Questions P3 (1203 questions)

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CAIE P3 2011 November Q4
4 During an experiment, the number of organisms present at time \(t\) days is denoted by \(N\), where \(N\) is treated as a continuous variable. It is given that $$\frac { \mathrm { d } N } { \mathrm {~d} t } = 1.2 \mathrm { e } ^ { - 0.02 t } N ^ { 0.5 }$$ When \(t = 0\), the number of organisms present is 100 .
  1. Find an expression for \(N\) in terms of \(t\).
  2. State what happens to the number of organisms present after a long time.
CAIE P3 2011 November Q5
5 It is given that \(\int _ { 1 } ^ { a } x \ln x \mathrm {~d} x = 22\), where \(a\) is a constant greater than 1 .
  1. Show that \(a = \sqrt { } \left( \frac { 87 } { 2 \ln a - 1 } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.
CAIE P3 2011 November Q6
6 The complex number \(w\) is defined by \(w = - 1 + \mathrm { i }\).
  1. Find the modulus and argument of \(w ^ { 2 }\) and \(w ^ { 3 }\), showing your working.
  2. The points in an Argand diagram representing \(w\) and \(w ^ { 2 }\) are the ends of a diameter of a circle. Find the equation of the circle, giving your answer in the form \(| z - ( a + b \mathrm { i } ) | = k\).
CAIE P3 2011 November Q7
7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - x ^ { 2 } + 4 x - a$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
  2. When \(a\) has the value found in part (i), express \(\frac { 8 x - 13 } { \mathrm { p } ( x ) }\) in partial fractions.
CAIE P3 2011 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{6025cf1d-525e-4f12-9517-f20ef5fff2fa-3_698_1006_758_571} The diagram shows the curve with parametric equations $$x = \sin t + \cos t , \quad y = \sin ^ { 3 } t + \cos ^ { 3 } t$$ for \(\frac { 1 } { 4 } \pi < t < \frac { 5 } { 4 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 3 \sin t \cos t\).
  2. Find the gradient of the curve at the origin.
  3. Find the values of \(t\) for which the gradient of the curve is 1 , giving your answers correct to 2 significant figures.
CAIE P3 2011 November Q9
9 The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } a
1
4 \end{array} \right) + \lambda \left( \begin{array} { r } 4
3
- 2 \end{array} \right)\), where \(a\) is a constant. The plane \(p\) has equation \(2 x - 2 y + z = 10\).
  1. Given that \(l\) does not lie in \(p\), show that \(l\) is parallel to \(p\).
  2. Find the value of \(a\) for which \(l\) lies in \(p\).
  3. It is now given that the distance between \(l\) and \(p\) is 6 . Find the possible values of \(a\).
CAIE P3 2011 November Q10
10
  1. Use the substitution \(u = \tan x\) to show that, for \(n \neq - 1\), $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { n + 2 } x + \tan ^ { n } x \right) \mathrm { d } x = \frac { 1 } { n + 1 }$$
  2. Hence find the exact value of
    (a) \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sec ^ { 4 } x - \sec ^ { 2 } x \right) \mathrm { d } x\),
    (b) \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 9 } x + 5 \tan ^ { 7 } x + 5 \tan ^ { 5 } x + \tan ^ { 3 } x \right) \mathrm { d } x\).
CAIE P3 2012 November Q1
1 Find the set of values of \(x\) satisfying the inequality \(3 | x - 1 | < | 2 x + 1 |\).
CAIE P3 2012 November Q2
2 Solve the equation $$5 ^ { x - 1 } = 5 ^ { x } - 5$$ giving your answer correct to 3 significant figures.
CAIE P3 2012 November Q3
3 Solve the equation $$\sin \left( \theta + 45 ^ { \circ } \right) = 2 \cos \left( \theta - 30 ^ { \circ } \right)$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2012 November Q4
4 When \(( 1 + a x ) ^ { - 2 }\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 3 }\) are equal.
  1. Find the exact value of \(a\).
  2. When \(a\) has this value, obtain the expansion up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2012 November Q5
5
  1. By differentiating \(\frac { 1 } { \cos x }\), show that if \(y = \sec x\) then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  2. Show that \(\frac { 1 } { \sec x - \tan x } \equiv \sec x + \tan x\).
