Questions P1 (1401 questions)

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CAIE P1 2023 November Q10
9 marks Standard +0.8
10 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-12_552_582_255_778} The diagram shows points \(A , B\) and \(C\) lying on a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is 2.8 radians. The shaded region is bounded by two arcs. The upper arc is part of the circle with centre \(O\) and radius \(r\). The lower arc is part of a circle with centre \(C\) and radius \(R\).
  1. State the size of angle \(A C O\) in radians.
  2. Find \(R\) in terms of \(r\).
  3. Find the area of the shaded region in terms of \(r\).
CAIE P1 2023 November Q11
10 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-14_693_782_267_669} The diagram shows part of the curve with equation \(y = x + \frac { 2 } { ( 2 x - 1 ) ^ { 2 } }\). The lines \(x = 1\) and \(x = 2\) intersect the curve at \(P\) and \(Q\) respectively and \(R\) is the stationary point on the curve.
  1. Verify that the \(x\)-coordinate of \(R\) is \(\frac { 3 } { 2 }\) and find the \(y\)-coordinate of \(R\).
  2. Find the exact value of the area of the shaded region.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2020 Specimen Q1
3 marks Easy -1.2
1 The following points $$A ( 0,1 ) , \quad B ( 1,6 ) , \quad C ( 1.5,7.75 ) , \quad D ( 1.9,8.79 ) \quad \text { and } \quad E ( 2,9 )$$ lie on the curve \(y = \mathrm { f } ( x )\). The table below shows the gradients of the chords \(A E\) and \(B E\).
Chord\(A E\)\(B E\)\(C E\)\(D E\)
Gradient of
chord
43
  1. Complete the table to show the gradients of \(C E\) and \(D E\).
  2. State what the values in the table indicate about the value of \(\mathrm { f } ^ { \prime } ( 2 )\).
CAIE P1 2020 Specimen Q2
4 marks Moderate -0.8
2 Functions \(f\) and \(g\) are defined by $$\begin{aligned} \mathrm { f } : x & \mapsto 3 x + 2 , \quad x \in \mathbb { R } , \\ \mathrm {~g} : x & \mapsto 4 x - 12 , \quad x \in \mathbb { R } . \end{aligned}$$ Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { gf } ( x )\).
CAIE P1 2020 Specimen Q3
4 marks Easy -1.2
3 An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 .
Find the value of \(n\).
CAIE P1 2020 Specimen Q4
5 marks Moderate -0.3
4 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }\) and that \(\mathrm { f } ( 3 ) = 1\). Find \(\mathrm { f } ( x )\).
CAIE P1 2020 Specimen Q5
5 marks Moderate -0.8
5
  1. The curve \(y = x ^ { 2 } + 3 x + 4\) is translated by \(\binom { 2 } { 0 }\).
    Find and simplify the equation of the translated curve.
  2. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 3 \mathrm { f } ( - x )\). Describe fully the two single transformations which have been combined to give the resulting transformation.
CAIE P1 2020 Specimen Q6
5 marks Moderate -0.5
6
  1. Find the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in the expansion of \(( 2 - x ) ^ { 6 }\).
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 x + 1 ) ( 2 - x ) ^ { 6 }\).
CAIE P1 2020 Specimen Q7
6 marks Standard +0.3
7
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2020 Specimen Q8
6 marks Standard +0.3
8 A curve has equation \(y = \frac { 12 } { 3 - 2 x }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
  2. Find the possible \(x\)-coordinates of \(A\).
CAIE P1 2020 Specimen Q9
7 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-14_713_912_258_573} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters CAD and BAE are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).
  1. Show that the radius of the larger circle is \(r \sqrt { 2 }\).
  2. Find the area of the shaded region in terms of \(r\).
CAIE P1 2020 Specimen Q10
8 marks Moderate -0.3
10 The circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 20 = 0\) has centre \(C\) and passes through points \(A\) and \(B\).
  1. State the coordinates of \(C\).
    It is given that the midpoint, \(D\), of \(A B\) has coordinates \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
  2. Find the equation of \(A B\), giving your answer in the form \(y = m x + c\).
  3. Find, by calculation, the \(x\)-coordinates of \(A\) and \(B\).
CAIE P1 2020 Specimen Q11
9 marks Moderate -0.8
11 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.
  1. It is given that \(a = 6\) and \(b = - 8\). Find the range of f .
  2. It is given instead that \(a = 5\) and that the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
  3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots then \(a ^ { 2 } < 4 ( b - a )\).
CAIE P1 2020 Specimen Q12
13 marks Moderate -0.5
12 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-20_524_972_274_548} 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. Calculate 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 \(m\). Calculate the value of \(m\).
