Questions P3 (1203 questions)

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CAIE P3 2017 June Q4
4 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \theta \sin \frac { 1 } { 2 } \theta \mathrm {~d} \theta\).
CAIE P3 2017 June Q5
5 A curve has equation \(y = \frac { 2 } { 3 } \ln \left( 1 + 3 \cos ^ { 2 } x \right)\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan x\).
  2. Hence find the \(x\)-coordinate of the point on the curve where the gradient is - 1 . Give your answer correct to 3 significant figures.
CAIE P3 2017 June Q6
6 The equation \(\cot x = 1 - x\) has one root in the interval \(0 < x < \pi\), denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) is greater than 2.5.
  2. Show that, if a sequence of values in the interval \(0 < x < \pi\) given by the iterative formula \(x _ { n + 1 } = \pi + \tan ^ { - 1 } \left( \frac { 1 } { 1 - x _ { n } } \right)\) converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2017 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{7f6f82c3-37d3-48da-9958-e4ef366a6467-10_389_488_258_831} The diagram shows a sketch of the curve \(y = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x }\) for \(x > 0\), and its minimum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Use the trapezium rule with two intervals to estimate the value of $$\int _ { 1 } ^ { 3 } \frac { \mathrm { e } ^ { \frac { 1 } { 2 } x } } { x } \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  3. The estimate found in part (ii) is denoted by \(E\). Explain, without further calculation, whether another estimate found using the trapezium rule with four intervals would be greater than \(E\) or less than \(E\).
CAIE P3 2017 June Q8
8 In a certain chemical reaction, a compound \(A\) is formed from a compound \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the reaction are \(x\) and \(y\) respectively and the sum of the masses is equal to 50 throughout the reaction. At any time the rate of increase of the mass of \(A\) is proportional to the mass of \(B\) at that time.
  1. Explain why \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 50 - x )\), where \(k\) is a constant.
    It is given that \(x = 0\) when \(t = 0\), and \(x = 25\) when \(t = 10\).
  2. Solve the differential equation in part (i) and express \(x\) in terms of \(t\).
CAIE P3 2017 June Q9
9 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x + 2 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 8 } \right) - 1\).
CAIE P3 2017 June Q10
10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + m \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } )\), where \(m\) is a constant.
  1. Given that the line \(l\) intersects the line passing through \(A\) and \(B\), find the value of \(m\).
  2. Find the equation of the plane which is parallel to \(\mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k }\) and contains the points \(A\) and \(B\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 June Q11
11 Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(z\) and \(w\) satisfy the equations $$z + ( 1 + \mathrm { i } ) w = \mathrm { i } \quad \text { and } \quad ( 1 - \mathrm { i } ) z + \mathrm { i } w = 1$$ Solve the equations for \(z\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. The complex numbers \(u\) and \(v\) are given by \(u = 1 + ( 2 \sqrt { 3 } ) \mathrm { i }\) and \(v = 3 + 2 \mathrm { i }\). In an Argand diagram, \(u\) and \(v\) are represented by the points \(A\) and \(B\). A third point \(C\) lies in the first quadrant and is such that \(B C = 2 A B\) and angle \(A B C = 90 ^ { \circ }\). Find the complex number \(z\) represented by \(C\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
CAIE P3 2018 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.
  1. Find the equations of \(A C\) and \(O B\).
  2. Find, by calculation, the coordinates of \(B\).
CAIE P3 2018 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-10_499_922_262_607} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circle and \(A T\) is a tangent to the circle. Angle \(A O B = \theta\) radians and \(O B T\) is a straight line.
  1. Express the area of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 3\) and \(\theta = 1.2\), find the perimeter of the shaded region.
CAIE P3 2018 June Q4
4 The function f is such that \(\mathrm { f } ( x ) = a + b \cos x\) for \(0 \leqslant x \leqslant 2 \pi\). It is given that \(\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5\) and \(\mathrm { f } ( \pi ) = 11\).
  1. Find the values of the constants \(a\) and \(b\).
    \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-05_63_1566_397_328}
  2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution.
