Questions P3 (1203 questions)

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CAIE P3 2021 June Q9
9 The equation of a curve is \(y = x ^ { - \frac { 2 } { 3 } } \ln x\) for \(x > 0\). The curve has one stationary point.
  1. Find the exact coordinates of the stationary point.
  2. Show that \(\int _ { 1 } ^ { 8 } y \mathrm {~d} x = 18 \ln 2 - 9\).
CAIE P3 2021 June Q10
10 The variables \(x\) and \(t\) satisfy the differential equation \(\frac { \mathrm { d } x } { \mathrm {~d} t } = x ^ { 2 } ( 1 + 2 x )\), and \(x = 1\) when \(t = 0\).
Using partial fractions, solve the differential equation, obtaining an expression for \(t\) in terms of \(x\).
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 2021 June Q1
1 Solve the inequality \(| 2 x - 1 | < 3 | x + 1 |\).
CAIE P3 2021 June Q2
2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 1 - i | \leqslant 1\) and \(\arg ( z - 1 ) \leqslant \frac { 3 } { 4 } \pi\).
CAIE P3 2021 June Q3
3 The variables \(x\) and \(y\) satisfy the equation \(x = A \left( 3 ^ { - y } \right)\), where \(A\) is a constant.
  1. Explain why the graph of \(y\) against \(\ln x\) is a straight line and state the exact value of the gradient of the line.
    It is given that the line intersects the \(y\)-axis at the point where \(y = 1.3\).
  2. Calculate the value of \(A\), giving your answer correct to 2 decimal places.
CAIE P3 2021 June Q4
4 Using integration by parts, find the exact value of \(\int _ { 0 } ^ { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P3 2021 June Q5
5 The complex number \(u\) is given by \(u = 10 - 4 \sqrt { 6 } \mathrm { i }\).
Find the two square roots of \(u\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and exact.
CAIE P3 2021 June Q6
6
  1. Prove that \(\operatorname { cosec } 2 \theta - \cot 2 \theta \equiv \tan \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } ( \operatorname { cosec } 2 \theta - \cot 2 \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln 2\).
CAIE P3 2021 June Q7
7 A curve is such that the gradient at a general point with coordinates \(( x , y )\) is proportional to \(\frac { y } { \sqrt { x + 1 } }\). The curve passes through the points with coordinates \(( 0,1 )\) and \(( 3 , e )\). By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).
CAIE P3 2021 June Q8
8 The equation of a curve is \(y = e ^ { - 5 x } \tan ^ { 2 } x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
Find the \(x\)-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.
CAIE P3 2021 June Q9
9 Let \(\mathrm { f } ( x ) = \frac { 14 - 3 x + 2 x ^ { 2 } } { ( 2 + x ) \left( 3 + x ^ { 2 } \right) }\).
  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 \(x ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{459b8403-a481-4ece-88c0-e7600a47c8e4-14_292_732_264_705} The diagram shows a trapezium \(A B C D\) in which \(A D = B C = r\) and \(A B = 2 r\). The acute angles \(B A D\) and \(A B C\) are both equal to \(x\) radians. Circular arcs of radius \(r\) with centres \(A\) and \(B\) meet at \(M\), the midpoint of \(A B\).
CAIE P3 2021 June Q11
11 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j }\) and \(\overrightarrow { O B } = \mathbf { j } - 2 \mathbf { k }\).
  1. Show that \(O A = O B\) and use a scalar product to calculate angle \(A O B\) in degrees.
    The midpoint of \(A B\) is \(M\). The point \(P\) on the line through \(O\) and \(M\) is such that \(P A : O A = \sqrt { 7 } : 1\).
  2. Find the possible position vectors of \(P\).
    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 2021 June Q1
1 Expand \(( 1 + 3 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2021 June Q2
2 Solve the equation \(4 ^ { x } = 3 + 4 ^ { - x }\). Give your answer correct to 3 decimal places.
CAIE P3 2021 June Q3
3 The parametric equations of a curve are $$x = t + \ln ( t + 2 ) , \quad y = ( t - 1 ) \mathrm { e } ^ { - 2 t }$$ where \(t > - 2\).