CAIE P1 2002 June Q1
4 marks Moderate -0.8
1 The line \(x + 2 y = 9\) intersects the curve \(x y + 18 = 0\) at the points \(A\) and \(B\). Find the coordinates of \(A\) and \(B\).
CAIE P1 2002 June Q2
5 marks Moderate -0.3
2
  1. Show that \(\sin x \tan x\) may be written as \(\frac { 1 - \cos ^ { 2 } x } { \cos x }\).
  2. Hence solve the equation \(2 \sin x \tan x = 3\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2002 June Q3
6 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-2_629_659_715_740} The diagram shows the curve \(y = 3 \sqrt { } x\) and the line \(y = x\) intersecting at \(O\) and \(P\). Find
  1. the coordinates of \(P\),
  2. the area of the shaded region.
CAIE P1 2002 June Q4
7 marks Moderate -0.8
4 A progression has a first term of 12 and a fifth term of 18.
  1. Find the sum of the first 25 terms if the progression is arithmetic.
  2. Find the 13th term if the progression is geometric.
CAIE P1 2002 June Q5
7 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-3_1070_754_255_699} The diagram shows a solid cylinder standing on a horizontal circular base, centre \(O\) and radius 4 units. The line \(B A\) is a diameter and the radius \(O C\) is at \(90 ^ { \circ }\) to \(O A\). Points \(O ^ { \prime } , A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\) lie on the upper surface of the cylinder such that \(O O ^ { \prime } , A A ^ { \prime } , B B ^ { \prime }\) and \(C C ^ { \prime }\) are all vertical and of length 12 units. The mid-point of \(B B ^ { \prime }\) is \(M\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O O ^ { \prime }\) respectively.
  1. Express each of the vectors \(\overrightarrow { M O }\) and \(\overrightarrow { M C ^ { \prime } }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Hence find the angle \(O M C ^ { \prime }\).
CAIE P1 2002 June Q6
7 marks Easy -1.2
6 The function f , where \(\mathrm { f } ( x ) = a \sin x + b\), is defined for the domain \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 2\) and that \(\mathrm { f } \left( \frac { 3 } { 2 } \pi \right) = - 8\),
  1. find the values of \(a\) and \(b\),
  2. find the values of \(x\) for which \(\mathrm { f } ( x ) = 0\), giving your answers in radians correct to 2 decimal places,
  3. sketch the graph of \(y = \mathrm { f } ( x )\).
CAIE P1 2002 June Q7
7 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-4_556_524_255_813} The diagram shows the circular cross-section of a uniform cylindrical log with centre \(O\) and radius 20 cm . The points \(A , X\) and \(B\) lie on the circumference of the cross-section and \(A B = 32 \mathrm {~cm}\).
  1. Show that angle \(A O B = 1.855\) radians, correct to 3 decimal places.
  2. Find the area of the sector \(A X B O\). The section \(A X B C D\), where \(A B C D\) is a rectangle with \(A D = 18 \mathrm {~cm}\), is removed.
  3. Find the area of the new cross-section (shown shaded in the diagram).
CAIE P1 2002 June Q8
10 marks Standard +0.3
8 A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is \(192 \pi \mathrm {~cm} ^ { 2 }\). The cylinder has a radius of \(r \mathrm {~cm}\) and a height of \(h \mathrm {~cm}\).
  1. Express \(h\) in terms of \(r\) and show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cylinder is given by $$V = \frac { 1 } { 2 } \pi \left( 192 r - r ^ { 3 } \right) .$$ Given that \(r\) can vary,
  2. find the value of \(r\) for which \(V\) has a stationary value,
  3. find this stationary value and determine whether it is a maximum or a minimum.
CAIE P1 2002 June Q9
11 marks Standard +0.3
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\) and \(P ( 1,5 )\) is a point on the curve.
  1. The normal to the curve at \(P\) crosses the \(x\)-axis at \(Q\). Find the coordinates of \(Q\).
  2. Find the equation of the curve.
  3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.3 units per second. Find the rate of increase of the \(y\)-coordinate when \(x = 1\).
CAIE P1 2002 June Q10
11 marks Moderate -0.3
10 The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 3 x + 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \frac { 6 } { 2 x + 3 } , & x \in \mathbb { R } , x \neq - 1.5 . \end{array}$$
  1. Find the value of \(x\) for which \(\operatorname { fg } ( x ) = 3\).
  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 two graphs.
  3. Express each of \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\), and solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2003 June Q1
3 marks Moderate -0.5
1 Find the value of the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( 2 x - \frac { 1 } { x } \right) ^ { 5 }\).