Questions — CAIE P1 (1202 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE P1 2012 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_502_663_255_740} The diagram shows part of the curve \(y = \frac { 9 } { 2 x + 3 }\), crossing the \(y\)-axis at the point \(B ( 0,3 )\). The point \(A\) on the curve has coordinates \(( 3,1 )\) and the tangent to the curve at \(A\) crosses the \(y\)-axis at \(C\).
  1. Find the equation of the tangent to the curve at \(A\).
  2. Determine, showing all necessary working, whether \(C\) is nearer to \(B\) or to \(O\).
  3. Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2012 November Q10
10 A curve is defined for \(x > 0\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x + \frac { 4 } { x ^ { 2 } }\). The point \(P ( 4,8 )\) lies on the curve.
  1. Find the equation of the curve.
  2. Show that the gradient of the curve has a minimum value when \(x = 2\) and state this minimum value.
CAIE P1 2012 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_611_668_1699_737} The diagram shows a sector of a circle with centre \(O\) and radius 20 cm . A circle with centre \(C\) and radius \(x \mathrm {~cm}\) lies within the sector and touches it at \(P , Q\) and \(R\). Angle \(P O R = 1.2\) radians.
  1. Show that \(x = 7.218\), correct to 3 decimal places.
  2. Find the total area of the three parts of the sector lying outside the circle with centre \(C\).
  3. Find the perimeter of the region \(O P S R\) bounded by the \(\operatorname { arc } P S R\) and the lines \(O P\) and \(O R\).
CAIE P1 2012 November Q1
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 2 - \frac { 1 } { 2 } x \right) ^ { 7 }\).
CAIE P1 2012 November Q2
2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.
CAIE P1 2012 November Q3
3 Solve the equation \(7 \cos x + 5 = 2 \sin ^ { 2 } x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-2_478_828_708_660} In the diagram, \(D\) lies on the side \(A B\) of triangle \(A B C\) and \(C D\) is an arc of a circle with centre \(A\) and radius 2 cm . The line \(B C\) is of length \(2 \sqrt { } 3 \mathrm {~cm}\) and is perpendicular to \(A C\). Find the area of the shaded region \(B D C\), giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
CAIE P1 2012 November Q5
5 The first term of a geometric progression is \(5 \frac { 1 } { 3 }\) and the fourth term is \(2 \frac { 1 } { 4 }\). Find
  1. the common ratio,
  2. the sum to infinity.
CAIE P1 2012 November Q6
6 The functions f and g are defined for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\) by $$\begin{aligned} & f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 6 } \pi
& g ( x ) = \cos x \end{aligned}$$ Solve the following equations for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. \(\operatorname { gf } ( x ) = 1\), giving your answer in terms of \(\pi\).
  2. \(\operatorname { fg } ( x ) = 1\), giving your answers correct to 2 decimal places.
CAIE P1 2012 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}
  1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
  2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p
    5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.
CAIE P1 2012 November Q8
8 A curve is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( 3 x + 4 ) ^ { \frac { 3 } { 2 } } - 6 x - 8 .$$
  1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Verify that the curve has a stationary point when \(x = - 1\) and determine its nature.
  3. It is now given that the stationary point on the curve has coordinates \(( - 1,5 )\). Find the equation of the curve.
CAIE P1 2012 November Q9
9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } p
1
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 4
2
p \end{array} \right)$$ where \(p\) is a constant.
  1. In the case where \(O A B\) is a straight line, state the value of \(p\) and find the unit vector in the direction of \(\overrightarrow { O A }\).
  2. In the case where \(O A\) is perpendicular to \(A B\), find the possible values of \(p\).
  3. In the case where \(p = 3\), the point \(C\) is such that \(O A B C\) is a parallelogram. Find the position vector of \(C\).
CAIE P1 2012 November Q10
10 A straight line has equation \(y = - 2 x + k\), where \(k\) is a constant, and a curve has equation \(y = \frac { 2 } { x - 3 }\).
  1. Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2 x ^ { 2 } - ( 6 + k ) x + ( 2 + 3 k ) = 0\).
