Questions P3 (1243 questions)

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CAIE P3 2019 June Q7
9 marks Standard +0.3
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
10 marks Standard +0.3
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 \(x ^ { 2 }\).
CAIE P3 2019 June Q9
10 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-14_666_703_260_721} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and four points \(A , B , C\) and \(D\) with position vectors \(\overrightarrow { O A } = 3 \mathbf { i } , \overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j } , \overrightarrow { O C } = \mathbf { i } + 3 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }\).
  1. Find the equation of the plane \(B C D\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the acute angle between the planes \(B C D\) and \(O A B C\).
CAIE P3 2019 June Q10
13 marks Standard +0.3
10 Throughout this question the use of a calculator is not permitted.
The complex number \(( \sqrt { } 3 ) + \mathrm { i }\) is denoted by \(u\).
  1. Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). Hence or otherwise state the exact values of the modulus and argument of \(u ^ { 4 }\).
  2. Verify that \(u\) is a root of the equation \(z ^ { 3 } - 8 z + 8 \sqrt { } 3 = 0\) and state the other complex root of this equation.
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\). 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
4 marks Standard +0.3
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - x ) ( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
CAIE P3 2019 June Q2
4 marks Moderate -0.3
2 Showing all necessary working, solve the equation \(9 ^ { x } = 3 ^ { x } + 12\). Give your answer correct to 2 decimal places.
CAIE P3 2019 June Q3
5 marks Standard +0.3
3 Showing all necessary working, solve the equation \(\cot 2 \theta = 2 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 June Q4
7 marks Standard +0.3
4 Find the exact coordinates of the point on the curve \(y = \frac { x } { 1 + \ln x }\) at which the gradient of the tangent is equal to \(\frac { 1 } { 4 }\).
CAIE P3 2019 June Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-08_492_812_260_664} In the diagram, \(A\) is the mid-point of the semicircle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the semicircle at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The area of the shaded region bounded by \(A B , A C\) and the arc with centre \(A\) is equal to half the area of the semicircle.
  1. Use triangle \(O A B\) to show that \(A B = 2 r \cos x\).
  2. Hence show that \(x = \cos ^ { - 1 } \sqrt { } \left( \frac { \pi } { 16 x } \right)\).
  3. Verify by calculation that \(x\) lies between 1 and 1.5.
  4. Use an iterative formula based on the equation in part (ii) to determine \(x\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 June Q7
8 marks Standard +0.3
7 The variables \(x\) and \(y\) satisfy the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { x + y }\). It is given that \(y = 0\) when \(x = 0\).
  1. Solve the differential equation, obtaining \(y\) in terms of \(x\).
  2. Explain why \(x\) can only take values that are less than 1 .
CAIE P3 2019 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 10 x + 9 } { ( 2 x + 1 ) ( 2 x + 3 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence show that \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 2 } \ln \frac { 9 } { 5 } + \frac { 1 } { 5 }\).
CAIE P3 2019 June Q9
10 marks Standard +0.3
9 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. The plane \(m\) is perpendicular to \(A B\) and passes through the mid-point of \(A B\). The plane \(m\) intersects the line \(l\) at the point \(P\). Find the equation of \(m\) and the position vector of \(P\).
CAIE P3 2019 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{772393d7-6e81-4b99-913a-63c9f87d1af2-16_524_689_260_726} The diagram shows the curve \(y = \sin 3 x \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) and its minimum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. By expanding \(\sin ( 3 x + x )\) and \(\sin ( 3 x - x )\) show that $$\sin 3 x \cos x = \frac { 1 } { 2 } ( \sin 4 x + \sin 2 x ) .$$
  2. Using the result of part (i) and showing all necessary working, find the exact area of the region \(R\).
  3. Using the result of part (i), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\cos 2 x\) and hence find the \(x\)-coordinate of \(M\), giving your answer correct to 2 decimal places.
