Questions P1 (1374 questions)

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CAIE P1 2011 June Q10
10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x - 4 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto 2 ( x - 1 ) ^ { 3 } + 8 , \quad x > 1 . \end{aligned}$$
  1. Evaluate fg(2).
  2. 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.
  3. Obtain an expression for \(\mathrm { g } ^ { \prime } ( x )\) and use your answer to explain why g has an inverse.
  4. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
CAIE P1 2012 June Q1
1 Solve the equation \(\sin 2 x = 2 \cos 2 x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2012 June Q2
2 Find the coefficient of \(x ^ { 6 }\) in the expansion of \(\left( 2 x ^ { 3 } - \frac { 1 } { x ^ { 2 } } \right) ^ { 7 }\).
CAIE P1 2012 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_584_659_575_742} In the diagram, \(A B C\) is an equilateral triangle of side 2 cm . The mid-point of \(B C\) is \(Q\). An arc of a circle with centre \(A\) touches \(B C\) at \(Q\), and meets \(A B\) at \(P\) and \(A C\) at \(R\). Find the total area of the shaded regions, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
CAIE P1 2012 June Q4
4 A watermelon is assumed to be spherical in shape while it is growing. Its mass, \(M \mathrm {~kg}\), and radius, \(r \mathrm {~cm}\), are related by the formula \(M = k r ^ { 3 }\), where \(k\) is a constant. It is also assumed that the radius is increasing at a constant rate of 0.1 centimetres per day. On a particular day the radius is 10 cm and the mass is 3.2 kg . Find the value of \(k\) and the rate at which the mass is increasing on this day.
CAIE P1 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-2_636_947_1738_598} The diagram shows the curve \(y = 7 \sqrt { } x\) and the line \(y = 6 x + k\), where \(k\) is a constant. The curve and the line intersect at the points \(A\) and \(B\).
  1. For the case where \(k = 2\), find the \(x\)-coordinates of \(A\) and \(B\).
  2. Find the value of \(k\) for which \(y = 6 x + k\) is a tangent to the curve \(y = 7 \sqrt { } x\).
CAIE P1 2012 June Q6
6 Two vectors \(\mathbf { u }\) and \(\mathbf { v }\) are such that \(\mathbf { u } = \left( \begin{array} { c } p ^ { 2 }
- 2
6 \end{array} \right)\) and \(\mathbf { v } = \left( \begin{array} { c } 2
p - 1
2 p + 1 \end{array} \right)\), where \(p\) is a constant.
  1. Find the values of \(p\) for which \(\mathbf { u }\) is perpendicular to \(\mathbf { v }\).
  2. For the case where \(p = 1\), find the angle between the directions of \(\mathbf { u }\) and \(\mathbf { v }\).
CAIE P1 2012 June Q7
7
  1. The first two terms of an arithmetic progression are 1 and \(\cos ^ { 2 } x\) respectively. Show that the sum of the first ten terms can be expressed in the form \(a - b \sin ^ { 2 } x\), where \(a\) and \(b\) are constants to be found.
  2. The first two terms of a geometric progression are 1 and \(\frac { 1 } { 3 } \tan ^ { 2 } \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
    1. Find the set of values of \(\theta\) for which the progression is convergent.
    2. Find the exact value of the sum to infinity when \(\theta = \frac { 1 } { 6 } \pi\).
CAIE P1 2012 June Q8
8 The function \(\mathrm { f } : x \mapsto x ^ { 2 } - 4 x + k\) is defined for the domain \(x \geqslant p\), where \(k\) and \(p\) are constants.
  1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b + k\), where \(a\) and \(b\) are constants.
  2. State the range of f in terms of \(k\).
  3. State the smallest value of \(p\) for which f is one-one.
  4. For the value of \(p\) found in part (iii), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain \(\mathrm { f } ^ { - 1 }\), giving your answers in terms of \(k\).
CAIE P1 2012 June Q9
9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).
  1. Find the equation of \(M B\) and the coordinates of \(B\).
  2. Show that \(A B\) is perpendicular to \(B C\).
  3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).
CAIE P1 2012 June Q10
10 It is given that a curve has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x\).
  1. Find the set of values of \(x\) for which the gradient of the curve is less than 5 .
  2. Find the values of \(\mathrm { f } ( x )\) at the two stationary points on the curve and determine the nature of each stationary point.
CAIE P1 2012 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-4_636_951_255_596} The diagram shows the line \(y = 1\) and part of the curve \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\).
  1. Show that the equation \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\) can be written in the form \(x = \frac { 4 } { y ^ { 2 } } - 1\).
  2. Find \(\int \left( \frac { 4 } { y ^ { 2 } } - 1 \right) \mathrm { d } y\). Hence find the area of the shaded region.
  3. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the exact value of the volume of revolution obtained.
