Questions — CAIE P3 (1070 questions)

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CAIE P3 2022 November Q7
7 The equation of a curve is \(y = \frac { x } { \cos ^ { 2 } x }\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\). At the point where \(x = a\), the tangent to the curve has gradient equal to 12 .
  1. Show that \(a = \cos ^ { - 1 } \left( \sqrt [ 3 ] { \frac { \cos a + 2 a \sin a } { 12 } } \right)\).
  2. Verify by calculation that \(a\) lies between 0.9 and 1 .
  3. Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 November Q8
8 In a certain chemical reaction the amount, \(x\) grams, of a substance is increasing. The differential equation satisfied by \(x\) and \(t\), the time in seconds since the reaction began, is $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k x \mathrm { e } ^ { - 0.1 t }$$ where \(k\) is a positive constant. It is given that \(x = 20\) at the start of the reaction.
  1. Solve the differential equation, obtaining a relation between \(x , t\) and \(k\).
  2. Given that \(x = 40\) when \(t = 10\), find the value of \(k\) and find the value approached by \(x\) as \(t\) becomes large.
CAIE P3 2022 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-14_535_1082_274_520} The diagram shows part of the curve \(y = ( 3 - x ) \mathrm { e } ^ { - \frac { 1 } { 3 } x }\) for \(x \geqslant 0\), and its minimum point \(M\).
  1. Find the exact coordinates of \(M\).
  2. Find the area of the shaded region bounded by the curve and the axes, giving your answer in terms of e.
CAIE P3 2022 November Q10
10 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 7 x + 8 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }\).
  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 \(x ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-18_737_1034_262_552} In the diagram, \(O A B C D\) is a solid figure in which \(O A = O B = 4\) units and \(O D = 3\) units. The edge \(O D\) is vertical, \(D C\) is parallel to \(O B\) and \(D C = 1\) unit. The base, \(O A B\), is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(B C\) is such that \(C N = 2 N B\).
  3. Express vectors \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  4. Calculate the angle in degrees between the directions of \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\).
  5. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\sqrt { \frac { 22 } { 5 } }\).
    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 2022 November Q1
1 Solve the equation \(2 ^ { 3 x - 1 } = 5 \left( 3 ^ { 1 - x } \right)\). Give your answer in the form \(\frac { \ln a } { \ln b }\) where \(a\) and \(b\) are integers.
CAIE P3 2022 November Q2
2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 3\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\).
CAIE P3 2022 November Q3
3 The equation of a curve is \(y = \sin x \sin 2 x\). The curve has a stationary point in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Find the \(x\)-coordinate of this point, giving your answer correct to 3 significant figures.
CAIE P3 2022 November Q4
4
  1. Express \(4 \cos x - \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(4 \cos 2 x - \sin 2 x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2022 November Q5
5
  1. Solve the equation \(z ^ { 2 } - 6 \mathrm { i } z - 12 = 0\), giving the answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
  2. On a sketch of an Argand diagram with origin \(O\), show points \(A\) and \(B\) representing the roots of the equation in part (a).
  3. Find the exact modulus and argument of each root.
  4. Hence show that the triangle \(O A B\) is equilateral.
CAIE P3 2022 November Q6
6 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1
3
1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3
1
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 5
3
- 2 \end{array} \right)$$
  1. Using a scalar product, find the cosine of angle \(B A C\).
  2. Hence find the area of triangle \(A B C\). Give your answer in a simplified exact form.
CAIE P3 2022 November Q7
7 The variables \(x\) and \(\theta\) satisfy the differential equation $$x \sin ^ { 2 } \theta \frac { \mathrm {~d} x } { \mathrm {~d} \theta } = \tan ^ { 2 } \theta - 2 \cot \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\) and \(x > 0\). It is given that \(x = 2\) when \(\theta = \frac { 1 } { 4 } \pi\).
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } \left( \cot ^ { 2 } \theta \right) = - \frac { 2 \cot \theta } { \sin ^ { 2 } \theta }\).
    (You may assume without proof that the derivative of \(\cot \theta\) with respect to \(\theta\) is \(- \operatorname { cosec } ^ { 2 } \theta\).)
