Questions — CAIE P3 (1070 questions)

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CAIE P3 2014 June Q8
8
  1. By sketching each of the graphs \(y = \operatorname { cosec } x\) and \(y = x ( \pi - x )\) for \(0 < x < \pi\), show that the equation $$\operatorname { cosec } x = x ( \pi - x )$$ has exactly two real roots in the interval \(0 < x < \pi\).
  2. Show that the equation \(\operatorname { cosec } x = x ( \pi - x )\) can be written in the form \(x = \frac { 1 + x ^ { 2 } \sin x } { \pi \sin x }\).
  3. The two real roots of the equation \(\operatorname { cosec } x = x ( \pi - x )\) in the interval \(0 < x < \pi\) are denoted by \(\alpha\) and \(\beta\), where \(\alpha < \beta\).
    (a) Use the iterative formula $$x _ { n + 1 } = \frac { 1 + x _ { n } ^ { 2 } \sin x _ { n } } { \pi \sin x _ { n } }$$ to find \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
    (b) Deduce the value of \(\beta\) correct to 2 decimal places.
CAIE P3 2014 June Q9
9
  1. Express \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 4 + 12 x + x ^ { 2 } } { ( 3 - x ) ( 1 + 2 x ) ^ { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
CAIE P3 2014 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{b6bede75-3da4-4dda-9303-a5a692fc2572-3_556_1093_1596_523} The diagram shows the curve \(y = 10 e ^ { - \frac { 1 } { 2 } x } \sin 4 x\) for \(x \geqslant 0\). The stationary points are labelled \(T _ { 1 } , T _ { 2 }\), \(T _ { 3 } , \ldots\) as shown.
  1. Find the \(x\)-coordinates of \(T _ { 1 }\) and \(T _ { 2 }\), giving each \(x\)-coordinate correct to 3 decimal places.
  2. It is given that the \(x\)-coordinate of \(T _ { n }\) is greater than 25 . Find the least possible value of \(n\).
CAIE P3 2014 June Q1
1 Find the set of values of \(x\) satisfying the inequality $$| x + 2 a | > 3 | x - a |$$ where \(a\) is a positive constant.
CAIE P3 2014 June Q2
2 Solve the equation $$2 \ln \left( 5 - \mathrm { e } ^ { - 2 x } \right) = 1$$ giving your answer correct to 3 significant figures.
CAIE P3 2014 June Q3
3 Solve the equation $$\cos \left( x + 30 ^ { \circ } \right) = 2 \cos x$$ giving all solutions in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2014 June Q4
5 marks
4 The parametric equations of a curve are $$x = t - \tan t , \quad y = \ln ( \cos t )$$ for \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).
  2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2 . Give your answer correct to 3 significant figures.
  3. The polynomial \(\mathrm { f } ( x )\) is of the form \(( x - 2 ) ^ { 2 } \mathrm {~g} ( x )\), where \(\mathrm { g } ( x )\) is another polynomial. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ^ { \prime } ( x )\).
  4. The polynomial \(x ^ { 5 } + a x ^ { 4 } + 3 x ^ { 3 } + b x ^ { 2 } + a\), where \(a\) and \(b\) are constants, has a factor \(( x - 2 ) ^ { 2 }\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]
CAIE P3 2014 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_551_519_260_813} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The shaded region is bounded by \(A B , A C\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.
  1. Show that \(x = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x } \right)\).
  2. Verify by calculation that \(x\) lies between 1 and 1.5.
  3. Use the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x _ { n } } \right)$$ to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 June Q7
7
  1. It is given that \(- 1 + ( \sqrt { } 5 ) \mathrm { i }\) is a root of the equation \(z ^ { 3 } + 2 z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation.
  2. The complex number \(w\) has modulus 1 and argument \(2 \theta\) radians. Show that \(\frac { w - 1 } { w + 1 } = \mathrm { i } \tan \theta\).
CAIE P3 2014 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_391_826_1946_657} The diagram shows the curve \(y = x \cos \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y + 4 \sin \frac { 1 } { 2 } x = 0\).
