Questions P1 (1401 questions)

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CAIE P1 2017 November Q5
6 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-08_446_844_260_648} The diagram shows an isosceles triangle \(A B C\) in which \(A C = 16 \mathrm {~cm}\) and \(A B = B C = 10 \mathrm {~cm}\). The circular arcs \(B E\) and \(B D\) have centres at \(A\) and \(C\) respectively, where \(D\) and \(E\) lie on \(A C\).
  1. Show that angle \(B A C = 0.6435\) radians, correct to 4 decimal places.
  2. Find the area of the shaded region.
CAIE P1 2017 November Q6
9 marks Moderate -0.3
6 The points \(A ( 1,1 )\) and \(B ( 5,9 )\) lie on the curve \(6 y = 5 x ^ { 2 } - 18 x + 19\).
  1. Show that the equation of the perpendicular bisector of \(A B\) is \(2 y = 13 - x\).
    The perpendicular bisector of \(A B\) meets the curve at \(C\) and \(D\).
  2. Find, by calculation, the distance \(C D\), giving your answer in the form \(\sqrt { } \left( \frac { p } { q } \right)\), where \(p\) and \(q\) are integers.
CAIE P1 2017 November Q7
9 marks Standard +0.3
7
  1. \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-12_499_568_267_826} The diagram shows part of the graph of \(y = a + b \sin x\). Find the values of the constants \(a\) and \(b\).
    1. Show that the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ may be expressed as \(3 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0\).
    2. Hence solve the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2017 November Q8
9 marks Moderate -0.3
8
  1. Relative to an origin \(O\), the position vectors of two points \(P\) and \(Q\) are \(\mathbf { p }\) and \(\mathbf { q }\) respectively. The point \(R\) is such that \(P Q R\) is a straight line with \(Q\) the mid-point of \(P R\). Find the position vector of \(R\) in terms of \(\mathbf { p }\) and \(\mathbf { q }\), simplifying your answer.
  2. The vector \(6 \mathbf { i } + a \mathbf { j } + b \mathbf { k }\) has magnitude 21 and is perpendicular to \(3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\). Find the possible values of \(a\) and \(b\), showing all necessary working.
CAIE P1 2017 November Q9
10 marks Moderate -0.8
9 Functions f and g are defined for \(x > 3\) by $$\begin{aligned} & \mathrm { f } : x \mapsto \frac { 1 } { x ^ { 2 } - 9 } \\ & \mathrm {~g} : x \mapsto 2 x - 3 \end{aligned}$$
  1. Find and simplify an expression for \(\operatorname { gg } ( x )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Solve the equation \(\operatorname { fg } ( x ) = \frac { 1 } { 7 }\).
CAIE P1 2017 November Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-18_401_584_264_776} The diagram shows part of the curve \(y = \frac { 1 } { 2 } \left( x ^ { 4 } - 1 \right)\), defined for \(x \geqslant 0\).
  1. Find, showing all necessary working, the area of the shaded region.
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  3. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.
CAIE P1 2017 November Q1
4 marks Moderate -0.3
1 Find the term independent of \(x\) in the expansion of \(\left( 2 x - \frac { 1 } { 4 x ^ { 2 } } \right) ^ { 9 }\).
CAIE P1 2017 November Q2
6 marks Moderate -0.8
2 A function f is defined by \(\mathrm { f } : x \mapsto 4 - 5 x\) for \(x \in \mathbb { R }\).
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and find the point of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\).
  2. Sketch, on the same diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
CAIE P1 2017 November Q3
6 marks Easy -1.2
3
  1. Each year, the value of a certain rare stamp increases by \(5 \%\) of its value at the beginning of the year. A collector bought the stamp for \(\\) 10000\( at the beginning of 2005. Find its value at the beginning of 2015 correct to the nearest \)\\( 100\).
  2. The sum of the first \(n\) terms of an arithmetic progression is \(\frac { 1 } { 2 } n ( 3 n + 7 )\). Find the 1 st term and the common difference of the progression.
CAIE P1 2017 November Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-06_401_698_255_721} The diagram shows a semicircle with centre \(O\) and radius 6 cm . The radius \(O C\) is perpendicular to the diameter \(A B\). The point \(D\) lies on \(A B\), and \(D C\) is an arc of a circle with centre \(B\).
