Questions P1 (1401 questions)

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CAIE P1 2012 November Q8
9 marks Moderate -0.3
8 A curve is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( 3 x + 4 ) ^ { \frac { 3 } { 2 } } - 6 x - 8 .$$
  1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Verify that the curve has a stationary point when \(x = - 1\) and determine its nature.
  3. It is now given that the stationary point on the curve has coordinates \(( - 1,5 )\). Find the equation of the curve.
CAIE P1 2012 November Q9
10 marks Moderate -0.3
9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } p \\ 1 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right)$$ where \(p\) is a constant.
  1. In the case where \(O A B\) is a straight line, state the value of \(p\) and find the unit vector in the direction of \(\overrightarrow { O A }\).
  2. In the case where \(O A\) is perpendicular to \(A B\), find the possible values of \(p\).
  3. In the case where \(p = 3\), the point \(C\) is such that \(O A B C\) is a parallelogram. Find the position vector of \(C\).
CAIE P1 2012 November Q10
10 marks Standard +0.3
10 A straight line has equation \(y = - 2 x + k\), where \(k\) is a constant, and a curve has equation \(y = \frac { 2 } { x - 3 }\).
  1. Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2 x ^ { 2 } - ( 6 + k ) x + ( 2 + 3 k ) = 0\).
  2. Find the two values of \(k\) for which the line is a tangent to the curve. The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).
  3. Find the coordinates of \(A\) and \(B\) and the equation of the line \(A B\).
CAIE P1 2012 November Q11
13 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-4_526_974_822_587} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.
  1. State the value of \(a\).
  2. Find the value of \(b\).
  3. Find the area of the shaded region.
  4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Find the value of \(m\).
CAIE P1 2013 November Q1
5 marks Moderate -0.8
1
  1. Find the first three terms when \(( 2 + 3 x ) ^ { 6 }\) is expanded in ascending powers of \(x\).
  2. In the expansion of \(( 1 + a x ) ( 2 + 3 x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is zero. Find the value of \(a\).
CAIE P1 2013 November Q2
5 marks Moderate -0.3
2 A curve has equation \(y = f ( x )\). It is given that \(f ^ { \prime } ( x ) = \frac { 1 } { \sqrt { } ( x + 6 ) } + \frac { 6 } { x ^ { 2 } }\) and that \(f ( 3 ) = 1\). Find \(f ( x )\).
CAIE P1 2013 November Q3
6 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-2_397_949_657_596} The diagram shows a pyramid \(O A B C D\) in which the vertical edge \(O D\) is 3 units in length. The point \(E\) is the centre of the horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 6 units and 4 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively.
  1. Express each of the vectors \(\overrightarrow { D B }\) and \(\overrightarrow { D E }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(B D E\).
CAIE P1 2013 November Q4
6 marks Moderate -0.3
4
  1. Solve the equation \(4 \sin ^ { 2 } x + 8 \cos x - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  2. Hence find the solution of the equation \(4 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) + 8 \cos \left( \frac { 1 } { 2 } \theta \right) - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2013 November Q5
6 marks Moderate -0.3
5 The function f is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } + 1 \text { for } x \geqslant 0$$
  1. Define in a similar way the inverse function \(\mathrm { f } ^ { - 1 }\).
  2. Solve the equation \(\operatorname { ff } ( x ) = \frac { 185 } { 16 }\).
CAIE P1 2013 November Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_412_629_258_758} The diagram shows a metal plate made by fixing together two pieces, \(O A B C D\) (shaded) and \(O A E D\) (unshaded). The piece \(O A B C D\) is a minor sector of a circle with centre \(O\) and radius \(2 r\). The piece \(O A E D\) is a major sector of a circle with centre \(O\) and radius \(r\). Angle \(A O D\) is \(\alpha\) radians. Simplifying your answers where possible, find, in terms of \(\alpha , \pi\) and \(r\),
  1. the perimeter of the metal plate,
  2. the area of the metal plate. It is now given that the shaded and unshaded pieces are equal in area.
  3. Find \(\alpha\) in terms of \(\pi\).
CAIE P1 2013 November Q7
9 marks Moderate -0.3
7 The point \(A\) has coordinates ( \(- 1,6\) ) and the point \(B\) has coordinates (7,2).
  1. Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(y = m x + c\).
  2. A point \(C\) on the perpendicular bisector has coordinates \(( p , q )\). The distance \(O C\) is 2 units, where \(O\) is the origin. Write down two equations involving \(p\) and \(q\) and hence find the coordinates of the possible positions of \(C\).
CAIE P1 2013 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_365_663_1813_740} The inside lane of a school running track consists of two straight sections each of length \(x\) metres, and two semicircular sections each of radius \(r\) metres, as shown in the diagram. The straight sections are perpendicular to the diameters of the semicircular sections. The perimeter of the inside lane is 400 metres.
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the region enclosed by the inside lane is given by \(A = 400 r - \pi r ^ { 2 }\).
