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CAIE P1 2013 November Q2
4 marks Moderate -0.8
2 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { - \frac { 3 } { 2 } } + 1\) and that \(\mathrm { f } ( 4 ) = 5\). Find \(\mathrm { f } ( x )\).
CAIE P1 2013 November Q3
5 marks Moderate -0.8
3 The point \(A\) has coordinates \(( 3,1 )\) and the point \(B\) has coordinates \(( - 21,11 )\). The point \(C\) is the mid-point of \(A B\).
  1. Find the equation of the line through \(A\) that is perpendicular to \(y = 2 x - 7\).
  2. Find the distance \(A C\).
CAIE P1 2013 November Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-2_711_643_900_753} The diagram shows a pyramid \(O A B C\) in which the edge \(O C\) is vertical. The horizontal base \(O A B\) is a triangle, right-angled at \(O\), and \(D\) is the mid-point of \(A B\). The edges \(O A , O B\) and \(O C\) have lengths of 8 units, 6 units and 10 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O B }\) and \(\overrightarrow { O C }\) respectively.
  1. Express each of the vectors \(\overrightarrow { O D }\) and \(\overrightarrow { C D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle ODC.
CAIE P1 2013 November Q5
6 marks Moderate -0.8
5
  1. In a geometric progression, the sum to infinity is equal to eight times the first term. Find the common ratio.
  2. In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference.
CAIE P1 2013 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-3_463_621_255_762} The diagram shows sector \(O A B\) with centre \(O\) and radius 11 cm . Angle \(A O B = \alpha\) radians. Points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. Arc \(C D\) has centre \(O\) and radius 5 cm .
  1. The area of the shaded region \(A B D C\) is equal to \(k\) times the area of the unshaded region \(O C D\). Find \(k\).
  2. The perimeter of the shaded region \(A B D C\) is equal to twice the perimeter of the unshaded region \(O C D\). Find the exact value of \(\alpha\).
CAIE P1 2013 November Q7
7 marks Moderate -0.3
7
  1. Find the possible values of \(x\) for which \(\sin ^ { - 1 } \left( x ^ { 2 } - 1 \right) = \frac { 1 } { 3 } \pi\), giving your answers correct to 3 decimal places.
  2. Solve the equation \(\sin \left( 2 \theta + \frac { 1 } { 3 } \pi \right) = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant \pi\), giving \(\theta\) in terms of \(\pi\) in your answers.
CAIE P1 2013 November Q8
8 marks Moderate -0.3
8
  1. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 4 }\).
  2. Find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left( x + 3 x ^ { 2 } \right) ^ { 5 }\).
  3. Hence find the coefficient of \(x ^ { 8 }\) in the expansion of \(\left[ 1 + \left( x + 3 x ^ { 2 } \right) \right] ^ { 5 }\).
CAIE P1 2013 November Q9
8 marks Standard +0.3
9 A curve has equation \(y = \frac { k ^ { 2 } } { x + 2 } + x\), where \(k\) is a positive constant. Find, in terms of \(k\), the values of \(x\) for which the curve has stationary points and determine the nature of each stationary point.
CAIE P1 2013 November Q10
9 marks Standard +0.3
10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant c\), where \(c\) is a constant. It is given that f is a one-one function.
  1. State the range of f in terms of \(c\) and find the smallest possible value of \(c\). The function g is defined by \(\mathrm { g } : x \mapsto a x + b\) for \(x \geqslant 0\), where \(a\) and \(b\) are positive constants. It is given that, when \(c = 0 , \operatorname { gf } ( 1 ) = 11\) and \(\operatorname { fg } ( 1 ) = 21\).
  2. Write down two equations in \(a\) and \(b\) and solve them to find the values of \(a\) and \(b\).
CAIE P1 2013 November Q11
12 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-4_547_1057_255_543} The diagram shows the curve \(y = \sqrt { } \left( x ^ { 4 } + 4 x + 4 \right)\).
  1. Find the equation of the tangent to the curve at the point ( 0,2 ).
  2. Show that the \(x\)-coordinates of the points of intersection of the line \(y = x + 2\) and the curve are given by the equation \(( x + 2 ) ^ { 2 } = x ^ { 4 } + 4 x + 4\). Hence find these \(x\)-coordinates.
