Questions — CAIE P3 (1110 questions)

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CAIE P3 2012 June Q5
7 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{4c71f68a-efb9-4408-bf03-874e0d4426d5-2_458_807_1786_667} The diagram shows the curve $$y = 8 \sin \frac { 1 } { 2 } x - \tan \frac { 1 } { 2 } x$$ for \(0 \leqslant x < \pi\). The \(x\)-coordinate of the maximum point is \(\alpha\) and the shaded region is enclosed by the curve and the lines \(x = \alpha\) and \(y = 0\).
  1. Show that \(\alpha = \frac { 2 } { 3 } \pi\).
  2. Find the exact value of the area of the shaded region.
CAIE P3 2012 June Q6
7 marks Standard +0.3
6 The equation of a curve is \(3 x ^ { 2 } - 4 x y + y ^ { 2 } = 45\).
  1. Find the gradient of the curve at the point \(( 2 , - 3 )\).
  2. Show that there are no points on the curve at which the gradient is 1 .
CAIE P3 2012 June Q7
8 marks Moderate -0.3
7 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x \mathrm { e } ^ { 3 x } } { y ^ { 2 } } .$$ It is given that \(y = 2\) when \(x = 0\). Solve the differential equation and hence find the value of \(y\) when \(x = 0.5\), giving your answer correct to 2 decimal places.
CAIE P3 2012 June Q8
9 marks Standard +0.3
8 The point \(P\) has coordinates \(( - 1,4,11 )\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\).
  1. Find the perpendicular distance from \(P\) to \(l\).
  2. Find the equation of the plane which contains \(P\) and \(l\), giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.
CAIE P3 2012 June Q9
10 marks Standard +0.8
9 By first expressing \(\frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 }\) in partial fractions, show that $$\int _ { 0 } ^ { 4 } \frac { 4 x ^ { 2 } + 5 x + 3 } { 2 x ^ { 2 } + 5 x + 2 } \mathrm {~d} x = 8 - \ln 9$$
CAIE P3 2012 June Q10
12 marks Standard +0.3
10
  1. It is given that \(2 \tan 2 x + 5 \tan ^ { 2 } x = 0\). Denoting \(\tan x\) by \(t\), form an equation in \(t\) and hence show that either \(t = 0\) or \(t = \sqrt [ 3 ] { } ( t + 0.8 )\).
  2. It is given that there is exactly one real value of \(t\) satisfying the equation \(t = \sqrt [ 3 ] { } ( t + 0.8 )\). Verify by calculation that this value lies between 1.2 and 1.3 .
  3. Use the iterative formula \(t _ { n + 1 } = \sqrt [ 3 ] { } \left( t _ { n } + 0.8 \right)\) to find the value of \(t\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  4. Using the values of \(t\) found in previous parts of the question, solve the equation $$2 \tan 2 x + 5 \tan ^ { 2 } x = 0$$ for \(- \pi \leqslant x \leqslant \pi\).
CAIE P3 2012 June Q1
4 marks Moderate -0.3
1 Solve the equation $$\ln ( 3 x + 4 ) = 2 \ln ( x + 1 )$$ giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q2
5 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-2_300_767_518_689} In the diagram, \(A B C\) is a triangle in which angle \(A B C\) is a right angle and \(B C = a\). A circular arc, with centre \(C\) and radius \(a\), joins \(B\) and the point \(M\) on \(A C\). The angle \(A C B\) is \(\theta\) radians. The area of the sector \(C M B\) is equal to one third of the area of the triangle \(A B C\).
  1. Show that \(\theta\) satisfies the equation $$\tan \theta = 3 \theta .$$
  2. This equation has one root in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). Use the iterative formula $$\theta _ { n + 1 } = \tan ^ { - 1 } \left( 3 \theta _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 June Q3
5 marks Moderate -0.3
3 Expand \(\sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2012 June Q4
6 marks Standard +0.8
4 Solve the equation $$\operatorname { cosec } 2 \theta = \sec \theta + \cot \theta$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2012 June Q5
6 marks Moderate -0.8
5 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 x + y }$$ and \(y = 0\) when \(x = 0\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2012 June Q6
8 marks Standard +0.8
6 The equation of a curve is \(y = 3 \sin x + 4 \cos ^ { 3 } x\).
  1. Find the \(x\)-coordinates of the stationary points of the curve in the interval \(0 < x < \pi\).
  2. Determine the nature of the stationary point in this interval for which \(x\) is least.
CAIE P3 2012 June Q8
10 marks Standard +0.8
8 Let \(I = \int _ { 2 } ^ { 5 } \frac { 5 } { x + \sqrt { } ( 6 - x ) } \mathrm { d } x\).
  1. Using the substitution \(u = \sqrt { } ( 6 - x )\), show that $$I = \int _ { 1 } ^ { 2 } \frac { 10 u } { ( 3 - u ) ( 2 + u ) } \mathrm { d } u$$
  2. Hence show that \(I = 2 \ln \left( \frac { 9 } { 2 } \right)\).
CAIE P3 2012 June Q9
11 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{d3f0b201-3004-497a-9b29-30c94d0bec5b-3_421_767_1567_689} The diagram shows the curve \(y = x ^ { \frac { 1 } { 2 } } \ln x\). The shaded region between the curve, the \(x\)-axis and the line \(x = \mathrm { e }\) is denoted by \(R\).
