Questions — CAIE P1 (1228 questions)

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CAIE P1 2010 June Q1
4 marks Moderate -0.8
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
5 marks Moderate -0.3
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
6 marks Moderate -0.8
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
6 marks Moderate -0.3
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
7 marks Moderate -0.3
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
7 marks Moderate -0.8
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
8 marks Standard +0.3
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
10 marks Standard +0.3
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
11 marks Moderate -0.8
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
11 marks Standard +0.3
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.
CAIE P1 2010 June Q1
5 marks Moderate -0.8
1 The first term of a geometric progression is 12 and the second term is - 6 . Find
  1. the tenth term of the progression,
  2. the sum to infinity.
CAIE P1 2010 June Q2
5 marks Moderate -0.8
2
  1. Find the first three terms, in descending powers of \(x\), in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 6 }\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }\).
CAIE P1 2010 June Q3
5 marks Moderate -0.8
3 The function \(\mathrm { f } : x \mapsto a + b \cos x\) is defined for \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } ( 0 ) = 10\) and that \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right) = 1\), find
  1. the values of \(a\) and \(b\),
  2. the range of \(f\),
  3. the exact value of \(\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)\).
CAIE P1 2010 June Q4
5 marks Standard +0.3
4
  1. Show that the equation \(2 \sin x \tan x + 3 = 0\) can be expressed as \(2 \cos ^ { 2 } x - 3 \cos x - 2 = 0\).
  2. Solve the equation \(2 \sin x \tan x + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2010 June Q5
7 marks Moderate -0.3
5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 3 x - 2 ) }\). Given that the curve passes through the point \(P ( 2,11 )\), find
  1. the equation of the normal to the curve at \(P\),
  2. the equation of the curve.
CAIE P1 2010 June Q6
8 marks Moderate -0.3
6 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 8 \mathbf { k } , \quad \overrightarrow { O C } = - \mathbf { i } - 2 \mathbf { j } + 10 \mathbf { k }$$
  1. Use a scalar product to find angle \(A B C\).
  2. Find the perimeter of triangle \(A B C\), giving your answer correct to 2 decimal places.
CAIE P1 2010 June Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_744_675_255_735} The diagram shows a metal plate \(A B C D E F\) which has been made by removing the two shaded regions from a circle of radius 10 cm and centre \(O\). The parallel edges \(A B\) and \(E D\) are both of length 12 cm .
  1. Show that angle \(D O E\) is 1.287 radians, correct to 4 significant figures.
  2. Find the perimeter of the metal plate.
  3. Find the area of the metal plate.
CAIE P1 2010 June Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726} The diagram shows a rhombus \(A B C D\) in which the point \(A\) is ( \(- 1,2\) ), the point \(C\) is ( 5,4 ) and the point \(B\) lies on the \(y\)-axis. Find
  1. the equation of the perpendicular bisector of \(A C\),
  2. the coordinates of \(B\) and \(D\),
  3. the area of the rhombus.
CAIE P1 2010 June Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-4_602_899_248_625} The diagram shows part of the curve \(y = x + \frac { 4 } { x }\) which has a minimum point at \(M\). The line \(y = 5\) intersects the curve at the points \(A\) and \(B\).
  1. Find the coordinates of \(A , B\) and \(M\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2010 June Q10
12 marks Moderate -0.3
10 The function \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \in \mathbb { R }\).
  1. Find the values of the constant \(k\) for which the line \(y + k x = 12\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  3. Find the range of f . The function \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \geqslant A\).
  4. Find the smallest value of \(A\) for which g has an inverse.
  5. For this value of \(A\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
CAIE P1 2011 June Q1
3 marks Moderate -0.3
1 Find the coefficient of \(x\) in the expansion of \(\left( x + \frac { 2 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2011 June Q2
4 marks Moderate -0.3
2 The volume of a spherical balloon is increasing at a constant rate of \(50 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the radius when the radius is 10 cm . [Volume of a sphere \(= \frac { 4 } { 3 } \pi r ^ { 3 }\).]
CAIE P1 2011 June Q3
5 marks Moderate -0.8
3
  1. Sketch the curve \(y = ( x - 2 ) ^ { 2 }\).
  2. The region enclosed by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained, giving your answer in terms of \(\pi\).
CAIE P1 2011 June Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-2_750_855_902_646} The diagram shows a prism \(A B C D P Q R S\) with a horizontal square base \(A P S D\) with sides of length 6 cm . The cross-section \(A B C D\) is a trapezium and is such that the vertical edges \(A B\) and \(D C\) are of lengths 5 cm and 2 cm respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(A D , A P\) and \(A B\) respectively.
  1. Express each of the vectors \(\overrightarrow { C P }\) and \(\overrightarrow { C Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to calculate angle \(P C Q\).
CAIE P1 2011 June Q5
6 marks Moderate -0.3
5
  1. Show that the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) can be written in the form $$2 \sin ^ { 4 } \theta + \sin ^ { 2 } \theta - 1 = 0 .$$
  2. Hence solve the equation \(2 \tan ^ { 2 } \theta \sin ^ { 2 } \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).