Questions P1 (1374 questions)

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CAIE P1 2009 November Q3
3
  1. Find the first 3 terms in the expansion of \(( 2 - x ) ^ { 6 }\) in ascending powers of \(x\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + 2 x + a x ^ { 2 } \right) ( 2 - x ) ^ { 6 }\) is 48 , find the value of the constant \(a\).
CAIE P1 2009 November Q4
4 The equation of a curve is \(y = x ^ { 4 } + 4 x + 9\).
  1. Find the coordinates of the stationary point on the curve and determine its nature.
  2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\).
CAIE P1 2009 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{7f66531c-2de6-44b7-9c48-7944acfea4d9-2_439_787_1174_678} The diagram shows a semicircle \(A B C\) with centre \(O\) and radius 6 cm . The point \(B\) is such that angle \(B O A\) is \(90 ^ { \circ }\) and \(B D\) is an arc of a circle with centre \(A\). Find
  1. the length of the \(\operatorname { arc } B D\),
  2. the area of the shaded region.
CAIE P1 2009 November Q6
6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k - 2 x\), where \(k\) is a constant.
  1. Given that the tangents to the curve at the points where \(x = 2\) and \(x = 3\) are perpendicular, find the value of \(k\).
  2. Given also that the curve passes through the point \(( 4,9 )\), find the equation of the curve.
CAIE P1 2009 November Q7
7 The equation of a curve is \(y = \frac { 12 } { x ^ { 2 } + 3 }\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the equation of the normal to the curve at the point \(P ( 1,3 )\).
  3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).
CAIE P1 2009 November Q8
8 The first term of an arithmetic progression is 8 and the common difference is \(d\), where \(d \neq 0\). The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is \(r\).
  1. Write down two equations connecting \(d\) and \(r\). Hence show that \(r = \frac { 3 } { 4 }\) and find the value of \(d\).
  2. Find the sum to infinity of the geometric progression.
  3. Find the sum of the first 8 terms of the arithmetic progression.
CAIE P1 2009 November Q9
9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
3
- 6 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 0
- 6
8 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } - 2
5
- 2 \end{array} \right)$$
  1. Find angle \(A O B\).
  2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and has magnitude 30 .
  3. Find the value of the constant \(p\) for which \(\overrightarrow { O A } + p \overrightarrow { O B }\) is perpendicular to \(\overrightarrow { O C }\).
CAIE P1 2009 November Q10
10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 , \quad x \in \mathbb { R } , \quad x > 0
& \mathrm {~g} : x \mapsto \frac { 2 x - 1 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$
  1. Solve the equation \(\operatorname { gf } ( x ) = x\).
  2. Express \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
  3. Show that the equation \(\mathrm { g } ^ { - 1 } ( x ) = x\) has no solutions.
  4. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
CAIE P1 2009 November Q1
1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { \sqrt { x } } - x\). Given that the curve passes through the point (4,6), find the equation of the curve.
CAIE P1 2009 November Q2
2
  1. Find, in terms of the non-zero constant \(k\), the first 4 terms in the expansion of \(( k + x ) ^ { 8 }\) in ascending powers of \(x\).
  2. Given that the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in this expansion are equal, find the value of \(k\).
CAIE P1 2009 November Q3
3 A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression in each of the following cases:
  1. the progression is arithmetic,
  2. the progression is geometric with a positive common ratio.
CAIE P1 2009 November Q4
4 The function f is defined by f : \(x \mapsto 5 - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. State, with a reason, whether f has an inverse.
CAIE P1 2009 November Q5
5
  1. Prove the identity \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x\).
  2. Solve the equation \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) = 9 \sin ^ { 3 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2009 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-2_590_666_1720_737} In the diagram, \(O A B C D E F G\) is a cube in which each side has length 6 . Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\) and the point \(Q\) is the mid-point of \(D F\).
