Questions — CAIE P1 (1228 questions)

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CAIE P1 2022 November Q5
6 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-06_743_750_269_687} The diagram shows a curve which has a maximum point at \(( 8,12 )\) and a minimum point at \(( 8,0 )\). The curve is the result of applying a combination of two transformations to a circle. The first transformation applied is a translation of \(\binom { 7 } { - 3 }\). The second transformation applied is a stretch in the \(y\)-direction.
  1. State the scale factor of the stretch.
  2. State the radius of the original circle.
  3. State the coordinates of the centre of the circle after the translation has been completed but before the stretch is applied.
  4. State the coordinates of the centre of the original circle.
CAIE P1 2022 November Q6
5 marks Moderate -0.3
6 It is given that \(\alpha = \cos ^ { - 1 } \left( \frac { 8 } { 17 } \right)\).
Find, without using the trigonometric functions on your calculator, the exact value of \(\frac { 1 } { \sin \alpha } + \frac { 1 } { \tan \alpha }\).
CAIE P1 2022 November Q7
7 marks Moderate -0.3
7 The curve \(y = \mathrm { f } ( x )\) is such that \(\mathrm { f } ^ { \prime } ( x ) = \frac { - 3 } { ( x + 2 ) ^ { 4 } }\).
  1. The tangent at a point on the curve where \(x = a\) has gradient \(- \frac { 16 } { 27 }\). Find the possible values of \(a\).
  2. Find \(\mathrm { f } ( x )\) given that the curve passes through the point \(( - 1,5 )\).
CAIE P1 2022 November Q8
7 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-10_492_888_255_625} The diagram shows two identical circles intersecting at points \(A\) and \(B\) and with centres at \(P\) and \(Q\). The radius of each circle is \(r\) and the distance \(P Q\) is \(\frac { 5 } { 3 } r\).
  1. Find the perimeter of the shaded region in terms of \(r\).
  2. Find the area of the shaded region in terms of \(r\).
CAIE P1 2022 November Q9
9 marks Standard +0.3
9 The first term of a geometric progression is 216 and the fourth term is 64.
  1. Find the sum to infinity of the progression.
    The second term of the geometric progression is equal to the second term of an arithmetic progression.
    The third term of the geometric progression is equal to the fifth term of the same arithmetic progression.
  2. Find the sum of the first 21 terms of the arithmetic progression. \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-14_798_786_269_667} The diagram shows the circle \(x ^ { 2 } + y ^ { 2 } = 2\) and the straight line \(y = 2 x - 1\) intersecting at the points \(A\) and \(B\). The point \(D\) on the \(x\)-axis is such that \(A D\) is perpendicular to the \(x\)-axis.
CAIE P1 2022 November Q11
11 marks Moderate -0.3
11 The coordinates of points \(A , B\) and \(C\) are \(A ( 5 , - 2 ) , B ( 10,3 )\) and \(C ( 2 p , p )\), where \(p\) is a constant.
  1. Given that \(A C\) and \(B C\) are equal in length, find the value of the fraction \(p\).
  2. It is now given instead that \(A C\) is perpendicular to \(B C\) and that \(p\) is an integer.
    1. Find the value of \(p\).
    2. Find the equation of the circle which passes through \(A , B\) and \(C\), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
      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 2023 November Q1
4 marks Moderate -0.8
1
  1. Expand \(( 1 + 3 x ) ^ { 6 }\) in ascending powers of \(x\) up to, and including, the term in \(x ^ { 2 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 - 7 x + x ^ { 2 } \right) ( 1 + 3 x ) ^ { 6 }\).
CAIE P1 2023 November Q2
4 marks Standard +0.3
2 A line has equation \(y = 2 c x + 3\) and a curve has equation \(y = c x ^ { 2 } + 3 x - c\), where \(c\) is a constant.
Showing all necessary working, determine which of the following statements is correct.
A The line and curve intersect only for a particular set of values of \(c\).
