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 2010 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-3_314_803_751_671} The diagram shows a metal plate consisting of a rectangle with sides \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and a quarter-circle of radius \(x \mathrm {~cm}\). The perimeter of the plate is 60 cm .
  1. Express \(y\) in terms of \(x\).
  2. Show that the area of the plate, \(A \mathrm {~cm} ^ { 2 }\), is given by \(A = 30 x - x ^ { 2 }\). Given that \(x\) can vary,
  3. find the value of \(x\) at which \(A\) is stationary,
  4. find this stationary value of \(A\), and determine whether it is a maximum or a minimum value.
CAIE P1 2010 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-4_837_1020_255_559} The diagram shows two circles, \(C _ { 1 }\) and \(C _ { 2 }\), touching at the point \(T\). Circle \(C _ { 1 }\) has centre \(P\) and radius 8 cm ; circle \(C _ { 2 }\) has centre \(Q\) and radius 2 cm . Points \(R\) and \(S\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively, and \(R S\) is a tangent to both circles.
  1. Show that \(R S = 8 \mathrm {~cm}\).
  2. Find angle \(R P Q\) in radians correct to 4 significant figures.
  3. Find the area of the shaded region.
CAIE P1 2010 November Q10
10 The equation of a curve is \(y = 3 + 4 x - x ^ { 2 }\).
  1. Show that the equation of the normal to the curve at the point \(( 3,6 )\) is \(2 y = x + 9\).
  2. Given that the normal meets the coordinate axes at points \(A\) and \(B\), find the coordinates of the mid-point of \(A B\).
  3. Find the coordinates of the point at which the normal meets the curve again.
CAIE P1 2010 November Q11
11 The equation of a curve is \(y = \frac { 9 } { 2 - x }\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine, with a reason, whether the curve has any stationary points.
  2. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  3. Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2010 November Q1
1
  1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 2 - x ^ { 2 } \right) \left( 1 - 2 x ^ { 2 } \right) ^ { 8 }\).
CAIE P1 2010 November Q2
2 Prove the identity $$\tan ^ { 2 } x - \sin ^ { 2 } x \equiv \tan ^ { 2 } x \sin ^ { 2 } x$$
CAIE P1 2010 November Q3
3 The length, \(x\) metres, of a Green Anaconda snake which is \(t\) years old is given approximately by the formula $$x = 0.7 \sqrt { } ( 2 t - 1 ) ,$$ where \(1 \leqslant t \leqslant 10\). Using this formula, find
  1. \(\frac { \mathrm { d } x } { \mathrm {~d} t }\),
  2. the rate of growth of a Green Anaconda snake which is 5 years old.
CAIE P1 2010 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-2_720_645_1183_751} The diagram shows points \(A , C , B , P\) on the circumference of a circle with centre \(O\) and radius 3 cm . Angle \(A O C =\) angle \(B O C = 2.3\) radians.
  1. Find angle \(A O B\) in radians, correct to 4 significant figures.
  2. Find the area of the shaded region \(A C B P\), correct to 3 significant figures.
CAIE P1 2010 November Q5
5
  1. The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum of the first \(m\) terms is zero. Find the value of \(m\).
  2. A geometric progression, in which all the terms are positive, has common ratio \(r\). The sum of the first \(n\) terms is less than \(90 \%\) of the sum to infinity. Show that \(r ^ { n } > 0.1\).
CAIE P1 2010 November Q6
6 A curve has equation \(y = k x ^ { 2 } + 1\) and a line has equation \(y = k x\), where \(k\) is a non-zero constant.
  1. Find the set of values of \(k\) for which the curve and the line have no common points.
  2. State the value of \(k\) for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.
CAIE P1 2010 November Q7
7 The function f is defined by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7 \text { for } x > 2$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\) and hence state the range of f .
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } ( x ) = x - 2 \text { for } x > 2$$ The function h is such that \(\mathrm { f } = \mathrm { hg }\) and the domain of h is \(x > 0\).
  3. Obtain an expression for \(\mathrm { h } ( x )\).
CAIE P1 2010 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-3_613_897_1311_623} The diagram shows part of the curve \(y = \frac { 2 } { 1 - x }\) and the line \(y = 3 x + 4\). The curve and the line meet at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\).
  2. Find the length of the line \(A B\) and the coordinates of the mid-point of \(A B\).
CAIE P1 2010 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_582_1072_255_541} The diagram shows a pyramid \(O A B C P\) in which the horizontal base \(O A B C\) is a square of side 10 cm and the vertex \(P\) is 10 cm vertically above \(O\). The points \(D , E , F , G\) lie on \(O P , A P , B P , C P\) respectively and \(D E F G\) is a horizontal square of side 6 cm . The height of \(D E F G\) above the base is \(a \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.
