Questions — CAIE P1 (1202 questions)

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CAIE P1 2008 June Q8
8 Functions f and g are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 4 x - 2 k & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, }
\mathrm { g } : x \mapsto \frac { 9 } { 2 - x } & \text { for } x \in \mathbb { R } , x \neq 2 . \end{array}$$
  1. Find the values of \(k\) for which the equation \(\mathrm { fg } ( x ) = x\) has two equal roots.
  2. Determine the roots of the equation \(\operatorname { fg } ( x ) = x\) for the values of \(k\) found in part (i).
CAIE P1 2008 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-3_791_885_1281_630} The diagram shows a curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { k } { x ^ { 3 } }\), where \(k\) is a constant. The curve passes through the points \(( 1,18 )\) and \(( 4,3 )\).
  1. Show, by integration, that the equation of the curve is \(y = \frac { 16 } { x ^ { 2 } } + 2\). The point \(P\) lies on the curve and has \(x\)-coordinate 1.6.
  2. Find the area of the shaded region.
CAIE P1 2008 June Q10
10 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } + p \mathbf { k }\) respectively.
  1. Find the value of \(p\) for which \(O A\) and \(O B\) are perpendicular.
  2. In the case where \(p = 6\), use a scalar product to find angle \(A O B\), correct to the nearest degree.
  3. Express the vector \(\overrightarrow { A B }\) is terms of \(p\) and hence find the values of \(p\) for which the length of \(A B\) is 3.5 units.
CAIE P1 2008 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).
  1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
  2. Find the coordinates of \(D\).
  3. Find, correct to 1 decimal place, the perimeter of \(A B C D\). \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 2009 June Q1
1 Prove the identity \(\frac { \sin x } { 1 - \sin x } - \frac { \sin x } { 1 + \sin x } \equiv 2 \tan ^ { 2 } x\).
CAIE P1 2009 June Q2
2 Find the set of values of \(k\) for which the line \(y = k x - 4\) intersects the curve \(y = x ^ { 2 } - 2 x\) at two distinct points.
CAIE P1 2009 June Q3
3
  1. Find the first 3 terms in the expansion of \(( 2 + 3 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Hence find the value of the constant \(a\) for which there is no term in \(x ^ { 2 }\) in the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 5 }\).
CAIE P1 2009 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_561_1210_895_465} The diagram shows the graph of \(y = a \sin ( b x ) + c\) for \(0 \leqslant x \leqslant 2 \pi\).
  1. Find the values of \(a , b\) and \(c\).
  2. Find the smallest value of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) for which \(y = 0\).
CAIE P1 2009 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_385_403_1866_872} The diagram shows a circle with centre \(O\). The circle is divided into two regions, \(R _ { 1 }\) and \(R _ { 2 }\), by the radii \(O A\) and \(O B\), where angle \(A O B = \theta\) radians. The perimeter of the region \(R _ { 1 }\) is equal to the length of the major \(\operatorname { arc } A B\).
  1. Show that \(\theta = \pi - 1\).
  2. Given that the area of region \(R _ { 1 }\) is \(30 \mathrm {~cm} ^ { 2 }\), find the area of region \(R _ { 2 }\), correct to 3 significant figures.
CAIE P1 2009 June Q6
6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 8 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$$
  1. Find the value of \(\overrightarrow { O A } \cdot \overrightarrow { O B }\) and hence state whether angle \(A O B\) is acute, obtuse or a right angle.
  2. The point \(X\) is such that \(\overrightarrow { A X } = \frac { 2 } { 5 } \overrightarrow { A B }\). Find the unit vector in the direction of \(O X\).
CAIE P1 2009 June Q7
7
  1. Find the sum to infinity of the geometric progression with first three terms \(0.5,0.5 ^ { 3 }\) and \(0.5 ^ { 5 }\).
  2. The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200 . Find the sum of all the terms in the progression.
CAIE P1 2009 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_599_716_1071_717} The diagram shows points \(A , B\) and \(C\) lying on the line \(2 y = x + 4\). The point \(A\) lies on the \(y\)-axis and \(A B = B C\). The line from \(D ( 10 , - 3 )\) to \(B\) is perpendicular to \(A C\). Calculate the coordinates of \(B\) and \(C\).
