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CAIE P1 2013 June Q10
10
  1. The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.
  2. The third term of a geometric progression is four times the first term. The sum of the first six terms is \(k\) times the first term. Find the possible values of \(k\).
CAIE P1 2013 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-4_598_789_255_678} The diagram shows the curve \(y = \sqrt { } ( 1 + 4 x )\), which intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(B\) meets the \(x\)-axis at \(C\). Find
  1. the equation of \(B C\),
  2. the area of the shaded region.
CAIE P1 2013 June Q1
1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 2 x + 5 )\) and \(( 2,5 )\) is a point on the curve. Find the equation of the curve.
CAIE P1 2013 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-2_501_641_461_753} The diagram shows a circle \(C\) with centre \(O\) and radius 3 cm . The radii \(O P\) and \(O Q\) are extended to \(S\) and \(R\) respectively so that \(O R S\) is a sector of a circle with centre \(O\). Given that \(P S = 6 \mathrm {~cm}\) and that the area of the shaded region is equal to the area of circle \(C\),
  1. show that angle \(P O Q = \frac { 1 } { 4 } \pi\) radians,
  2. find the perimeter of the shaded region.
CAIE P1 2013 June Q3
3
  1. Express the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) as a quadratic equation in \(\cos ^ { 2 } \theta\).
  2. Solve the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 \leqslant \theta \leqslant \pi\), giving solutions in terms of \(\pi\).
CAIE P1 2013 June Q4
4
  1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).
CAIE P1 2013 June Q5
5
  1. Sketch, on the same diagram, the curves \(y = \sin 2 x\) and \(y = \cos x - 1\) for \(0 \leqslant x \leqslant 2 \pi\).
  2. Hence state the number of solutions, in the interval \(0 \leqslant x \leqslant 2 \pi\), of the equations
    (a) \(2 \sin 2 x + 1 = 0\),
    (b) \(\sin 2 x - \cos x + 1 = 0\).
CAIE P1 2013 June Q6
6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.
CAIE P1 2013 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_465_554_255_794} The diagram shows three points \(A ( 2,14 ) , B ( 14,6 )\) and \(C ( 7,2 )\). The point \(X\) lies on \(A B\), and \(C X\) is perpendicular to \(A B\). Find, by calculation,
  1. the coordinates of \(X\),
  2. the ratio \(A X : X B\).
CAIE P1 2013 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_716_437_1137_854} The diagram shows a parallelogram \(O A B C\) in which $$\overrightarrow { O A } = \left( \begin{array} { r } 3
3
- 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 5
0
2 \end{array} \right)$$
  1. Use a scalar product to find angle \(B O C\).
  2. Find a vector which has magnitude 35 and is parallel to the vector \(\overrightarrow { O C }\).
CAIE P1 2013 June Q9
9
  1. In an arithmetic progression, the sum, \(S _ { n }\), of the first \(n\) terms is given by \(S _ { n } = 2 n ^ { 2 } + 8 n\). Find the first term and the common difference of the progression.
  2. The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the \(n\)th term respectively of an arithmetic progression. Find the value of \(n\).
CAIE P1 2013 June Q10
10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x + k , x \in \mathbb { R }\), where \(k\) is a constant.
  1. In the case where \(k = 3\), solve the equation \(\mathrm { ff } ( x ) = 25\). The function g is defined by \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x + 8 , x \in \mathbb { R }\).
  2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has no real solutions. The function \(h\) is defined by \(h : x \mapsto x ^ { 2 } - 6 x + 8 , x > 3\).
  3. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2013 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-4_643_570_849_790} The diagram shows part of the curve \(y = \frac { 8 } { \sqrt { } x } - x\) and points \(A ( 1,7 )\) and \(B ( 4,0 )\) which lie on the curve. The tangent to the curve at \(B\) intersects the line \(x = 1\) at the point \(C\).
  1. Find the coordinates of \(C\).
  2. Find the area of the shaded region.
CAIE P1 2014 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-2_750_1287_258_427} The diagram shows part of the graph of \(y = a + b \sin x\). State the values of the constants \(a\) and \(b\). [2
CAIE P1 2014 June Q2
2
  1. Express \(4 x ^ { 2 } - 12 x\) in the form \(( 2 x + a ) ^ { 2 } + b\).
  2. Hence, or otherwise, find the set of values of \(x\) satisfying \(4 x ^ { 2 } - 12 x > 7\).
CAIE P1 2014 June Q3
3 Find the term independent of \(x\) in the expansion of \(\left( 4 x ^ { 3 } + \frac { 1 } { 2 x } \right) ^ { 8 }\).
CAIE P1 2014 June Q4
4 A curve has equation \(y = \frac { 4 } { ( 3 x + 1 ) ^ { 2 } }\). Find the equation of the tangent to the curve at the point where the line \(x = - 1\) intersects the curve.
CAIE P1 2014 June Q5
5 An arithmetic progression has first term \(a\) and common difference \(d\). It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.
  1. Find \(d\) in terms of \(a\).
  2. Find the 100th term in terms of \(a\).
CAIE P1 2014 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-3_625_897_260_623} The diagram shows triangle \(A B C\) in which \(A B\) is perpendicular to \(B C\). The length of \(A B\) is 4 cm and angle \(C A B\) is \(\alpha\) radians. The arc \(D E\) with centre \(A\) and radius 2 cm meets \(A C\) at \(D\) and \(A B\) at \(E\). Find, in terms of \(\alpha\),
  1. the area of the shaded region,
  2. the perimeter of the shaded region.
CAIE P1 2014 June Q7
7 The coordinates of points \(A\) and \(B\) are \(( a , 2 )\) and \(( 3 , b )\) respectively, where \(a\) and \(b\) are constants. The distance \(A B\) is \(\sqrt { } ( 125 )\) units and the gradient of the line \(A B\) is 2 . Find the possible values of \(a\) and of \(b\).
CAIE P1 2014 June Q8
8 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 p
4
p ^ { 2 } \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { c } - p
- 1
p ^ { 2 } \end{array} \right)$$
  1. Find the values of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
  2. For the case where \(p = 3\), find the unit vector in the direction of \(\overrightarrow { B A }\).
CAIE P1 2014 June Q9
9
  1. Prove the identity \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } \equiv \frac { 1 } { \tan \theta }\).
  2. Hence solve the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 4 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2014 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-4_819_812_255_662} The diagram shows the function f defined for \(- 1 \leqslant x \leqslant 4\), where $$f ( x ) = \begin{cases} 3 x - 2 & \text { for } - 1 \leqslant x \leqslant 1
\frac { 4 } { 5 - x } & \text { for } 1 < x \leqslant 4 \end{cases}$$
  1. State the range of f .
  2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
  3. Obtain expressions to define the function \(\mathrm { f } ^ { - 1 }\), giving also the set of values for which each expression is valid.
CAIE P1 2014 June Q11
11 A line has equation \(y = 2 x + c\) and a curve has equation \(y = 8 - 2 x - x ^ { 2 }\).
  1. For the case where the line is a tangent to the curve, find the value of the constant \(c\).
  2. For the case where \(c = 11\), find the \(x\)-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.
CAIE P1 2014 June Q12
12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).
  1. Find the equation of the curve.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. Find the coordinates of the stationary point and determine its nature.