) = \mathrm { gf } ( a )\).
  3. Find the value of \(b ( b \neq a )\) for which \(\mathrm { g } ( b ) = b\).
  4. Find and simplify an expression for \(\mathrm { f } ^ { - 1 } \mathrm {~g} ( x )\). The function h is defined by $$\mathrm { h } : x \mapsto x ^ { 2 } - 2 , \quad \text { for } x \leqslant 0$$
  5. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2011 June Q1
1 Find \(\int \left( x ^ { 3 } + \frac { 1 } { x ^ { 3 } } \right) \mathrm { d } x\).
CAIE P1 2011 June Q2
2
  1. Find the terms in \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(\left( 1 - \frac { 3 } { 2 } x \right) ^ { 6 }\).
  2. Given that there is no term in \(x ^ { 3 }\) in the expansion of \(( k + 2 x ) \left( 1 - \frac { 3 } { 2 } x \right) ^ { 6 }\), find the value of the constant \(k\).
CAIE P1 2011 June Q3
3 The equation \(x ^ { 2 } + p x + q = 0\), where \(p\) and \(q\) are constants, has roots - 3 and 5 .
  1. Find the values of \(p\) and \(q\).
  2. Using these values of \(p\) and \(q\), find the value of the constant \(r\) for which the equation \(x ^ { 2 } + p x + q + r = 0\) has equal roots.
CAIE P1 2011 June Q4
4 A curve has equation \(y = \frac { 4 } { 3 x - 4 }\) and \(P ( 2,2 )\) is a point on the curve.
  1. Find the equation of the tangent to the curve at \(P\).
  2. Find the angle that this tangent makes with the \(x\)-axis.
CAIE P1 2011 June Q5
5
  1. Prove the identity \(\frac { \cos \theta } { \tan \theta ( 1 - \sin \theta ) } \equiv 1 + \frac { 1 } { \sin \theta }\).
  2. Hence solve the equation \(\frac { \cos \theta } { \tan \theta ( 1 - \sin \theta ) } = 4\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2011 June Q6
6 The function f is defined by \(\mathrm { f } : x \mapsto \frac { x + 3 } { 2 x - 1 } , x \in \mathbb { R } , x \neq \frac { 1 } { 2 }\).
  1. Show that \(\operatorname { ff } ( x ) = x\).
  2. Hence, or otherwise, obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2011 June Q7
7 The line \(L _ { 1 }\) passes through the points \(A ( 2,5 )\) and \(B ( 10,9 )\). The line \(L _ { 2 }\) is parallel to \(L _ { 1 }\) and passes through the origin. The point \(C\) lies on \(L _ { 2 }\) such that \(A C\) is perpendicular to \(L _ { 2 }\). Find
  1. the coordinates of \(C\),
  2. the distance \(A C\).
CAIE P1 2011 June Q8
8 Relative to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2
3
5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
2
3 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }
CAIE P1 2011 June Q10
10
0
6 \end{array} \right)$$
  1. Find angle \(A B C\). The point \(D\) is such that \(A B C D\) is a parallelogram.
  2. Find the position vector of \(D\). 9 The function f is such that \(\mathrm { f } ( x ) = 3 - 4 \cos ^ { k } x\), for \(0 \leqslant x \leqslant \pi\), where \(k\) is a constant.
  3. In the case where \(k = 2\),
    (a) find the range of f,
    (b) find the exact solutions of the equation \(\mathrm { f } ( x ) = 1\).
  4. In the case where \(k = 1\),
    (a) sketch the graph of \(y = \mathrm { f } ( x )\),
    (b) state, with a reason, whether f has an inverse. 10 (a) A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm , find the perimeter of the smallest sector.
    (b) The first, second and third terms of a geometric progression are \(2 k + 3 , k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
  5. the value of the constant \(k\),
  6. the sum to infinity of the progression.
CAIE P1 2011 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{5e2a6be8-b260-4b03-90c5-7485adbc39cc-3_513_1023_1838_561} The diagram shows part of the curve \(y = 4 \sqrt { } x - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).
  1. Find the coordinates of \(A\) and \(M\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).
CAIE P1 2011 June Q1
1 The coefficient of \(x ^ { 3 }\) in the expansion of \(( a + x ) ^ { 5 } + ( 1 - 2 x ) ^ { 6 }\), where \(a\) is positive, is 90 . Find the value of \(a\).
CAIE P1 2011 June Q2
2 Find the set of values of \(m\) for which the line \(y = m x + 4\) intersects the curve \(y = 3 x ^ { 2 } - 4 x + 7\) at two distinct points.
CAIE P1 2011 June Q3
3 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Given that \(P Q = \sqrt { } ( 45 )\) and that the gradient of the line \(P Q\) is \(- \frac { 1 } { 2 }\), find the values of \(a\) and \(b\).
CAIE P1 2011 June Q4
4
  1. Differentiate \(\frac { 2 x ^ { 3 } + 5 } { x }\) with respect to \(x\).
  2. Find \(\int ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\) and hence find the value of \(\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\).
CAIE P1 2011 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-2_748_1155_1146_495} In the diagram, \(O A B C D E F G\) is a rectangular block in which \(O A = O D = 6 \mathrm {~cm}\) and \(A B = 12 \mathrm {~cm}\). The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is the mid-point of \(D G , Q\) is the centre of the square face \(C B F G\) and \(R\) lies on \(A B\) such that \(A R = 4 \mathrm {~cm}\).
  1. Express each of the vectors \(\overrightarrow { P Q }\) and \(\overrightarrow { R Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(R Q P\).
CAIE P1 2011 June Q6
6
  1. A geometric progression has a third term of 20 and a sum to infinity which is three times the first term. Find the first term.
  2. An arithmetic progression is such that the eighth term is three times the third term. Show that the sum of the first eight terms is four times the sum of the first four terms.
CAIE P1 2011 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-3_462_956_258_593} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius 6 cm , and angle \(A O B = \frac { 1 } { 3 } \pi\) radians. The line \(A X\) is a tangent to the circle at \(A\), and \(O B X\) is a straight line.
  1. Show that the exact length of \(A X\) is \(6 \sqrt { } 3 \mathrm {~cm}\). Find, in terms of \(\pi\) and \(\sqrt { } 3\),
  2. the area of the shaded region,
  3. the perimeter of the shaded region.
CAIE P1 2011 June Q8
8
  1. Prove the identity \(\left( \frac { 1 } { \sin \theta } - \frac { 1 } { \tan \theta } \right) ^ { 2 } \equiv \frac { 1 - \cos \theta } { 1 + \cos \theta }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \sin \theta } - \frac { 1 } { \tan \theta } \right) ^ { 2 } = \frac { 2 } { 5 }\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2011 June Q9
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { } x } - 1\) and \(P ( 9,5 )\) is a point on the curve.
  1. Find the equation of the curve.
  2. Find the coordinates of the stationary point on the curve.
  3. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and determine the nature of the stationary point.
  4. The normal to the curve at \(P\) makes an angle of \(\tan ^ { - 1 } k\) with the positive \(x\)-axis. Find the value of \(k\).