Questions P3 (1243 questions)

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CAIE P3 2019 November Q3
4 marks Standard +0.3
3 Showing all necessary working, solve the equation \(\frac { 3 ^ { 2 x } + 3 ^ { - x } } { 3 ^ { 2 x } - 3 ^ { - x } } = 4\). Give your answer correct to 3 decimal places.
CAIE P3 2019 November Q4
7 marks Standard +0.8
4
  1. By first expanding \(\tan ( 2 x + x )\), show that the equation \(\tan 3 x = 3 \cot x\) can be written in the form \(\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0\).
  2. Hence solve the equation \(\tan 3 x = 3 \cot x\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P3 2019 November Q5
7 marks Moderate -0.3
5
  1. By sketching a suitable pair of graphs, show that the equation \(\ln ( x + 2 ) = 4 \mathrm { e } ^ { - x }\) has exactly one real root.
  2. Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).
  3. Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 4 } { \ln \left( x _ { n } + 2 \right) } \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2019 November Q7
9 marks Standard +0.3
7 The plane \(m\) has equation \(x + 4 y - 8 z = 2\). The plane \(n\) is parallel to \(m\) and passes through the point \(P\) with coordinates \(( 5,2 , - 2 )\).
  1. Find the equation of \(n\), giving your answer in the form \(a x + b y + c z = d\).
  2. Calculate the perpendicular distance between \(m\) and \(n\).
  3. The line \(l\) lies in the plane \(n\), passes through the point \(P\) and is perpendicular to \(O P\), where \(O\) is the origin. Find a vector equation for \(l\).
CAIE P3 2019 November Q8
10 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-14_604_497_262_822} The diagram shows the graph of \(y = \sec x\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Use the trapezium rule with 2 intervals to estimate the value of \(\int _ { 0 } ^ { 1.2 } \sec x \mathrm {~d} x\), giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, whether the trapezium rule gives an overestimate or an underestimate of the true value of the integral in part (i).
  3. \(P\) is the point on the part of the curve \(y = \sec x\) for \(0 \leqslant x < \frac { 1 } { 2 } \pi\) at which the gradient is 2 . By first differentiating \(\frac { 1 } { \cos x }\), find the \(x\)-coordinate of \(P\), giving your answer correct to 3 decimal places.
CAIE P3 2019 November Q9
10 marks Standard +0.3
9 The variables \(x\) and \(t\) satisfy the differential equation \(5 \frac { \mathrm {~d} x } { \mathrm {~d} t } = ( 20 - x ) ( 40 - x )\). It is given that \(x = 10\) when \(t = 0\).
  1. Using partial fractions, solve the differential equation, obtaining an expression for \(x\) in terms of \(t\). [9]
  2. State what happens to the value of \(x\) when \(t\) becomes large.
CAIE P3 2019 November Q10
12 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{5b5ed7d1-028e-4f9a-ae9e-26071d0df678-18_449_787_262_678} The diagram shows the graph of \(y = \mathrm { e } ^ { \cos x } \sin ^ { 3 } x\) for \(0 \leqslant x \leqslant \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
  1. Find the \(x\)-coordinate of \(M\). Show all necessary working and give your answer correct to 2 decimal places.
  2. By first using the substitution \(u = \cos x\), find the exact value of the area of \(R\).
    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 Specimen Q1
4 marks Standard +0.8
1 Solve the inequality \(| 2 x - 5 | > 3 | 2 x + 1 |\).
CAIE P3 Specimen Q2
5 marks Moderate -0.3
2 Using the substitution \(u = 3 ^ { x }\), solve the equation \(3 ^ { x } + 3 ^ { 2 x } = 3 ^ { 3 x }\) giving your answer correct to 3 significant figures.
CAIE P3 Specimen Q3
6 marks Standard +0.8
3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).
CAIE P3 Specimen Q4
7 marks Standard +0.3
4 The equation \(x ^ { 3 } - x ^ { 2 } - 6 = 0\) has one real root, denoted by \(\alpha\).
  1. Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
  2. Show that, if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( x _ { n } + \frac { 6 } { x _ { n } } \right)$$ converges, then it converges to \(\alpha\).
  3. Use this iterative formula to determine \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P3 Specimen Q5
7 marks Standard +0.3
5 The equation of a curve is \(y = \mathrm { e } ^ { - 2 x } \tan x\), for \(0 \leqslant x < \frac { 1 } { 2 } \pi\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that it can be written in the form \(\mathrm { e } ^ { - 2 x } ( a + b \tan x ) ^ { 2 }\), where \(a\) and \(b\) are constants.
  2. Explain why the gradient of the curve is never negative.
  3. Find the value of \(x\) for which the gradient is least.
CAIE P3 Specimen Q6
8 marks Moderate -0.8
6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x + 1\) ) the remainder is 1 .
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 Specimen Q7
9 marks Standard +0.3
7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).
  1. Find a vector equation for the line passing through \(A\) and \(B\).
  2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
  3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).