  3. The region shaded in the diagram is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of revolution.
CAIE P1 2014 November Q1
3 marks Moderate -0.8
1 In the expansion of \(( 2 + a x ) ^ { 7 }\), the coefficient of \(x\) is equal to the coefficient of \(x ^ { 2 }\). Find the value of the non-zero constant \(a\).
CAIE P1 2014 November Q2
3 marks Standard +0.8
2 Find the value of \(x\) satisfying the equation \(\sin ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } ( 3 )\).
CAIE P1 2014 November Q3
4 marks Standard +0.8
3 Solve the equation \(\frac { 13 \sin ^ { 2 } \theta } { 2 + \cos \theta } + \cos \theta = 2\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2014 November Q4
5 marks Easy -1.2
4 The line \(4 x + k y = 20\) passes through the points \(A ( 8 , - 4 )\) and \(B ( b , 2 b )\), where \(k\) and \(b\) are constants.
  1. Find the values of \(k\) and \(b\).
  2. Find the coordinates of the mid-point of \(A B\).
CAIE P1 2014 November Q5
5 marks Standard +0.3
5 Find the set of values of \(k\) for which the line \(y = 2 x - k\) meets the curve \(y = x ^ { 2 } + k x - 2\) at two distinct points.
CAIE P1 2014 November Q6
7 marks Moderate -0.3
6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).
  1. Show that angle \(O A B\) is a right angle.
  2. Find the area of triangle \(O A B\).
CAIE P1 2014 November Q7
7 marks Standard +0.3
7
  1. A geometric progression has first term \(a ( a \neq 0 )\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(2 r\) and sum to infinity \(3 S\). Find the value of \(r\).
  2. An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 . Find the value of \(n\).
CAIE P1 2014 November Q8
8 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-3_408_686_264_731} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius 4 cm . Angle \(A O B\) is \(\alpha\) radians. The point \(D\) on \(O B\) is such that \(A D\) is perpendicular to \(O B\). The arc \(D C\), with centre \(O\), meets \(O A\) at \(C\).
  1. Find an expression in terms of \(\alpha\) for the perimeter of the shaded region \(A B D C\).
  2. For the case where \(\alpha = \frac { 1 } { 6 } \pi\), find the area of the shaded region \(A B D C\), giving your answer in the form \(k \pi\), where \(k\) is a constant to be determined.
CAIE P1 2014 November Q9
11 marks Moderate -0.8
9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).
  1. Find the equation of the normal to the curve at \(P\).
  2. Find the equation of the curve.
  3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.
CAIE P1 2014 November Q10
10 marks Moderate -0.8
10
  1. Express \(x ^ { 2 } - 2 x - 15\) in the form \(( x + a ) ^ { 2 } + b\). The function f is defined for \(p \leqslant x \leqslant q\), where \(p\) and \(q\) are positive constants, by $$f : x \mapsto x ^ { 2 } - 2 x - 15$$ The range of f is given by \(c \leqslant \mathrm { f } ( x ) \leqslant d\), where \(c\) and \(d\) are constants.
  2. State the smallest possible value of \(c\). For the case where \(c = 9\) and \(d = 65\),
  3. find \(p\) and \(q\),
  4. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2014 November Q11
12 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-4_995_905_260_621} The diagram shows parts of the curves \(y = ( 4 x + 1 ) ^ { \frac { 1 } { 2 } }\) and \(y = \frac { 1 } { 2 } x ^ { 2 } + 1\) intersecting at points \(P ( 0,1 )\) and \(Q ( 2,3 )\). The angle between the tangents to the two curves at \(Q\) is \(\alpha\).
  1. Find \(\alpha\), giving your answer in degrees correct to 3 significant figures.
  2. Find by integration the area of the shaded region.
CAIE P1 2015 November Q1
3 marks Standard +0.3
1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.
CAIE P1 2015 November Q2
3 marks Easy -1.3
2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).
CAIE P1 2015 November Q3
4 marks Standard +0.3
3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).
CAIE P1 2015 November Q4
6 marks Moderate -0.3
4
  1. Show that the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) can be expressed as $$4 \sin ^ { 2 } \theta - 15 \sin \theta - 4 = 0$$
  2. Hence solve the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).