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 2004 November Q3
3 In the diagram, \(A C\) is an arc of a circle, centre \(O\) and radius 6 cm . The line \(B C\) is perpendicular to \(O C\) and \(O A B\) is a straight line. Angle \(A O C = \frac { 1 } { 3 } \pi\) radians. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
CAIE P1 2004 November Q4
4
  1. Sketch and label, on the same diagram, the graphs of \(y = 2 \sin x\) and \(y = \cos 2 x\), for the interval \(0 \leqslant x \leqslant \pi\).
  2. Hence state the number of solutions of the equation \(2 \sin x = \cos 2 x\) in the interval \(0 \leqslant x \leqslant \pi\).
CAIE P1 2004 November Q5
5 The equation of a curve is \(y = x ^ { 2 } - 4 x + 7\) and the equation of a line is \(y + 3 x = 9\). The curve and the line intersect at the points \(A\) and \(B\).
  1. The mid-point of \(A B\) is \(M\). Show that the coordinates of \(M\) are \(\left( \frac { 1 } { 2 } , 7 \frac { 1 } { 2 } \right)\).
  2. Find the coordinates of the point \(Q\) on the curve at which the tangent is parallel to the line \(y + 3 x = 9\).
  3. Find the distance \(M Q\).
CAIE P1 2004 November Q6
6 The function \(\mathrm { f } : x \mapsto 5 \sin ^ { 2 } x + 3 \cos ^ { 2 } x\) is defined for the domain \(0 \leqslant x \leqslant \pi\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a + b \sin ^ { 2 } x\), stating the values of \(a\) and \(b\).
  2. Hence find the values of \(x\) for which \(\mathrm { f } ( x ) = 7 \sin x\).
  3. State the range of f .
CAIE P1 2004 November Q7
7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 4 x - 3 ) }\) and \(P ( 3,3 )\) is a point on the curve.
  1. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\).
  2. Find the equation of the curve.
CAIE P1 2004 November Q8
8 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) and \(- 5 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) respectively, relative to an origin \(O\).
  1. Use a scalar product to calculate angle \(A O B\), giving your answer in radians correct to 3 significant figures.
  2. The point \(C\) is such that \(\overrightarrow { A B } = 2 \overrightarrow { B C }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).
CAIE P1 2004 November Q9
9 The function f : \(x \mapsto 2 x - a\), where \(a\) is a constant, is defined for all real \(x\).
  1. In the case where \(a = 3\), solve the equation \(\mathrm { ff } ( x ) = 11\). The function \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x\) is defined for all real \(x\).
  2. Find the value of \(a\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real solution. The function \(\mathrm { h } : x \mapsto x ^ { 2 } - 6 x\) is defined for the domain \(x \geqslant 3\).
  3. Express \(x ^ { 2 } - 6 x\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are constants.
  4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { h } ^ { - 1 }\).
CAIE P1 2004 November Q10
10 A curve has equation \(y = x ^ { 2 } + \frac { 2 } { x }\).
  1. Write down expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the coordinates of the stationary point on the curve and determine its nature.
  3. Find the volume of the solid formed when the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated completely about the \(x\)-axis.
CAIE P1 2005 November Q1
1 Solve the equation \(3 \sin ^ { 2 } \theta - 2 \cos \theta - 3 = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2005 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{933cdfe1-27bb-450d-8b9a-b494916242cb-2_625_582_397_778} In the diagram, \(O A B\) and \(O C D\) are radii of a circle, centre \(O\) and radius 16 cm . Angle \(A O C = \alpha\) radians. \(A C\) and \(B D\) are arcs of circles, centre \(O\) and radii 10 cm and 16 cm respectively.
  1. In the case where \(\alpha = 0.8\), find the area of the shaded region.
  2. Find the value of \(\alpha\) for which the perimeter of the shaded region is 28.9 cm .
CAIE P1 2005 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{933cdfe1-27bb-450d-8b9a-b494916242cb-2_737_693_1484_726} In the diagram, \(A B E D\) is a trapezium with right angles at \(E\) and \(D\), and \(C E D\) is a straight line. The lengths of \(A B\) and \(B C\) are \(2 d\) and \(( 2 \sqrt { 3 } ) d\) respectively, and angles \(B A D\) and \(C B E\) are \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively.
  1. Find the length of \(C D\) in terms of \(d\).
  2. Show that angle \(C A D = \tan ^ { - 1 } \left( \frac { 2 } { \sqrt { 3 } } \right)\).
CAIE P1 2005 November Q4
4 Relative to an origin \(O\), the position vectors of points \(P\) and \(Q\) are given by $$\overrightarrow { O P } = \left( \begin{array} { r } - 2
3
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O Q } = \left( \begin{array} { l } 2
1
q \end{array} \right)$$ where \(q\) is a constant.
  1. In the case where \(q = 3\), use a scalar product to show that \(\cos P O Q = \frac { 1 } { 7 }\).
  2. Find the values of \(q\) for which the length of \(\overrightarrow { P Q }\) is 6 units.
    \includegraphics[max width=\textwidth, alt={}]{933cdfe1-27bb-450d-8b9a-b494916242cb-3_647_741_845_699}
    The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm , and the cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone.