  2. Find the two values of \(k\) for which the line is a tangent to the curve. The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).
  3. Find the coordinates of \(A\) and \(B\) and the equation of the line \(A B\).
CAIE P1 2012 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-4_526_974_822_587} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.
  1. State the value of \(a\).
  2. Find the value of \(b\).
  3. Find the area of the shaded region.
  4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Find the value of \(m\).
CAIE P1 2013 November Q1
1
  1. Find the first three terms when \(( 2 + 3 x ) ^ { 6 }\) is expanded in ascending powers of \(x\).
  2. In the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is zero. Find the value of \(a\).
CAIE P1 2013 November Q2
2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).
CAIE P1 2013 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-2_397_949_657_596} The diagram shows a pyramid \(O A B C D\) in which the vertical edge \(O D\) is 3 units in length. The point \(E\) is the centre of the horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 6 units and 4 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively.
  1. Express each of the vectors \(\overrightarrow { D B }\) and \(\overrightarrow { D E }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(B D E\).
CAIE P1 2013 November Q4
4
  1. Solve the equation \(4 \sin ^ { 2 } x + 8 \cos x - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Hence find the solution of the equation \(4 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) + 8 \cos \left( \frac { 1 } { 2 } \theta \right) - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2013 November Q5
5 The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } + 1 \text { for } x \geqslant 0$$
  1. Define in a similar way the inverse function \(\mathrm { f } ^ { - 1 }\).
  2. Solve the equation \(\operatorname { ff } ( x ) = \frac { 185 } { 16 }\).
CAIE P1 2013 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_412_629_258_758} The diagram shows a metal plate made by fixing together two pieces, \(O A B C D\) (shaded) and \(O A E D\) (unshaded). The piece \(O A B C D\) is a minor sector of a circle with centre \(O\) and radius \(2 r\). The piece \(O A E D\) is a major sector of a circle with centre \(O\) and radius \(r\). Angle \(A O D\) is \(\alpha\) radians. Simplifying your answers where possible, find, in terms of \(\alpha , \pi\) and \(r\),
  1. the perimeter of the metal plate,
  2. the area of the metal plate. It is now given that the shaded and unshaded pieces are equal in area.
  3. Find \(\alpha\) in terms of \(\pi\).
CAIE P1 2013 November Q7
7 The point \(A\) has coordinates ( \(- 1,6\) ) and the point \(B\) has coordinates (7,2).
  1. Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(y = m x + c\).
  2. A point \(C\) on the perpendicular bisector has coordinates \(( p , q )\). The distance \(O C\) is 2 units, where \(O\) is the origin. Write down two equations involving \(p\) and \(q\) and hence find the coordinates of the possible positions of \(C\).
CAIE P1 2013 November Q8
5 marks
8
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_365_663_1813_740} The inside lane of a school running track consists of two straight sections each of length \(x\) metres, and two semicircular sections each of radius \(r\) metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres.
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the region enclosed by the inside lane is given by \(A = 400 r - \pi r ^ { 2 }\).
  2. Given that \(x\) and \(r\) can vary, show that, when \(A\) has a stationary value, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum. [5]
CAIE P1 2013 November Q9
9
  1. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000 . Find the common difference and the first term.
  2. A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity 6. A second geometric progression has first term \(2 a\), common ratio \(r ^ { 2 }\) and sum to infinity 7 . Find the values of \(a\) and \(r\).
CAIE P1 2013 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-4_654_974_614_587} The diagram shows the curve \(y = ( 3 - 2 x ) ^ { 3 }\) and the tangent to the curve at the point \(\left( \frac { 1 } { 2 } , 8 \right)\).
  1. Find the equation of this tangent, giving your answer in the form \(y = m x + c\).
  2. Find the area of the shaded region.
CAIE P1 2013 November Q1
1 Given that \(\cos x = p\), where \(x\) is an acute angle in degrees, find, in terms of \(p\),
  1. \(\sin x\),
  2. \(\tan x\),
  3. \(\tan \left( 90 ^ { \circ } - x \right)\).