  2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis.
CAIE P3 2014 June Q9
9 The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \(( 1 - 0.01 N )\). When \(t = 0 , N = 20\) and \(\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.32\).
  1. Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation $$\frac { \mathrm { d } N } { \mathrm {~d} t } = 0.02 N ( 1 - 0.01 N )$$
  2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\).
  3. Find the time at which the population will be double its value at \(t = 0\).
CAIE P3 2014 June Q10
10 Referred to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \quad \overrightarrow { O B } = 2 \mathbf { i } + 4 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 3 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }$$
  1. Find the exact value of the cosine of angle \(B A C\).
  2. Hence find the exact value of the area of triangle \(A B C\).
  3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2014 June Q1
1 Solve the equation \(\log _ { 10 } ( x + 9 ) = 2 + \log _ { 10 } x\).
CAIE P3 2014 June Q2
2 Expand \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
CAIE P3 2014 June Q3
3
  1. Show that the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ can be written in the form $$2 \tan ^ { 2 } x + ( \sqrt { } 3 ) \tan x - 1 = 0$$
  2. Hence solve the equation $$\tan \left( x - 60 ^ { \circ } \right) + \cot x = \sqrt { } 3$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2014 June Q4
4 The equation \(x = \frac { 10 } { \mathrm { e } ^ { 2 x } - 1 }\) has one positive real root, denoted by \(\alpha\).
  1. Show that \(\alpha\) lies between \(x = 1\) and \(x = 2\).
  2. Show that if a sequence of positive values given by the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( 1 + \frac { 10 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2014 June Q5
5 The variables \(x\) and \(\theta\) satisfy the differential equation $$2 \cos ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \sqrt { } ( 2 x + 1 )$$ and \(x = 0\) when \(\theta = \frac { 1 } { 4 } \pi\). Solve the differential equation and obtain an expression for \(x\) in terms of \(\theta\).
CAIE P3 2014 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).
CAIE P3 2014 June Q7
7
  1. The complex number \(\frac { 3 - 5 \mathrm { i } } { 1 + 4 \mathrm { i } }\) is denoted by \(u\). Showing your working, express \(u\) 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 \(| z - 2 - \mathrm { i } | \leqslant 1\) and \(| z - \mathrm { i } | \leqslant | z - 2 |\).
    2. Calculate the maximum value of \(\arg z\) for points lying in the shaded region.
CAIE P3 2014 June Q8
8 Let \(f ( x ) = \frac { 6 + 6 x } { ( 2 - x ) \left( 2 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(\frac { A } { 2 - x } + \frac { B x + C } { 2 + x ^ { 2 } }\).
  2. Show that \(\int _ { - 1 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 3 \ln 3\).
CAIE P3 2014 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_639_387_1749_879} The diagram shows the curve \(y = \mathrm { e } ^ { 2 \sin x } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Using the substitution \(u = \sin x\), find the exact value of the area of the shaded region bounded by the curve and the axes.
  2. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.
CAIE P3 2014 June Q10
10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } + \lambda ( 3 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\) and the plane \(p\) has equation \(2 x + 3 y - 5 z = 18\).
  1. Find the position vector of the point of intersection of \(l\) and \(p\).
  2. Find the acute angle between \(l\) and \(p\).
  3. A second plane \(q\) is perpendicular to the plane \(p\) and contains the line \(l\). Find the equation of \(q\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2015 June Q1
1 Use logarithms to solve the equation \(2 ^ { 5 x } = 3 ^ { 2 x + 1 }\), giving the answer correct to 3 significant figures.
CAIE P3 2015 June Q2
2 Use the trapezium rule with three intervals to find an approximation to $$\int _ { 0 } ^ { 3 } \left| 3 ^ { x } - 10 \right| \mathrm { d } x$$
CAIE P3 2015 June Q3
3 Show that, for small values of \(x ^ { 2 }\), $$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$ where the value of the constant \(k\) is to be determined.