CAIE P1 2012 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-2_618_533_246_808} The diagram shows the region enclosed by the curve \(y = \frac { 6 } { 2 x - 3 }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2012 June Q2
2 The equation of a curve is \(y = 4 \sqrt { } x + \frac { 2 } { \sqrt { } x }\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.12 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\).
CAIE P1 2012 June Q3
3 The coefficient of \(x ^ { 3 }\) in the expansion of \(( a + x ) ^ { 5 } + ( 2 - x ) ^ { 6 }\) is 90 . Find the value of the positive constant \(a\).
CAIE P1 2012 June Q4
4 The point \(A\) has coordinates \(( - 1 , - 5 )\) and the point \(B\) has coordinates \(( 7,1 )\). The perpendicular bisector of \(A B\) meets the \(x\)-axis at \(C\) and the \(y\)-axis at \(D\). Calculate the length of \(C D\).
  1. Prove the identity \(\tan x + \frac { 1 } { \tan x } \equiv \frac { 1 } { \sin x \cos x }\).
  2. Solve the equation \(\frac { 2 } { \sin x \cos x } = 1 + 3 \tan x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2012 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-3_446_645_258_751} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius 8 cm . The line \(A B\) is a chord of the circle and angle \(A O B = 2.4\) radians. Find
  1. the length of \(A B\),
  2. the perimeter of the plate,
  3. the area of the plate.
CAIE P1 2012 June Q7
7
  1. In an arithmetic progression, the sum of the first \(n\) terms, denoted by \(S _ { n }\), is given by $$S _ { n } = n ^ { 2 } + 8 n .$$ Find the first term and the common difference.
  2. In a geometric progression, the second term is 9 less than the first term. The sum of the second and third terms is 30 . Given that all the terms of the progression are positive, find the first term.
CAIE P1 2012 June Q8
8
  1. Find the angle between the vectors \(3 \mathbf { i } - 4 \mathbf { k }\) and \(2 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k }\). The vector \(\overrightarrow { O A }\) has a magnitude of 15 units and is in the same direction as the vector \(3 \mathbf { i } - 4 \mathbf { k }\). The vector \(\overrightarrow { O B }\) has a magnitude of 14 units and is in the same direction as the vector \(2 \mathbf { i } + 3 \mathbf { j } - 6 \mathbf { k }\).
  2. Express \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  3. Find the unit vector in the direction of \(\overrightarrow { A B }\).
CAIE P1 2012 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{fa90db86-a73a-40db-b416-3c9f470fa207-4_762_848_255_646} The diagram shows part of the curve \(y = - x ^ { 2 } + 8 x - 10\) which passes through the points \(A\) and \(B\). The curve has a maximum point at \(A\) and the gradient of the line \(B A\) is 2 .
  1. Find the coordinates of \(A\) and \(B\).
  2. Find \(\int y \mathrm {~d} x\) and hence evaluate the area of the shaded region.
CAIE P1 2012 June Q10
10 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + 5 & \text { for } x \in \mathbb { R } ,
\mathrm {~g} : x \mapsto \frac { 8 } { x - 3 } & \text { for } x \in \mathbb { R } , x \neq 3 \end{array}$$
  1. Obtain expressions, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\), stating the value of \(x\) for which \(\mathrm { g } ^ { - 1 } ( x )\) is not defined.
  2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram, making clear the relationship between the two graphs.
  3. Given that the equation \(\operatorname { fg } ( x ) = 5 - k x\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\).
CAIE P1 2012 June Q1
1
  1. Prove the identity \(\tan ^ { 2 } \theta - \sin ^ { 2 } \theta \equiv \tan ^ { 2 } \theta \sin ^ { 2 } \theta\).
  2. Use this result to explain why \(\tan \theta > \sin \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P1 2012 June Q2
2 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
- 1
CAIE P1 2012 June Q4
4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4
2
- 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1
3
p \end{array} \right)$$ Find
  1. the unit vector in the direction of \(\overrightarrow { A B }\),
  2. the value of the constant \(p\) for which angle \(B O C = 90 ^ { \circ }\). 3 The first three terms in the expansion of \(( 1 - 2 x ) ^ { 2 } ( 1 + a x ) ^ { 6 }\), in ascending powers of \(x\), are \(1 - x + b x ^ { 2 }\). Find the values of the constants \(a\) and \(b\). 4
  3. Solve the equation \(\sin 2 x + 3 \cos 2 x = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  4. How many solutions has the equation \(\sin 2 x + 3 \cos 2 x = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 1080 ^ { \circ }\) ?
CAIE P1 2012 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{1b5d8cb1-fd1b-4fcf-8975-b5d020991c9a-2_570_1050_1393_550} The diagram shows part of the curve \(x = \frac { 8 } { y ^ { 2 } } - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B ( 6,1 )\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.