CAIE P1 2005 November Q6
6 A small trading company made a profit of \(
) 250000\( in the year 2000. The company considered two different plans, plan \)A\( and plan \)B$, for increasing its profits. Under plan \(A\), the annual profit would increase each year by \(5 \%\) of its value in the preceding year. Find, for plan \(A\),
  1. the profit for the year 2008,
  2. the total profit for the 10 years 2000 to 2009 inclusive. Under plan \(B\), the annual profit would increase each year by a constant amount \(
    ) D\(.
  3. Find the value of \)D$ for which the total profit for the 10 years 2000 to 2009 inclusive would be the same for both plans.
CAIE P1 2005 November Q7
7 Three points have coordinates \(A ( 2,6 ) , B ( 8,10 )\) and \(C ( 6,0 )\). The perpendicular bisector of \(A B\) meets the line \(B C\) at \(D\). Find
  1. the equation of the perpendicular bisector of \(A B\) in the form \(a x + b y = c\),
  2. the coordinates of \(D\).
CAIE P1 2005 November Q8
8 A function f is defined by \(\mathrm { f } : x \mapsto ( 2 x - 3 ) ^ { 3 } - 8\), for \(2 \leqslant x \leqslant 4\).
  1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and show that f is an increasing function.
  2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2005 November Q9
9 The equation of a curve is \(x y = 12\) and the equation of a line \(l\) is \(2 x + y = k\), where \(k\) is a constant.
  1. In the case where \(k = 11\), find the coordinates of the points of intersection of \(l\) and the curve.
  2. Find the set of values of \(k\) for which \(l\) does not intersect the curve.
  3. In the case where \(k = 10\), one of the points of intersection is \(P ( 2,6 )\). Find the angle, in degrees correct to 1 decimal place, between \(l\) and the tangent to the curve at \(P\).
CAIE P1 2005 November Q10
10 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 16 } { x ^ { 3 } }\), and \(( 1,4 )\) is a point on the curve.
  1. Find the equation of the curve.
  2. A line with gradient \(- \frac { 1 } { 2 }\) is a normal to the curve. Find the equation of this normal, giving your answer in the form \(a x + b y = c\).
  3. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\). \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed.
    University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2006 November Q1
1 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2006 November Q2
2 Given that \(x = \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)\), find the exact value of
  1. \(\cos ^ { 2 } x\),
  2. \(\tan ^ { 2 } x\).
CAIE P1 2006 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-2_536_606_735_772} In the diagram, \(A O B\) is a sector of a circle with centre \(O\) and radius 12 cm . The point \(A\) lies on the side \(C D\) of the rectangle \(O C D B\). Angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express the area of the shaded region in the form \(a ( \sqrt { } 3 ) - b \pi\), stating the values of the integers \(a\) and \(b\).
CAIE P1 2006 November Q4
4 The position vectors of points \(A\) and \(B\) are \(\left( \begin{array} { r } - 3
6
3 \end{array} \right)\) and \(\left( \begin{array} { r } - 1
2
4 \end{array} \right)\) respectively, relative to an origin \(O\).
  1. Calculate angle \(A O B\).
  2. The point \(C\) is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the unit vector in the direction of \(\overrightarrow { O C }\).
CAIE P1 2006 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_684_771_260_685} The three points \(A ( 1,3 ) , B ( 13,11 )\) and \(C ( 6,15 )\) are shown in the diagram. The perpendicular from \(C\) to \(A B\) meets \(A B\) at the point \(D\). Find
  1. the equation of \(C D\),
  2. the coordinates of \(D\).
CAIE P1 2006 November Q6
6
  1. Find the sum of all the integers between 100 and 400 that are divisible by 7 .
  2. The first three terms in a geometric progression are \(144 , x\) and 64 respectively, where \(x\) is positive. Find
    1. the value of \(x\),
    2. the sum to infinity of the progression.
CAIE P1 2006 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_490_665_1793_740} The diagram shows the curve \(y = x ( x - 1 ) ( x - 2 )\), which crosses the \(x\)-axis at the points \(O ( 0,0 )\), \(A ( 1,0 )\) and \(B ( 2,0 )\).
  1. The tangents to the curve at the points \(A\) and \(B\) meet at the point \(C\). Find the \(x\)-coordinate of \(C\).
  2. Show by integration that the area of the shaded region \(R _ { 1 }\) is the same as the area of the shaded region \(R _ { 2 }\).
CAIE P1 2006 November Q8
8 The equation of a curve is \(y = \frac { 6 } { 5 - 2 x }\).
  1. Calculate the gradient of the curve at the point where \(x = 1\).
  2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(y\) has a constant value of 0.02 units per second. Find the rate of increase of \(x\) when \(x = 1\).
  3. The region between the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume obtained is \(\frac { 12 } { 5 } \pi\).