  2. Solve the differential equation and find the value of \(x\) when \(\theta = \frac { 1 } { 6 } \pi\).
CAIE P3 2022 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{3c63c42a-2658-4984-93e8-b2a8d18eb37a-12_473_839_274_644} The diagram shows part of the curve \(y = \sin \sqrt { x }\). This part of the curve intersects the \(x\)-axis at the point where \(x = a\).
  1. State the exact value of \(a\).
  2. Using the substitution \(u = \sqrt { x }\), find the exact area of the shaded region in the first quadrant bounded by this part of the curve and the \(x\)-axis.
CAIE P3 2022 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3c63c42a-2658-4984-93e8-b2a8d18eb37a-14_407_734_267_699} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The shaded region is the minor segment on the chord \(A C\) and its area is one third of the area of the semicircle. The angle \(C A B\) is \(\theta\) radians.
  1. Show that \(\theta = \frac { 1 } { 3 } ( \pi - 1.5 \sin 2 \theta )\).
  2. Verify by calculation that \(0.5 < \theta < 0.7\).
  3. Use an iterative formula based on the equation in part (a) to determine \(\theta\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 2022 November Q10
10 Let \(\mathrm { f } ( x ) = \frac { 4 - x + x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Find the exact value of \(\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer as a single logarithm.
    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 2022 November Q1
1 Solve the equation \(\ln ( 2 x - 1 ) = 2 \ln ( x + 1 ) - \ln x\). Give your answer correct to 3 decimal places.
CAIE P3 2022 November Q2
2 Expand \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
CAIE P3 2022 November Q3
3 Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \sec ^ { 2 } x \mathrm {~d} x\).
CAIE P3 2022 November Q4
4 The parametric equations of a curve are $$x = 2 t - \tan t , \quad y = \ln ( \sin 2 t )$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).
CAIE P3 2022 November Q5
5
  1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z + 2 | \leqslant 2\) and \(\operatorname { Im } z \geqslant 1\).
  2. Find the greatest value of \(\arg z\) for points in the shaded region.
CAIE P3 2022 November Q6
6 Solve the quadratic equation \(( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2022 November Q7
4 marks
7
  1. Show that the equation \(\sqrt { 5 } \sec x + \tan x = 4\) can be expressed as \(R \cos ( x + \alpha ) = \sqrt { 5 }\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [4]
  2. Hence solve the equation \(\sqrt { 5 } \sec 2 x + \tan 2 x = 4\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2022 November Q8
8 The curve with equation \(y = \frac { x ^ { 3 } } { \mathrm { e } ^ { x } - 1 }\) has a stationary point at \(x = p\), where \(p > 0\).
  1. Show that \(p = 3 \left( 1 - \mathrm { e } ^ { - p } \right)\).
  2. Verify by calculation that \(p\) lies between 2.5 and 3 .
  3. Use an iterative formula based on the equation in part (a) to determine \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2022 November Q9
9 With respect to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 0
5
2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1
0
1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 4
- 3
- 2 \end{array} \right)$$ The midpoint of \(A C\) is \(M\) and the point \(N\) lies on \(B C\), between \(B\) and \(C\), and is such that \(B N = 2 N C\).
  1. Find the position vectors of \(M\) and \(N\).
  2. Find a vector equation for the line through \(M\) and \(N\).
  3. Find the position vector of the point \(Q\) where the line through \(M\) and \(N\) intersects the line through \(A\) and \(B\).
CAIE P3 2022 November Q10
10 A gardener is filling an ornamental pool with water, using a hose that delivers 30 litres of water per minute. Initially the pool is empty. At time \(t\) minutes after filling begins the volume of water in the pool is \(V\) litres. The pool has a small leak and loses water at a rate of \(0.01 V\) litres per minute. The differential equation satisfied by \(V\) and \(t\) is of the form \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a - b V\).
  1. Write down the values of the constants \(a\) and \(b\).
  2. Solve the differential equation and find the value of \(t\) when \(V = 1000\).
  3. Obtain an expression for \(V\) in terms of \(t\) and hence state what happens to \(V\) as \(t\) becomes large.
CAIE P3 2022 November Q11
11 Let \(\mathrm { f } ( x ) = \frac { 5 - x + 6 x ^ { 2 } } { ( 3 - x ) \left( 1 + 3 x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x\), simplifying your answer.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.