CAIE P3 Specimen Q8
9 marks Moderate -0.3
8 The variables \(x\) and \(\theta\) satisfy the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = ( x + 2 ) \sin ^ { 2 } 2 \theta$$ and it is given that \(x = 0\) when \(\theta = 0\). Solve the differential equation and calculate the value of \(x\) when \(\theta = \frac { 1 } { 4 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 Specimen Q9
10 marks Standard +0.3
9 The complex number \(3 - \mathrm { i }\) is denoted by \(u\). Its complex conjugate is denoted by \(u ^ { * }\).
  1. On an Argand diagram with origin \(O\), show the points \(A , B\) and \(C\) representing the complex numbers \(u , u ^ { * }\) and \(u ^ { * } - u\) respectively. What type of quadrilateral is \(O A B C\) ?
  2. Showing your working and without using a calculator, express \(\frac { u ^ { * } } { u }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  3. By considering the argument of \(\frac { u ^ { * } } { u }\), prove that $$\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right) = 2 \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) .$$ \includegraphics[max width=\textwidth, alt={}, center]{d4a7604c-9e2c-47ef-a496-8697bc88fdd4-18_360_758_260_689} The diagram shows the curve \(y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }\) for \(x \geqslant 0\), and its maximum point \(M\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = p\).
  4. Find the exact value of the \(x\)-coordinate of \(M\).
  5. Calculate the value of \(p\) for which the area of \(R\) is equal to 1 . Give your answer correct to 3 significant figures.
CAIE P3 2020 June Q1
4 marks Standard +0.3
1 Find the set of values of \(x\) for which \(2 \left( 3 ^ { 1 - 2 x } \right) < 5 ^ { x }\). Give your answer in a simplified exact form. [4]
CAIE P3 2020 June Q2
5 marks Moderate -0.8
2
  1. Expand \(( 2 - 3 x ) ^ { - 2 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
  2. State the set of values of \(x\) for which the expansion is valid.
CAIE P3 2020 June Q3
6 marks Standard +0.3
3 Express the equation \(\tan \left( \theta + 60 ^ { \circ } \right) = 2 + \tan \left( 60 ^ { \circ } - \theta \right)\) as a quadratic equation in \(\tan \theta\), and hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2020 June Q4
6 marks Standard +0.8
4 The curve with equation \(y = \mathrm { e } ^ { 2 x } ( \sin x + 3 \cos x )\) has a stationary point in the interval \(0 \leqslant x \leqslant \pi\).
  1. Find the \(x\)-coordinate of this point, giving your answer correct to 2 decimal places.
  2. Determine whether the stationary point is a maximum or a minimum.
CAIE P3 2020 June Q5
8 marks Standard +0.3
5
  1. Find the quotient and remainder when \(2 x ^ { 3 } - x ^ { 2 } + 6 x + 3\) is divided by \(x ^ { 2 } + 3\).
  2. Using your answer to part (a), find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } - x ^ { 2 } + 6 x + 3 } { x ^ { 2 } + 3 } \mathrm {~d} x\).
CAIE P3 2020 June Q6
8 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{da8dedae-4714-408e-a983-90ece63d9e91-08_501_1086_262_525} The diagram shows a circle with centre \(O\) and radius \(r\). The tangents to the circle at the points \(A\) and \(B\) meet at \(T\), and angle \(A O B\) is \(2 x\) radians. The shaded region is bounded by the tangents \(A T\) and \(B T\), and by the minor \(\operatorname { arc } A B\). The area of the shaded region is equal to the area of the circle.
  1. Show that \(x\) satisfies the equation \(\tan x = \pi + x\).
  2. This equation has one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\). Verify by calculation that this root lies between 1 and 1.4.
  3. Use the iterative formula $$x _ { n + 1 } = \tan ^ { - 1 } \left( \pi + x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2020 June Q7
8 marks Standard +0.3
7 Let \(\mathrm { f } ( x ) = \frac { \cos x } { 1 + \sin x }\).
  1. Show that \(\mathrm { f } ^ { \prime } ( x ) < 0\) for all \(x\) in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 3 } { 2 } \pi\).
  2. Find \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in a simplified exact form.
CAIE P3 2020 June Q8
9 marks Standard +0.3
8 A certain curve is such that its gradient at a point \(( x , y )\) is proportional to \(\frac { y } { x \sqrt { x } }\). The curve passes through the points with coordinates \(( 1,1 )\) and \(( 4 , \mathrm { e } )\).
  1. By setting up and solving a differential equation, find the equation of the curve, expressing \(y\) in terms of \(x\).
  2. Describe what happens to \(y\) as \(x\) tends to infinity.
CAIE P3 2020 June Q9
9 marks Standard +0.3
9 With respect to the origin \(O\), the vertices of a triangle \(A B C\) have position vectors $$\overrightarrow { O A } = 2 \mathbf { i } + 5 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } + \mathbf { k }$$
  1. Using a scalar product, show that angle \(A B C\) is a right angle.
  2. Show that triangle \(A B C\) is isosceles.
  3. Find the exact length of the perpendicular from \(O\) to the line through \(B\) and \(C\).