B The line and curve intersect for all values of \(c\).
C The line and curve do not intersect for any values of \(c\).
CAIE P1 2023 November Q3
3 marks Moderate -0.5
3 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-05_424_529_248_815} The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, \(x \mathrm {~cm}\), of each edge of the container increases at the constant rate of 0.01 cm per minute. The volume of the container at time \(t\) minutes is \(V \mathrm {~cm} ^ { 3 }\). Find the rate of increase of \(V\) when \(x = 20\).
CAIE P1 2023 November Q4
6 marks Moderate -0.3
4 The transformation R denotes a reflection in the \(x\)-axis and the transformation T denotes a translation of \(\binom { 3 } { - 1 }\).
  1. Find the equation, \(y = \mathrm { g } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations R followed by T .
  2. Find the equation, \(y = \mathrm { h } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations T followed by R .
  3. State fully the transformation that maps the curve \(y = \mathrm { g } ( x )\) onto the curve \(y = \mathrm { h } ( x )\).
CAIE P1 2023 November Q5
6 marks Standard +0.3
5
  1. Show that the equation $$4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0$$ may be expressed in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers to be found.
  2. Hence solve the equation \(4 \sin x + \frac { 5 } { \tan x } + \frac { 2 } { \sin x } = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2023 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-08_534_506_255_815} The diagram shows a motif formed by the major arc \(A B\) of a circle with radius \(r\) and centre \(O\), and the minor arc \(A O B\) of a circle, also with radius \(r\) but with centre \(C\). The point \(C\) lies on the circle with centre \(O\).
  1. Given that angle \(A C B = k \pi\) radians, state the value of the fraction \(k\).
  2. State the perimeter of the shaded motif in terms of \(\pi\) and \(r\).
  3. Find the area of the shaded motif, giving your answer in terms of \(\pi , r\) and \(\sqrt { 3 }\).
CAIE P1 2023 November Q7
7 marks Standard +0.3
7 The sum of the first two terms of a geometric progression is 15 and the sum to infinity is \(\frac { 125 } { 7 }\). The common ratio of the progression is negative. Find the third term of the progression.
CAIE P1 2023 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-12_684_776_274_680} The diagram shows the curves with equations \(y = 2 ( 2 x - 3 ) ^ { 4 }\) and \(y = ( 2 x - 3 ) ^ { 2 } + 1\) meeting at points \(A\) and \(B\).
  1. By using the substitution \(u = 2 x - 3\) find, by calculation, the coordinates of \(A\) and \(B\).
  2. Find the exact area of the shaded region.
CAIE P1 2023 November Q9
9 marks Standard +0.3
9
  1. Express \(4 x ^ { 2 } - 12 x + 13\) in the form \(( 2 x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
    The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(p < x < q\), where \(p\) and \(q\) are constants. The function g is defined by \(\mathrm { g } ( x ) = 3 x + 1\) for \(x < 8\).
  2. Given that it is possible to form the composite function gf , find the least possible value of \(p\) and the greatest possible value of \(q\).
  3. Find an expression for \(\operatorname { gf } ( x )\).
    The function h is defined by \(\mathrm { h } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(x < 0\).
  4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2023 November Q10
11 marks Moderate -0.3
10 A curve has a stationary point at \(( 2 , - 10 )\) and is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Find the equation of the curve.
  3. Find the coordinates of the other stationary point and determine its nature.
  4. Find the equation of the tangent to the curve at the point where the curve crosses the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-18_689_828_276_646} The diagram shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 40\). Parallel tangents, each with gradient 1 , touch the circle at points \(A\) and \(B\).
    1. Find the equation of the line \(A B\), giving the answer in the form \(y = m x + c\).
    2. Find the coordinates of \(A\), giving each coordinate in surd form.
    3. Find the equation of the tangent at \(A\), giving the answer in the form \(y = m x + c\), where \(c\) is in surd form.