    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
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(5 ^ { 3 - 2 x } = 4 \left( 7 ^ { x } \right)\), giving your answer correct to 3 decimal places.
CAIE P3 2019 June Q2
5 marks Standard +0.3
2 Show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x ^ { 2 } \cos 2 x \mathrm {~d} x = \frac { 1 } { 32 } \left( \pi ^ { 2 } - 8 \right)\).
CAIE P3 2019 June Q3
7 marks Standard +0.3
3 Let \(f ( \theta ) = \frac { 1 - \cos 2 \theta + \sin 2 \theta } { 1 + \cos 2 \theta + \sin 2 \theta }\).
  1. Show that \(\mathrm { f } ( \theta ) = \tan \theta\).
  2. Hence show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( \theta ) \mathrm { d } \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).
CAIE P3 2019 June Q4
7 marks Standard +0.3
4 The equation of a curve is \(y = \frac { 1 + \mathrm { e } ^ { - x } } { 1 - \mathrm { e } ^ { - x } }\), for \(x > 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is always negative.
  2. The gradient of the curve is equal to - 1 when \(x = a\). Show that \(a\) satisfies the equation \(\mathrm { e } ^ { 2 a } - 4 \mathrm { e } ^ { a } + 1 = 0\). Hence find the exact value of \(a\).
CAIE P3 2019 June Q5
7 marks Standard +0.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 5$$ It is given that \(y = 2\) when \(x = 0\). Solve the differential equation obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
CAIE P3 2019 June Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{87392b1c-3683-45b4-8d55-36760b5f0cc1-10_547_531_260_806} The diagram shows the curve \(y = x ^ { 4 } - 2 x ^ { 3 } - 7 x - 6\). The curve intersects the \(x\)-axis at the points \(( a , 0 )\) and \(( b , 0 )\), where \(a < b\). It is given that \(b\) is an integer.
  1. Find the value of \(b\).
  2. Hence show that \(a\) satisfies the equation \(a = - \frac { 1 } { 3 } \left( 2 + a ^ { 2 } + a ^ { 3 } \right)\).
  3. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2019 June Q7
7 marks Standard +0.3
7 The curve \(y = \sin \left( x + \frac { 1 } { 3 } \pi \right) \cos x\) has two stationary points in the interval \(0 \leqslant x \leqslant \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. By considering the formula for \(\cos ( A + B )\), show that, at the stationary points on the curve, \(\cos \left( 2 x + \frac { 1 } { 3 } \pi \right) = 0\).
  3. Hence find the exact \(x\)-coordinates of the stationary points.
CAIE P3 2019 June Q8
9 marks Standard +0.3
8 Throughout this question the use of a calculator is not permitted.
The complex number \(u\) is defined by $$u = \frac { 4 \mathrm { i } } { 1 - ( \sqrt { } 3 ) \mathrm { i } }$$
  1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
  2. Find the exact modulus and argument of \(u\).
  3. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z | < 2\) and \(| z - u | < | z |\).
CAIE P3 2019 June Q9
10 marks Standard +0.8
9 Let \(\mathrm { f } ( x ) = \frac { 2 x ( 5 - x ) } { ( 3 + x ) ( 1 - x ) ^ { 2 } }\).
  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 \(x ^ { 3 }\).
CAIE P3 2019 June Q10
11 marks Standard +0.3
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\).
  1. The point \(P\) has position vector \(4 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\). Find the length of the perpendicular from \(P\) to \(l\).
  2. It is given that \(l\) lies in the plane with equation \(a x + b y + 2 z = 13\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).
    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 2016 March Q1
3 marks Standard +0.3
1 Solve the equation \(\ln \left( x ^ { 2 } + 4 \right) = 2 \ln x + \ln 4\), giving your answer in an exact form.
CAIE P3 2016 March Q2
6 marks Standard +0.3
2 Express the equation \(\tan \left( \theta + 45 ^ { \circ } \right) - 2 \tan \left( \theta - 45 ^ { \circ } \right) = 4\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).