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
  2. Find the exact \(y\)-coordinate of the stationary point of the curve.
CAIE P3 2021 June Q4
4 Let \(f ( x ) = \frac { 15 - 6 x } { ( 1 + 2 x ) ( 4 - x ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\), giving your answer in the form \(\ln \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.
CAIE P3 2021 June Q5
5
  1. By first expanding \(\tan ( 2 \theta + 2 \theta )\), show that the equation \(\tan 4 \theta = \frac { 1 } { 2 } \tan \theta\) may be expressed as \(\tan ^ { 4 } \theta + 2 \tan ^ { 2 } \theta - 7 = 0\).
  2. Hence solve the equation \(\tan 4 \theta = \frac { 1 } { 2 } \tan \theta\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2021 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation \(\cot \frac { 1 } { 2 } x = 1 + \mathrm { e } ^ { - x }\) has exactly one root in the interval \(0 < x \leqslant \pi\).
  2. Verify by calculation that this root lies between 1 and 1.5.
  3. Use the iterative formula \(x _ { n + 1 } = 2 \tan ^ { - 1 } \left( \frac { 1 } { 1 + \mathrm { e } ^ { - x _ { n } } } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2021 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{1990cbac-d96f-4484-be4b-67dab35b3147-10_647_519_260_813} For the curve shown in the diagram, the normal to the curve at the point \(P\) with coordinates \(( x , y )\) meets the \(x\)-axis at \(N\). The point \(M\) is the foot of the perpendicular from \(P\) to the \(x\)-axis. The curve is such that for all values of \(x\) in the interval \(0 \leqslant x < \frac { 1 } { 2 } \pi\), the area of triangle \(P M N\) is equal to \(\tan x\).
    1. Show that \(\frac { M N } { y } = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Hence show that \(x\) and \(y\) satisfy the differential equation \(\frac { 1 } { 2 } y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \tan x\).
  2. Given that \(y = 1\) when \(x = 0\), solve this differential equation to find the equation of the curve, expressing \(y\) in terms of \(x\).
CAIE P3 2021 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{1990cbac-d96f-4484-be4b-67dab35b3147-12_458_725_262_708} The diagram shows the curve \(y = \frac { \ln x } { x ^ { 4 } }\) and its maximum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. By using integration by parts, show that for all \(a > 1 , \int _ { 1 } ^ { a } \frac { \ln x } { x ^ { 4 } } \mathrm {~d} x < \frac { 1 } { 9 }\).
CAIE P3 2021 June Q9
9 The quadrilateral \(A B C D\) is a trapezium in which \(A B\) and \(D C\) are parallel. With respect to the origin \(O\), the position vectors of \(A , B\) and \(C\) are given by \(\overrightarrow { O A } = - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \overrightarrow { O B } = \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\).
  1. Given that \(\overrightarrow { D C } = 3 \overrightarrow { A B }\), find the position vector of \(D\).
  2. State a vector equation for the line through \(A\) and \(B\).
  3. Find the distance between the parallel sides and hence find the area of the trapezium.
CAIE P3 2021 June Q10
10
  1. Verify that \(- 1 + \sqrt { 2 } \mathrm { i }\) is a root of the equation \(z ^ { 4 } + 3 z ^ { 2 } + 2 z + 12 = 0\).
  2. Find the other roots of this equation.
    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 2022 June Q1
1 Solve the equation \(2 \left( 3 ^ { 2 x - 1 } \right) = 4 ^ { x + 1 }\), giving your answer correct to 2 decimal places.
CAIE P3 2022 June Q2
2
  1. Expand \(\left( 2 - x ^ { 2 } \right) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2022 June Q3
3 Solve the equation \(2 \cot 2 x + 3 \cot x = 5\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).