CAIE P1 2009 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_391_595_1978_774} The diagram shows part of the curve \(y = \frac { 6 } { 3 x - 2 }\).
  1. Find the gradient of the curve at the point where \(x = 2\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).
CAIE P1 2009 June Q10
10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(0 \leqslant x \leqslant A\), where \(A\) is a constant.
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  2. State the value of \(A\) for which the graph of \(y = \mathrm { f } ( x )\) has a line of symmetry.
  3. When \(A\) has this value, find the range of f . The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant 4\).
  4. Explain why \(g\) has an inverse.
  5. Obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2009 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-4_686_805_950_669} The diagram shows the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\) for \(x \geqslant 0\). The curve has a maximum point at \(A\) and a minimum point on the \(x\)-axis at \(B\). The normal to the curve at \(C ( 2,2 )\) meets the normal to the curve at \(B\) at the point \(D\).
  1. Find the coordinates of \(A\) and \(B\).
  2. Find the equation of the normal to the curve at \(C\).
  3. Find the area of the shaded region. \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 June Q1
1 The acute angle \(x\) radians is such that \(\tan x = k\), where \(k\) is a positive constant. Express, in terms of \(k\),
  1. \(\tan ( \pi - x )\),
  2. \(\tan \left( \frac { 1 } { 2 } \pi - x \right)\),
  3. \(\sin x\).
CAIE P1 2010 June Q2
2
  1. Find the first 3 terms in the expansion of \(\left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\) in descending powers of \(x\).
  2. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 2 } { x ^ { 2 } } \right) \left( 2 x - \frac { 3 } { x } \right) ^ { 5 }\).
CAIE P1 2010 June Q3
3 The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49 .
  1. Find the first term of the progression and the common difference. The \(n\)th term of the progression is 46 .
  2. Find the value of \(n\).
CAIE P1 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-2_428_550_1343_794} The diagram shows the curve \(y = 6 x - x ^ { 2 }\) and the line \(y = 5\). Find the area of the shaded region.
CAIE P1 2010 June Q5
5 The function f is such that \(\mathrm { f } ( x ) = 2 \sin ^ { 2 } x - 3 \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a + b \cos ^ { 2 } x\), stating the values of \(a\) and \(b\).
  2. State the greatest and least values of \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) + 1 = 0\).
CAIE P1 2010 June Q6
6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6\) and the point \(( 9,2 )\) lies on the curve.
  1. Find the equation of the curve.
  2. Find the \(x\)-coordinate of the stationary point on the curve and determine the nature of the stationary point.
CAIE P1 2010 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_766_589_251_778} The diagram shows part of the curve \(y = 2 - \frac { 18 } { 2 x + 3 }\), which crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(A\) crosses the \(y\)-axis at \(C\).
  1. Show that the equation of the line \(A C\) is \(9 x + 4 y = 27\).
  2. Find the length of \(B C\).
CAIE P1 2010 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_625_547_1489_797} The diagram shows a triangle \(A B C\) in which \(A\) is \(( 3 , - 2 )\) and \(B\) is \(( 15,22 )\). The gradients of \(A B , A C\) and \(B C\) are \(2 m , - 2 m\) and \(m\) respectively, where \(m\) is a positive constant.
  1. Find the gradient of \(A B\) and deduce the value of \(m\).
  2. Find the coordinates of \(C\). The perpendicular bisector of \(A B\) meets \(B C\) at \(D\).
  3. Find the coordinates of \(D\).
CAIE P1 2010 June Q9
9 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 7\) for \(x \in \mathbb { R }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } - c\).
  2. State the range of f .
  3. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) < 21\). The function g is defined by \(\mathrm { g } : x \mapsto 2 x + k\) for \(x \in \mathbb { R }\).
  4. Find the value of the constant \(k\) for which the equation \(\operatorname { gf } ( x ) = 0\) has two equal roots.
CAIE P1 2010 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-4_552_629_842_758} The diagram shows the parallelogram \(O A B C\). Given that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O C } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k }\), find
  1. the unit vector in the direction of \(\overrightarrow { O B }\),
  2. the acute angle between the diagonals of the parallelogram,
  3. the perimeter of the parallelogram, correct to 1 decimal place. \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. }