  1. Calculate the length of the \(\operatorname { arc } D C\).
  2. Find the value of \(\frac { \text { area of region } P } { \text { area of region } Q }\),
    giving your answer correct to 3 significant figures.
CAIE P1 2017 November Q5
7 marks Standard +0.3
5
  1. Show that the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) can be expressed as $$2 \cos ^ { 2 } 2 x + 3 \cos 2 x + 1 = 0$$
  2. Hence solve the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2017 November Q6
8 marks Moderate -0.8
6
  1. The function f , defined by \(\mathrm { f } : x \mapsto a + b \sin x\) for \(x \in \mathbb { R }\), is such that \(\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3\).
    1. Find the values of the constants \(a\) and \(b\).
    2. Evaluate \(\mathrm { ff } ( 0 )\).
  2. The function g is defined by \(\mathrm { g } : x \mapsto c + d \sin x\) for \(x \in \mathbb { R }\). The range of g is given by \(- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10\). Find the values of the constants \(c\) and \(d\).
CAIE P1 2017 November Q7
8 marks Standard +0.3
7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.
  1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
  2. Find the set of values of \(m\) for which \(L\) does not meet the curve.
CAIE P1 2017 November Q8
9 marks Moderate -0.8
8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 2 } + 5 x - 4\).
  1. Find the \(x\)-coordinate of each of the stationary points of the curve.
  2. Obtain an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence or otherwise find the nature of each of the stationary points.
  3. Given that the curve passes through the point \(( 6,2 )\), find the equation of the curve.
CAIE P1 2017 November Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 2 \\ - 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6 \\ 1 \\ 1 \end{array} \right)\).
  1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
    The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
  2. Find the position vector of \(C\).
  3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.
CAIE P1 2017 November Q10
11 marks Moderate -0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-18_551_689_260_726} The diagram shows part of the curve \(y = \sqrt { } ( 5 x - 1 )\) and the normal to the curve at the point \(P ( 2,3 )\). This normal meets the \(x\)-axis at \(Q\).
  1. Find the equation of the normal at \(P\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2017 November Q1
4 marks Standard +0.3
1 An arithmetic progression has first term - 12 and common difference 6 . The sum of the first \(n\) terms exceeds 3000 . Calculate the least possible value of \(n\).
CAIE P1 2017 November Q2
4 marks Standard +0.8
2 Find the set of values of \(a\) for which the curve \(y = - \frac { 2 } { x }\) and the straight line \(y = a x + 3 a\) meet at two distinct points.
CAIE P1 2017 November Q3
5 marks Standard +0.3
3
  1. Find the term independent of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 6 }\).
  2. Find the value of \(a\) for which there is no term independent of \(x\) in the expansion of $$\left( 1 + a x ^ { 2 } \right) \left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$$
CAIE P1 2017 November Q4
5 marks Standard +0.3
4 The function f is such that \(\mathrm { f } ( x ) = ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 6 x\) for \(\frac { 1 } { 2 } < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which f is a decreasing function.
CAIE P1 2017 November Q5
7 marks Moderate -0.3
5
  1. Show that the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) may be expressed as \(5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0\).
    [0pt] [3]
  2. Hence solve the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2017 November Q6
7 marks Standard +0.3
6 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } - 1 } \text { for } x < - 1 \\ & \mathrm {~g} ( x ) = x ^ { 2 } + 1 \text { for } x > 0 \end{aligned}$$
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Solve the equation \(\operatorname { gf } ( x ) = 5\).
CAIE P1 2017 November Q7
7 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-10_401_561_260_790} The diagram shows a rectangle \(A B C D\) in which \(A B = 5\) units and \(B C = 3\) units. Point \(P\) lies on \(D C\) and \(A P\) is an arc of a circle with centre \(B\). Point \(Q\) lies on \(D C\) and \(A Q\) is an arc of a circle with centre \(D\).
  1. Show that angle \(A B P = 0.6435\) radians, correct to 4 decimal places.
  2. Calculate the areas of the sectors \(B A P\) and \(D A Q\).
  3. Calculate the area of the shaded region.
CAIE P1 2017 November Q8
8 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-12_485_570_262_790} The diagram shows parts of the graphs of \(y = 3 - 2 x\) and \(y = 4 - 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).
  1. Find by calculation the \(x\)-coordinates of \(A\) and \(B\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2017 November Q9
9 marks Standard +0.3
9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.
  1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
  2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).