Questions P3 (1203 questions)

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CAIE P3 2011 June Q7
7
  1. The complex number \(u\) is defined by \(u = \frac { 5 } { a + 2 \mathrm { i } }\), where the constant \(a\) is real.
    1. Express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    2. Find the value of \(a\) for which \(\arg \left( u ^ { * } \right) = \frac { 3 } { 4 } \pi\), where \(u ^ { * }\) denotes the complex conjugate of \(u\).
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z | < 2\) and \(| z | < | z - 2 - 2 \mathrm { i } |\).
CAIE P3 2011 June Q8
8
  1. Express \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in partial fractions.
  2. Hence obtain the expansion of \(\frac { 5 x - x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
CAIE P3 2011 June Q9
9 Two planes have equations \(x + 2 y - 2 z = 7\) and \(2 x + y + 3 z = 5\).
  1. Calculate the acute angle between the planes.
  2. Find a vector equation for the line of intersection of the planes.
CAIE P3 2011 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{76371b0f-0145-4cc4-a147-27bcd749816a-3_451_933_1777_605} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).
  1. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\) is equal to \(2 - \frac { 17 } { \mathrm { e } ^ { 3 } }\).
  2. Find the \(x\)-coordinate of the maximum point \(M\) on the curve.
  3. Find the \(x\)-coordinate of the point \(P\) at which the tangent to the curve passes through the origin.
CAIE P3 2011 June Q1
1 Use logarithms to solve the equation \(5 ^ { 2 x - 1 } = 2 \left( 3 ^ { x } \right)\), giving your answer correct to 3 significant figures.
CAIE P3 2011 June Q2
2 The curve \(y = \frac { \ln x } { x ^ { 3 } }\) has one stationary point. Find the \(x\)-coordinate of this point.
CAIE P3 2011 June Q3
3 Show that \(\int _ { 0 } ^ { 1 } ( 1 - x ) \mathrm { e } ^ { - \frac { 1 } { 2 } x } \mathrm {~d} x = 4 \mathrm { e } ^ { - \frac { 1 } { 2 } } - 2\).
CAIE P3 2011 June Q4
4
  1. Show that the equation $$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = k$$ can be written in the form $$( 2 \sqrt { } 3 ) \left( 1 + \tan ^ { 2 } \theta \right) = k \left( 1 - 3 \tan ^ { 2 } \theta \right)$$
  2. Hence solve the equation $$\tan \left( 60 ^ { \circ } + \theta \right) + \tan \left( 60 ^ { \circ } - \theta \right) = 3 \sqrt { } 3$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2011 June Q5
5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + 5 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the quadratic factor of \(\mathrm { p } ( x )\).
CAIE P3 2011 June Q6
6
  1. By sketching a suitable pair of graphs, show that the equation $$\cot x = 1 + x ^ { 2 }$$ where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
  2. Verify by calculation that this root lies between 0.5 and 0.8.
  3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 1 } { 1 + x _ { n } ^ { 2 } } \right)$$ to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2011 June Q7
7
  1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. State the modulus and argument of each root.
  3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$
CAIE P3 2011 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{5b219e1c-e5a0-4f75-910d-fca9761e5088-3_435_895_799_625} The diagram shows the curve \(y = 5 \sin ^ { 3 } x \cos ^ { 2 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
  1. Find the \(x\)-coordinate of \(M\).
  2. Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.
CAIE P3 2011 June Q9
9 In a chemical reaction, a compound \(X\) is formed from two compounds \(Y\) and \(Z\). The masses in grams of \(X , Y\) and \(Z\) present at time \(t\) seconds after the start of the reaction are \(x , 10 - x\) and \(20 - x\) respectively. At any time the rate of formation of \(X\) is proportional to the product of the masses of \(Y\) and \(Z\) present at the time. When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\).
  1. Show that \(x\) and \(t\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.01 ( 10 - x ) ( 20 - x )$$
  2. Solve this differential equation and obtain an expression for \(x\) in terms of \(t\).
  3. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE P3 2011 June Q10
10 With respect to the origin \(O\), the lines \(l\) and \(m\) have vector equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively.
  1. Prove that \(l\) and \(m\) do not intersect.
  2. Calculate the acute angle between the directions of \(l\) and \(m\).
  3. Find the equation of the plane which is parallel to \(l\) and contains \(m\), giving your answer in the form \(a x + b y + c z = d\).
CAIE P3 2012 June Q1
1 Solve the equation \(\left| 4 - 2 ^ { x } \right| = 10\), giving your answer correct to 3 significant figures.
CAIE P3 2012 June Q2
2
  1. Expand \(\frac { 1 } { \sqrt { } ( 1 - 4 x ) }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { 1 + 2 x } { \sqrt { } ( 4 - 16 x ) }\).
CAIE P3 2012 June Q3
3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } - 3 a x + 4 a$$ where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
  2. When \(a\) has this value,
    (a) factorise \(\mathrm { p } ( x )\) completely,
    (b) find all the roots of the equation \(\mathrm { p } \left( x ^ { 2 } \right) = 0\).
CAIE P3 2012 June Q4
4 The complex number \(u\) is defined by \(u = \frac { ( 1 + 2 \mathrm { i } ) ^ { 2 } } { 2 + \mathrm { i } }\).
  1. Without using a calculator and showing your working, express \(u\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. Sketch an Argand diagram showing the locus of the complex number \(z\) such that \(| z - u | = | u |\).
CAIE P3 2012 June Q5
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
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
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
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
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
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
1 Solve the equation $$\ln ( 3 x + 4 ) = 2 \ln ( x + 1 )$$ giving your answer correct to 3 significant figures.