  1. Express each of the vectors \(\overrightarrow { O Q }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Find the angle \(O Q P\).
CAIE P1 2009 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_301_485_264_829} A piece of wire of length 50 cm is bent to form the perimeter of a sector \(P O Q\) of a circle. The radius of the circle is \(r \mathrm {~cm}\) and the angle \(P O Q\) is \(\theta\) radians (see diagram).
  1. Express \(\theta\) in terms of \(r\) and show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the sector is given by $$A = 25 r - r ^ { 2 } .$$
  2. Given that \(r\) can vary, find the stationary value of \(A\) and determine its nature.
CAIE P1 2009 November Q8
8 The function f is such that \(\mathrm { f } ( x ) = \frac { 3 } { 2 x + 5 }\) for \(x \in \mathbb { R } , x \neq - 2.5\).
  1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and explain why f is a decreasing function.
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  3. A curve has the equation \(y = \mathrm { f } ( x )\). Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2009 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_554_723_1557_712} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 0 , - 2 )\) and \(C\) is \(( 12,14 )\). The diagonal \(B D\) is parallel to the \(x\)-axis.
  1. Explain why the \(y\)-coordinate of \(D\) is 6 . The \(x\)-coordinate of \(D\) is \(h\).
  2. Express the gradients of \(A D\) and \(C D\) in terms of \(h\).
  3. Calculate the \(x\)-coordinates of \(D\) and \(B\).
  4. Calculate the area of the rectangle \(A B C D\).
CAIE P1 2009 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-4_702_625_260_758}
  1. The diagram shows the line \(2 y = x + 5\) and the curve \(y = x ^ { 2 } - 4 x + 7\), which intersect at the points \(A\) and \(B\). Find
    (a) the \(x\)-coordinates of \(A\) and \(B\),
    (b) the equation of the tangent to the curve at \(B\),
    (c) the acute angle, in degrees correct to 1 decimal place, between this tangent and the line \(2 y = x + 5\).
  2. Determine the set of values of \(k\) for which the line \(2 y = x + k\) does not intersect the curve \(y = x ^ { 2 } - 4 x + 7\). \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 Find \(\int \left( x + \frac { 1 } { x } \right) ^ { 2 } \mathrm {~d} x\).
CAIE P1 2010 November Q2
2 In the expansion of \(( 1 + a x ) ^ { 6 }\), where \(a\) is a constant, the coefficient of \(x\) is - 30 . Find the coefficient of \(x ^ { 3 }\).
CAIE P1 2010 November Q3
3 Functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 3
& \mathrm {~g} : x \mapsto x ^ { 2 } - 2 x \end{aligned}$$ Express \(\operatorname { gf } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
CAIE P1 2010 November Q4
4
  1. Prove the identity \(\frac { \sin x \tan x } { 1 - \cos x } \equiv 1 + \frac { 1 } { \cos x }\).
  2. Hence solve the equation \(\frac { \sin x \tan x } { 1 - \cos x } + 2 = 0\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2010 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-2_741_533_1279_808} The diagram shows a pyramid \(O A B C\) with a horizontal base \(O A B\) where \(O A = 6 \mathrm {~cm} , O B = 8 \mathrm {~cm}\) and angle \(A O B = 90 ^ { \circ }\). The point \(C\) is vertically above \(O\) and \(O C = 10 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) as shown. Use a scalar product to find angle \(A C B\).
CAIE P1 2010 November Q6
6
  1. The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75 . Find the first term and the common difference.
  2. The first term of a geometric progression is 16 and the fourth term is \(\frac { 27 } { 4 }\). Find the sum to infinity of the progression.
CAIE P1 2010 November Q7
7 A function f is defined by f : \(x \mapsto 3 - 2 \tan \left( \frac { 1 } { 2 } x \right)\) for \(0 \leqslant x < \pi\).
  1. State the range of f .
  2. State the exact value of \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right)\).
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Obtain an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\).