  2. Given that \(x\) and \(r\) can vary, show that, when \(A\) has a stationary value, there are no straight sections in the track. Determine whether the stationary value is a maximum or a minimum. [5]
CAIE P1 2013 November Q9
10 marks Standard +0.3
9
  1. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000 . Find the common difference and the first term.
  2. A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity 6. A second geometric progression has first term \(2 a\), common ratio \(r ^ { 2 }\) and sum to infinity 7 . Find the values of \(a\) and \(r\).
CAIE P1 2013 November Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-4_654_974_614_587} The diagram shows the curve \(y = ( 3 - 2 x ) ^ { 3 }\) and the tangent to the curve at the point \(\left( \frac { 1 } { 2 } , 8 \right)\).
  1. Find the equation of this tangent, giving your answer in the form \(y = m x + c\).
  2. Find the area of the shaded region.
CAIE P1 2013 November Q1
3 marks Easy -1.2
1 Given that \(\cos x = p\), where \(x\) is an acute angle in degrees, find, in terms of \(p\),
  1. \(\sin x\),
  2. \(\tan x\),
  3. \(\tan \left( 90 ^ { \circ } - x \right)\).
CAIE P1 2013 November Q2
6 marks Standard +0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f66324-e1fc-40e1-98e7-625187e24d3d-2_579_556_600_301} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f66324-e1fc-40e1-98e7-625187e24d3d-2_579_876_605_973} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 1 shows a hollow cone with no base, made of paper. The radius of the cone is 6 cm and the height is 8 cm . The paper is cut from \(A\) to \(O\) and opened out to form the sector shown in Fig. 2. The circular bottom edge of the cone in Fig. 1 becomes the arc of the sector in Fig. 2. The angle of the sector is \(\theta\) radians. Calculate
  1. the value of \(\theta\),
  2. the area of paper needed to make the cone.
CAIE P1 2013 November Q3
7 marks Moderate -0.8
3 The equation of a curve is \(y = \frac { 2 } { \sqrt { } ( 5 x - 6 ) }\).
  1. Find the gradient of the curve at the point where \(x = 2\).
  2. Find \(\int \frac { 2 } { \sqrt { } ( 5 x - 6 ) } \mathrm { d } x\) and hence evaluate \(\int _ { 2 } ^ { 3 } \frac { 2 } { \sqrt { } ( 5 x - 6 ) } \mathrm { d } x\).
CAIE P1 2013 November Q4
7 marks Standard +0.3
4 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } + p \mathbf { k } .$$
  1. In the case where \(p = 6\), find the unit vector in the direction of \(\overrightarrow { A B }\).
  2. Find the values of \(p\) for which angle \(A O B = \cos ^ { - 1 } \left( \frac { 1 } { 5 } \right)\).
CAIE P1 2013 November Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_636_811_255_667} The diagram shows a rectangle \(A B C D\) in which point \(A\) is ( 0,8 ) and point \(B\) is ( 4,0 ). The diagonal \(A C\) has equation \(8 y + x = 64\). Find, by calculation, the coordinates of \(C\) and \(D\).
CAIE P1 2013 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-3_465_663_1160_740} In the diagram, \(S\) is the point ( 0,12 ) and \(T\) is the point ( 16,0 ). The point \(Q\) lies on \(S T\), between \(S\) and \(T\), and has coordinates \(( x , y )\). The points \(P\) and \(R\) lie on the \(x\)-axis and \(y\)-axis respectively and \(O P Q R\) is a rectangle.
  1. Show that the area, \(A\), of the rectangle \(O P Q R\) is given by \(A = 12 x - \frac { 3 } { 4 } x ^ { 2 }\).
  2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.
CAIE P1 2013 November Q7
8 marks Moderate -0.8
7
  1. An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile.
    1. Given that the \(n\)th mile takes 9 minutes, find the value of \(n\).
    2. Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon.
  2. The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
CAIE P1 2013 November Q8
10 marks Moderate -0.3
8 A function f is defined by \(\mathrm { f } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant 2 \pi\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find the range of f .
  3. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant k\).
  4. State the maximum value of \(k\) for which g has an inverse.
  5. Obtain an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2013 November Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{d5f66324-e1fc-40e1-98e7-625187e24d3d-4_584_670_881_740} The diagram shows part of the curve \(y = \frac { 8 } { x } + 2 x\) and three points \(A , B\) and \(C\) on the curve with \(x\)-coordinates 1, 2 and 5 respectively.
  1. A point \(P\) moves along the curve in such a way that its \(x\)-coordinate increases at a constant rate of 0.04 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing as \(P\) passes through \(A\).
  2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2013 November Q10
10 marks Moderate -0.3
10 A curve has equation \(y = 2 x ^ { 2 } - 3 x\).
  1. Find the set of values of \(x\) for which \(y > 9\).
  2. Express \(2 x ^ { 2 } - 3 x\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 2 } - 3 x \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + k$$ where \(k\) is a constant.
  3. Find the value of \(k\) for which the equation \(\mathrm { gf } ( x ) = 0\) has equal roots.
CAIE P1 2013 November Q1
3 marks Easy -1.2
1 Solve the inequality \(x ^ { 2 } - x - 2 > 0\).