  1. Show that \(a = 4\).
  2. Express the vector \(\overrightarrow { B G }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  3. Use a scalar product to find angle \(G B A\).
CAIE P1 2010 November Q10
5 marks
10
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_433_969_1475_587} The diagram shows an open rectangular tank of height \(h\) metres covered with a lid. The base of the tank has sides of length \(x\) metres and \(\frac { 1 } { 2 } x\) metres and the lid is a rectangle with sides of length \(\frac { 5 } { 4 } x\) metres and \(\frac { 4 } { 5 } x\) metres. When full the tank holds \(4 \mathrm {~m} ^ { 3 }\) of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is \(A \mathrm {~m} ^ { 2 }\).
  1. Express \(h\) in terms of \(x\) and hence show that \(A = \frac { 3 } { 2 } x ^ { 2 } + \frac { 24 } { x }\).
  2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) is a minimum, showing clearly that \(A\) is a minimum and not a maximum.
    [0pt] [5]
CAIE P1 2010 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-5_609_897_255_625} The diagram shows part of the curve \(y = \frac { 1 } { ( 3 x + 1 ) ^ { \frac { 1 } { 4 } } }\). The curve cuts the \(y\)-axis at \(A\) and the line \(x = 5\) at \(B\).
  1. Show that the equation of the line \(A B\) is \(y = - \frac { 1 } { 10 } x + 1\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2010 November Q1
1 Find the term independent of \(x\) in the expansion of \(\left( x - \frac { 1 } { x ^ { 2 } } \right) ^ { 9 }\).
CAIE P1 2010 November Q2
2 Points \(A , B\) and \(C\) have coordinates \(( 2,5 ) , ( 5 , - 1 )\) and \(( 8,6 )\) respectively.
  1. Find the coordinates of the mid-point of \(A B\).
  2. Find the equation of the line through \(C\) perpendicular to \(A B\). Give your answer in the form \(a x + b y + c = 0\).
CAIE P1 2010 November Q3
3 Solve the equation \(15 \sin ^ { 2 } x = 13 + \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2010 November Q4
4
  1. Sketch the curve \(y = 2 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
  2. By adding a suitable straight line to your sketch, determine the number of real roots of the equation $$2 \pi \sin x = \pi - x$$ State the equation of the straight line.
CAIE P1 2010 November Q5
5 A curve has equation \(y = \frac { 1 } { x - 3 } + x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the coordinates of the maximum point \(A\) and the minimum point \(B\) on the curve.
CAIE P1 2010 November Q6
6 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x - 5\).
  1. Find the set of values of \(x\) for which f is an increasing function.
  2. Given that the curve passes through \(( 1,3 )\), find \(\mathrm { f } ( x )\).
CAIE P1 2010 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_778_816_255_662} The diagram shows the function f defined for \(0 \leqslant x \leqslant 6\) by $$\begin{aligned} & x \mapsto \frac { 1 } { 2 } x ^ { 2 } \quad \text { for } 0 \leqslant x \leqslant 2 ,
& x \mapsto \frac { 1 } { 2 } x + 1 \text { for } 2 < x \leqslant 6 . \end{aligned}$$
  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 \(\mathrm { f } ^ { - 1 } ( x )\), giving the set of values of \(x\) for which each expression is valid.
CAIE P1 2010 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_600_883_1731_630} The diagram shows a rhombus \(A B C D\). Points \(P\) and \(Q\) lie on the diagonal \(A C\) such that \(B P D\) is an arc of a circle with centre \(C\) and \(B Q D\) is an arc of a circle with centre \(A\). Each side of the rhombus has length 5 cm and angle \(B A D = 1.2\) radians.
  1. Find the area of the shaded region \(B P D Q\).
  2. Find the length of \(P Q\).
CAIE P1 2010 November Q9
9
  1. A geometric progression has first term 100 and sum to infinity 2000. Find the second term.
  2. An arithmetic progression has third term 90 and fifth term 80 .
    1. Find the first term and the common difference.
    2. Find the value of \(m\) given that the sum of the first \(m\) terms is equal to the sum of the first ( \(m + 1\) ) terms.
    3. Find the value of \(n\) given that the sum of the first \(n\) terms is zero.
CAIE P1 2010 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-4_561_599_744_774} The diagram shows triangle \(O A B\), in which the position vectors of \(A\) and \(B\) with respect to \(O\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = - 3 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k } .$$ \(C\) is a point on \(O A\) such that \(\overrightarrow { O C } = p \overrightarrow { O A }\), where \(p\) is a constant.
  1. Find angle \(A O B\).
  2. Find \(\overrightarrow { B C }\) in terms of \(p\) and vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  3. Find the value of \(p\) given that \(B C\) is perpendicular to \(O A\).