  1. Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = m x + c\).
  2. Find by integration the volume of the solid obtained when the region \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and e.
CAIE P3 2012 June Q10
12 marks Standard +0.8
10 Two planes, \(m\) and \(n\), have equations \(x + 2 y - 2 z = 1\) and \(2 x - 2 y + z = 7\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\).
  1. Show that \(l\) is parallel to \(m\).
  2. Find the position vector of the point of intersection of \(l\) and \(n\).
  3. A point \(P\) lying on \(l\) is such that its perpendicular distances from \(m\) and \(n\) are equal. Find the position vectors of the two possible positions for \(P\) and calculate the distance between them.
    [0pt] [The perpendicular distance of a point with position vector \(x _ { 1 } \mathbf { i } + y _ { 1 } \mathbf { j } + z _ { 1 } \mathbf { k }\) from the plane \(a x + b y + c z = d\) is \(\frac { \left| a x _ { 1 } + b y _ { 1 } + c z _ { 1 } - d \right| } { \sqrt { } \left( a ^ { 2 } + b ^ { 2 } + c ^ { 2 } \right) }\).]
CAIE P3 2012 June Q1
4 marks Moderate -0.3
1 Expand \(\frac { 1 } { \sqrt { } ( 4 + 3 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2012 June Q2
4 marks Moderate -0.3
2 Solve the equation \(\ln ( 2 x + 3 ) = 2 \ln x + \ln 3\), giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q3
5 marks Moderate -0.3
3 The parametric equations of a curve are $$x = \sin 2 \theta - \theta , \quad y = \cos 2 \theta + 2 \sin \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \cos \theta } { 1 + 2 \sin \theta }\).
CAIE P3 2012 June Q4
6 marks Standard +0.3
4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { x ^ { 3 } }\) has one stationary point.
  1. Find the \(x\)-coordinate of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2012 June Q5
8 marks Standard +0.3
5 In a certain chemical process a substance \(A\) reacts with another substance \(B\). The masses in grams of \(A\) and \(B\) present at time \(t\) seconds after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 0.6 x y\) and \(x = 5 \mathrm { e } ^ { - 3 t }\). When \(t = 0 , y = 70\).
  1. Form a differential equation in \(y\) and \(t\). Solve this differential equation and obtain an expression for \(y\) in terms of \(t\).
  2. The percentage of the initial mass of \(B\) remaining at time \(t\) is denoted by \(p\). Find the exact value approached by \(p\) as \(t\) becomes large.
CAIE P3 2012 June Q6
8 marks Standard +0.3
6 It is given that \(\tan 3 x = k \tan x\), where \(k\) is a constant and \(\tan x \neq 0\).
  1. By first expanding \(\tan ( 2 x + x )\), show that $$( 3 k - 1 ) \tan ^ { 2 } x = k - 3$$
  2. Hence solve the equation \(\tan 3 x = k \tan x\) when \(k = 4\), giving all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
  3. Show that the equation \(\tan 3 x = k \tan x\) has no root in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\) when \(k = 2\).
CAIE P3 2012 June Q7
9 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{e2cc23d2-f3ac-488b-97e1-79e2a98a87ba-3_421_885_251_628} The diagram shows part of the curve \(y = \cos ( \sqrt { } x )\) for \(x \geqslant 0\), where \(x\) is in radians. The shaded region between the curve, the axes and the line \(x = p ^ { 2 }\), where \(p > 0\), is denoted by \(R\). The area of \(R\) is equal to 1 .
  1. Use the substitution \(x = u ^ { 2 }\) to find \(\int _ { 0 } ^ { p ^ { 2 } } \cos ( \sqrt { } x ) \mathrm { d } x\). Hence show that \(\sin p = \frac { 3 - 2 \cos p } { 2 p }\).
  2. Use the iterative formula \(p _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 3 - 2 \cos p _ { n } } { 2 p _ { n } } \right)\), with initial value \(p _ { 1 } = 1\), to find the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 June Q8
10 marks Standard +0.3
8 Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } - 7 x - 1 } { ( x + 1 ) ( 2 x - 3 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that \(\int _ { 2 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln \left( \frac { 49 } { 3 } \right)\).
CAIE P3 2012 June Q10
11 marks Standard +0.3
10
  1. The complex numbers \(u\) and \(w\) satisfy the equations $$u - w = 4 \mathrm { i } \quad \text { and } \quad u w = 5$$ Solve the equations for \(u\) and \(w\), giving all answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 + 2 \mathrm { i } | \leqslant 2 , \arg z \leqslant - \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \geqslant 1\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
    2. Calculate the greatest possible value of \(\operatorname { Re } z\) for points lying in the shaded region.
CAIE P3 2013 June Q1
3 marks Easy -1.8
1 Find the quotient and remainder when \(2 x ^ { 2 }\) is divided by \(x + 2\).