      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 2023 November Q1
4 marks Moderate -0.8
1 A curve is such that its gradient at a point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x - 3 x ^ { - \frac { 1 } { 2 } }\). It is given that the curve passes through the point \(( 4,1 )\). Find the equation of the curve.
CAIE P1 2023 November Q2
4 marks Moderate -0.8
2 The circle with equation \(( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 40\) intersects the \(y\)-axis at points \(A\) and \(B\).
  1. Find the \(y\)-coordinates of \(A\) and \(B\), expressing your answers in terms of surds.
  2. Find the equation of the circle which has \(A B\) as its diameter.
CAIE P1 2023 November Q3
7 marks Moderate -0.3
3
  1. Show that the equation $$5 \cos \theta - \sin \theta \tan \theta + 1 = 0$$ may be expressed in the form \(a \cos ^ { 2 } \theta + b \cos \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be found.
  2. Hence solve the equation \(5 \cos \theta - \sin \theta \tan \theta + 1 = 0\) for \(0 < \theta < 2 \pi\).
CAIE P1 2023 November Q4
7 marks Moderate -0.3
4
  1. Expand the following in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\).
    1. \(( 1 + 2 x ) ^ { 5 }\).
    2. \(( 1 - a x ) ^ { 6 }\), where \(a\) is a constant.
      In the expansion of \(( 1 + 2 x ) ^ { 5 } ( 1 - a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is - 5 .
  2. Find the possible values of \(a\).
CAIE P1 2023 November Q5
7 marks Moderate -0.3
5 The first, second and third terms of a geometric progression are \(2 p + 6,5 p\) and \(8 p + 2\) respectively.
  1. Find the possible values of the constant \(p\).
  2. One of the values of \(p\) found in (a) is a negative fraction. Use this value of \(p\) to find the sum to infinity of this progression.
CAIE P1 2023 November Q6
7 marks Standard +0.3
6 A line has equation \(y = 6 x - c\) and a curve has equation \(y = c x ^ { 2 } + 2 x - 3\), where \(c\) is a constant. The line is a tangent to the curve at point \(P\). Find the possible values of \(c\) and the corresponding coordinates of \(P\).
CAIE P1 2023 November Q7
7 marks Moderate -0.3
7 The function f is defined by \(\mathrm { f } ( x ) = 1 + \frac { 3 } { x - 2 }\) for \(x > 2\).
  1. State the range of f.
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
    The function g is defined by \(\mathrm { g } ( x ) = 2 x - 2\) for \(x > 0\).
  3. Obtain a simplified expression for \(\mathrm { gf } ( x )\).
CAIE P1 2023 November Q8
5 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-10_515_936_274_577} The diagram shows part of the graph of \(y = \sin ( a ( x + b ) )\), where \(a\) and \(b\) are positive constants.
  1. State the value of \(a\) and one possible value of \(b\).
    Another curve, with equation \(y = \mathrm { f } ( x )\), has a single stationary point at the point \(( p , q )\), where \(p\) and \(q\) are constants. This curve is transformed to a curve with equation $$y = - 3 f \left( \frac { 1 } { 4 } ( x + 8 ) \right) .$$
  2. For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of \(p\) and \(q\).
CAIE P1 2023 November Q9
8 marks Standard +0.3
9 A curve has equation \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\).
  1. Find the equation of the normal to the curve at the point \(A ( 4,3 )\), giving your answer in the form \(y = m x + c\).
    A point is moving along the curve \(y = 2 x ^ { \frac { 1 } { 2 } } - 1\) in such a way that at \(A\) the rate of increase of the \(x\)-coordinate is \(3 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate of increase of the \(y\)-coordinate at \(A\).
    At \(A\) the moving point suddenly changes direction and speed, and moves down the normal in such a way that the rate of decrease of the \(y\)-coordinate is constant at \(5 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
  3. As the point moves down the normal, find the rate of change of its \(x\)-coordinate.