Questions P1 (1374 questions)

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CAIE P1 2023 November Q10
10 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 4 x - 3 ) ^ { \frac { 5 } { 3 } } - \frac { 20 } { 3 } x\).
  1. Find the \(x\)-coordinates of the stationary points of the curve and determine their nature.
  2. State the set of values for which the function f is increasing.
CAIE P1 2023 November Q11
11 The coordinates of points \(A , B\) and \(C\) are (6, 4), ( \(p , 7\) ) and (14, 18) respectively, where \(p\) is a constant. The line \(A B\) is perpendicular to the line \(B C\).
  1. Given that \(p < 10\), find the value of \(p\).
    A circle passes through the points \(A , B\) and \(C\).
  2. Find the equation of the circle.
  3. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(d x + e y + f = 0\), where \(d , e\) and \(f\) are integers.
    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
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
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
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
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
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
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 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
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
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.
CAIE P1 2023 November Q10
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
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 2024 November Q1
1 In the expansion of \(\left( k x + \frac { 2 } { x } \right) ^ { 4 }\) ,where \(k\) is a positive constant,the term independent of \(x\) is equal to 150 .
Find the value of \(k\) and hence determine the coefficient of \(x ^ { 2 }\) in the expansion.
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-02_2718_35_107_2012}
CAIE P1 2024 November Q2
2 The curve \(y = x ^ { 2 } - \frac { a } { x }\) has a stationary point at \(( - 3 , b )\).
Find the values of the constants \(a\) and \(b\).
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-04_403_517_251_776} The diagram shows a sector of a circle, centre \(O\), where \(O B = O C = 15 \mathrm {~cm}\). The size of angle \(B O C\) is \(\frac { 2 } { 5 } \pi\) radians. Points \(A\) and \(D\) on the lines \(O B\) and \(O C\) respectively are joined by an arc \(A D\) of circle with centre \(O\). The shaded region is bounded by the \(\operatorname { arcs } A D\) and \(B C\) and by the straight lines \(A B\) and \(D C\). It is given that the area of the shaded region is \(\frac { 209 } { 5 } \pi \mathrm {~cm} ^ { 2 }\). Find the perimeter of the shaded region. Give your answer in terms of \(\pi\).
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-04_2720_38_105_2010}
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-05_2723_35_101_20}
CAIE P1 2024 November Q4
4 Show that the curve with equation \(x ^ { 2 } - 3 x y - 40 = 0\) and the line with equation \(3 x + y + k = 0\) meet for all values of the constant \(k\).
CAIE P1 2024 November Q5
5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x - 3 \sqrt { x } + 1\).
  1. Find the \(x\)-coordinate of the point on the curve at which the gradient is \(\frac { 11 } { 2 }\).
  2. Given that the curve passes through the point \(( 4,11 )\), find the equation of the curve.
    \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-06_2715_31_106_2016}
CAIE P1 2024 November Q6
6 Circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 18 = 0 \text { and } ( x - 9 ) ^ { 2 } + ( y + 4 ) ^ { 2 } - 64 = 0$$ respectively.
  1. Find the distance between the centres of the circles.
    \(P\) and \(Q\) are points on \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
  2. Find the greatest and least possible values of \(d\).
CAIE P1 2024 November Q7
4 marks
7
\includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-08_534_462_248_804} The diagram shows part of the curve with equation \(y = \frac { 12 } { \sqrt [ 3 ] { 2 x + 1 } }\). The point \(A\) on the curve has coordinates \(\left( \frac { 7 } { 2 } , 6 \right)\).
  1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = m x + c\). [4]
    \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-08_2716_38_109_2012}
  2. Find the area of the region bounded by the curve and the lines \(x = 0 , x = \frac { 7 } { 2 }\) and \(y = 0\).
CAIE P1 2024 November Q8
8
  1. It is given that \(\beta\) is an angle between \(90 ^ { \circ }\) and \(180 ^ { \circ }\) such that \(\sin \beta = a\).
    Express \(\tan ^ { 2 } \beta - 3 \sin \beta \cos \beta\) in terms of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-11_2726_35_97_20}
  2. Solve the equation \(\sin ^ { 2 } \theta + 2 \cos ^ { 2 } \theta = 4 \sin \theta + 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P1 2024 November Q9
9 The equation of a curve is \(y = 4 + 5 x + 6 x ^ { 2 } - 3 x ^ { 3 }\).
  1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases.
  2. It is given that \(y = 9 x + k\) is a tangent to the curve. Find the value of the constant \(k\).
CAIE P1 2024 November Q10
10 An arithmetic progression has first term 5 and common difference \(d\), where \(d > 0\). The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.
  1. Find the value of \(d\).
  2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S _ { 77 }\). The sum of the first 10 terms of the geometric progression is denoted by \(G _ { 10 }\). Find the value of \(S _ { 77 } - G _ { 10 }\).
CAIE P1 2024 November Q11
11 The function f is defined by \(\mathrm { f } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \in \mathbb { R }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are constants, and state the range of f .
  2. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = \mathrm { h } ( x )\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \mathrm { h } ( x )\) has a minimum point at the origin. Give details of the reflection and translation involved.
    \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-16_2715_38_109_2009} The function g is defined by \(\mathrm { g } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \leqslant 0\).
  3. Sketch the graph of \(y = \mathrm { g } ( x )\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes.
  4. Sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\) on your diagram in (c), and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). You should label the two graphs in your diagram appropriately and show any relevant mirror line.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-18_2714_38_109_2010}
CAIE P1 2024 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{39393c20-34df-4167-a8bf-e4067371de81-02_778_1195_260_475} The diagram shows the curve with equation \(y = a \sin ( b x ) + c\) for \(0 \leqslant x \leqslant 2 \pi\), where \(a , b\) and \(c\) are positive constants.
  1. State the values of \(a , b\) and \(c\).
  2. For these values of \(a , b\) and \(c\), determine the number of solutions in the interval \(0 \leqslant x \leqslant 2 \pi\) for each of the following equations:
    1. \(a \sin ( b x ) + c = 7 - x\)
    2. \(\quad a \sin ( b x ) + c = 2 \pi ( x - 1 )\).
CAIE P1 2024 November Q2
2 The first term of an arithmetic progression is - 20 and the common difference is 5 .
  1. Find the sum of the first 20 terms of the progression.
    It is given that the sum of the first \(2 k\) terms is 10 times the sum of the first \(k\) terms.
  2. Find the value of \(k\).