  3. Deduce that \(\frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \equiv 2 \sec ^ { 2 } x - 1 + 2 \sec x \tan x\).
  4. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 } { ( \sec x - \tan x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } ( 8 \sqrt { } 2 - \pi )\).
CAIE P3 2012 November Q6
6 The variables \(x\) and \(y\) are related by the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 - y ^ { 2 }$$ When \(x = 2 , y = 0\). Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2012 November Q7
7 The equation of a curve is \(\ln ( x y ) - y ^ { 3 } = 1\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x \left( 3 y ^ { 3 } - 1 \right) }\).
  2. Find the coordinates of the point where the tangent to the curve is parallel to the \(y\)-axis, giving each coordinate correct to 3 significant figures.
CAIE P3 2012 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{7fe27759-d014-4bc6-8391-342d9df8280e-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
  3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 November Q9
9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).
  1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
  2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).
CAIE P3 2012 November Q10
10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
- 2
4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
- 1
7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1
- 5
- 3 \end{array} \right) .$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).
  1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).
CAIE P3 2012 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{346e8866-ca23-4ea6-81bf-bf62502a16d1-3_397_750_255_699} The diagram shows the curve \(y = \mathrm { e } ^ { - \frac { 1 } { 2 } x ^ { 2 } } \sqrt { } \left( 1 + 2 x ^ { 2 } \right)\) for \(x \geqslant 0\), and its maximum point \(M\).
  1. Find the exact value of the \(x\)-coordinate of \(M\).
  2. The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \ln \left( 4 + 8 x _ { n } ^ { 2 } \right) \right) ,$$ with initial value \(x _ { 1 } = 2\), converges to a certain value \(\alpha\). State an equation satisfied by \(\alpha\) and hence show that \(\alpha\) is the \(x\)-coordinate of a point on the curve where \(y = 0.5\).
  3. Use the iterative formula to determine \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2012 November Q10
10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
- 2
4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2
- 1
7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1
- 5
- 3 \end{array} \right)$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).
  1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).
CAIE P3 2012 November Q1
1 Solve the equation $$\ln ( x + 5 ) = 1 + \ln x$$ giving your answer in terms of e.
CAIE P3 2012 November Q2
2
  1. Express \(24 \sin \theta - 7 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$24 \sin \theta - 7 \cos \theta = 17$$
CAIE P3 2012 November Q3
3 The parametric equations of a curve are $$x = \frac { 4 t } { 2 t + 3 } , \quad y = 2 \ln ( 2 t + 3 )$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
  2. Find the gradient of the curve at the point for which \(x = 1\).
CAIE P3 2012 November Q4
4 The variables \(x\) and \(y\) are related by the differential equation $$\left( x ^ { 2 } + 4 \right) \frac { d y } { d x } = 6 x y$$ It is given that \(y = 32\) when \(x = 0\). Find an expression for \(y\) in terms of \(x\).
CAIE P3 2012 November Q5
5 The expression \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 3 x \mathrm { e } ^ { - 2 x }\).
  1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( - \frac { 1 } { 2 } \right)\).
  2. Find the exact value of \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x\).
CAIE P3 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_561_732_255_705} The diagram shows the curve \(y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x - 16\), which crosses the \(x\)-axis at the points \(( \alpha , 0 )\) and \(( \beta , 0 )\) where \(\alpha < \beta\). It is given that \(\alpha\) is an integer.
  1. Find the value of \(\alpha\).
  2. Show that \(\beta\) satisfies the equation \(x = \sqrt [ 3 ] { } ( 8 - 2 x )\).
  3. Use an iteration process based on the equation in part (ii) to find the value of \(\beta\) correct to 2 decimal places. Show the result of each iteration to 4 decimal places.