    \includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-06_622_878_260_632} The diagram shows a three-dimensional shape. The base \(O A B\) is a horizontal triangle in which angle \(A O B\) is \(90 ^ { \circ }\). The side \(O B C D\) is a rectangle and the side \(O A D\) lies in a vertical plane. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O B\) respectively and the unit vector \(\mathbf { k }\) is vertical. The position vectors of \(A , B\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).
CAIE P3 2018 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-08_454_684_255_726} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor \(\operatorname { arc } A B\) and the lines \(A T\) and \(B T\). Angle \(A O B\) is \(2 \theta\) radians.
  1. In the case where the area of the sector \(A O B\) is the same as the area of the shaded region, show that \(\tan \theta = 2 \theta\).
  2. In the case where \(r = 8 \mathrm {~cm}\) and the length of the minor \(\operatorname { arc } A B\) is 19.2 cm , find the area of the shaded region.
CAIE P3 2018 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{8c1580a7-6e79-4cd0-b59a-a1c33bd76b0c-18_643_969_258_587} The diagram shows part of the curve \(y = \frac { x } { 2 } + \frac { 6 } { x }\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).
  1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in terms of \(\pi\).
    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 P3 2018 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.
CAIE P3 2018 June Q7
3 marks
7
    1. Express \(\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 }\) in the form \(a \sin ^ { 2 } \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 } = \frac { 1 } { 4 }$$ for \(- 90 ^ { \circ } \leqslant \theta \leqslant 0 ^ { \circ }\).

  2. \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-11_549_796_267_717} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\).
    2. Find the \(y\)-coordinate of \(B\).
CAIE P3 2018 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.
  1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
  2. Use a scalar product to find angle \(B D O\).
CAIE P3 2018 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.
  1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
    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 P3 2019 June Q1
1 Use the trapezium rule with 3 intervals to estimate the value of $$\int _ { 0 } ^ { 3 } \left| 2 ^ { x } - 4 \right| \mathrm { d } x$$
CAIE P3 2019 June Q2
2 Showing all necessary working, solve the equation \(\ln ( 2 x - 3 ) = 2 \ln x - \ln ( x - 1 )\). Give your answer correct to 2 decimal places.
CAIE P3 2019 June Q3
3 Find the gradient of the curve \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 1\) at the point with coordinates \(( 1,3 )\).
CAIE P3 2019 June Q4
4 By first expressing the equation \(\cot \theta - \cot \left( \theta + 45 ^ { \circ } \right) = 3\) as a quadratic equation in \(\tan \theta\), solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 June Q5
5
  1. Differentiate \(\frac { 1 } { \sin ^ { 2 } \theta }\) with respect to \(\theta\).
  2. The variables \(x\) and \(\theta\) satisfy the differential equation $$x \tan \theta \frac { d x } { d \theta } + \operatorname { cosec } ^ { 2 } \theta = 0$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\) and \(x > 0\). It is given that \(x = 4\) when \(\theta = \frac { 1 } { 6 } \pi\). Solve the differential equation, obtaining an expression for \(x\) in terms of \(\theta\).
CAIE P3 2019 June Q6
6
  1. By first expanding \(\sin ( 2 x + x )\), show that \(\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x\).
  2. Hence, showing all necessary working, find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x \mathrm {~d} x\).
CAIE P3 2019 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-10_702_597_258_772} The diagram shows the curves \(y = 4 \cos \frac { 1 } { 2 } x\) and \(y = \frac { 1 } { 4 - x }\), for \(0 \leqslant x < 4\). When \(x = a\), the tangents to the curves are perpendicular.
  1. Show that \(a = 4 - \sqrt { } \left( 2 \sin \frac { 1 } { 2 } a \right)\).
  2. Verify by calculation that \(a\) lies between 2 and 3 .
  3. Use an iterative formula based on the equation in part (i) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 June Q8
8 Let \(f ( x ) = \frac { 16 - 17 x } { ( 2 + x ) ( 3 - x ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).