Questions — CAIE P3 (1070 questions)

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CAIE P3 2016 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{84df6b9a-6118-44a2-9c18-512039ded4fd-3_416_677_258_733} The diagram shows part of the curve \(y = \left( 2 x - x ^ { 2 } \right) \mathrm { e } ^ { \frac { 1 } { 2 } x }\) and its maximum point \(M\).
  1. Find the exact \(x\)-coordinate of \(M\).
  2. Find the exact value of the area of the shaded region bounded by the curve and the positive \(x\)-axis.
CAIE P3 2016 November Q1
1 It is given that \(z = \ln ( y + 2 ) - \ln ( y + 1 )\). Express \(y\) in terms of \(z\).
CAIE P3 2016 November Q2
2 The equation of a curve is \(y = \frac { \sin x } { 1 + \cos x }\), for \(- \pi < x < \pi\). Show that the gradient of the curve is positive for all \(x\) in the given interval.
CAIE P3 2016 November Q3
3 Express the equation \(\cot 2 \theta = 1 + \tan \theta\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2016 November Q4
4 The polynomial \(4 x ^ { 4 } + a x ^ { 2 } + 11 x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 2\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).
CAIE P3 2016 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-2_346_437_1155_854} The diagram shows a variable point \(P\) with coordinates \(( x , y )\) and the point \(N\) which is the foot of the perpendicular from \(P\) to the \(x\)-axis. \(P\) moves on a curve such that, for all \(x \geqslant 0\), the gradient of the curve is equal in value to the area of the triangle \(O P N\), where \(O\) is the origin.
  1. State a differential equation satisfied by \(x\) and \(y\). The point with coordinates \(( 0,2 )\) lies on the curve.
  2. Solve the differential equation to obtain the equation of the curve, expressing \(y\) in terms of \(x\).
  3. Sketch the curve.
CAIE P3 2016 November Q6
6 Let \(I = \int _ { 1 } ^ { 4 } \frac { ( \sqrt { } x ) - 1 } { 2 ( x + \sqrt { } x ) } \mathrm { d } x\).
  1. Using the substitution \(u = \sqrt { } x\), show that \(I = \int _ { 1 } ^ { 2 } \frac { u - 1 } { u + 1 } \mathrm {~d} u\).
  2. Hence show that \(I = 1 + \ln \frac { 4 } { 9 }\).
CAIE P3 2016 November Q8
8 Let \(\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } + x + 6 } { ( x + 2 ) \left( x ^ { 2 } + 4 \right) }\).
  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 \(x ^ { 2 }\).
CAIE P3 2016 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-3_481_483_1434_831} The diagram shows the curves \(y = x \cos x\) and \(y = \frac { k } { x }\), where \(k\) is a constant, for \(0 < x \leqslant \frac { 1 } { 2 } \pi\). The curves touch at the point where \(x = a\).
  1. Show that \(a\) satisfies the equation \(\tan a = \frac { 2 } { a }\).
  2. Use the iterative formula \(a _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { a _ { n } } \right)\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  3. Hence find the value of \(k\) correct to 2 decimal places.
CAIE P3 2016 November Q10
10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).
  1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
  2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).
CAIE P3 2017 November Q1
1 Find the quotient and remainder when \(x ^ { 4 }\) is divided by \(x ^ { 2 } + 2 x - 1\).
CAIE P3 2017 November Q2
2 Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C \left( a ^ { x } \right)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\).
\(x\)0.91.62.43.2
\(\ln y\)1.71.92.32.6
By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{be0fb208-2ef1-4fae-84ff-ad2e8bf2dcc5-03_759_944_749_596}
CAIE P3 2017 November Q3
3 The equation \(x ^ { 3 } = 3 x + 7\) has one real root, denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) lies between 2 and 3 .
    Two iterative formulae, \(A\) and \(B\), derived from this equation are as follows: $$\begin{aligned} & x _ { n + 1 } = \left( 3 x _ { n } + 7 \right) ^ { \frac { 1 } { 3 } }
    & x _ { n + 1 } = \frac { x _ { n } ^ { 3 } - 7 } { 3 } \end{aligned}$$ Each formula is used with initial value \(x _ { 1 } = 2.5\).
  2. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2017 November Q4
4
  1. Prove the identity \(\tan \left( 45 ^ { \circ } + x \right) + \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \sec 2 x\).
  2. Hence sketch the graph of \(y = \tan \left( 45 ^ { \circ } + x \right) + \tan \left( 45 ^ { \circ } - x \right)\) for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P3 2017 November Q5
5 The equation of a curve is \(2 x ^ { 4 } + x y ^ { 3 } + y ^ { 4 } = 10\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 x ^ { 3 } + y ^ { 3 } } { 3 x y ^ { 2 } + 4 y ^ { 3 } }\).
  2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.
CAIE P3 2017 November Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 \cos ^ { 2 } y \tan x ,$$ for \(0 \leqslant x < \frac { 1 } { 2 } \pi\), and \(x = 0\) when \(y = \frac { 1 } { 4 } \pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac { 1 } { 3 } \pi\).
CAIE P3 2017 November Q7
4 marks
7
  1. The complex number \(u\) is given by \(u = 8 - 15 \mathrm { i }\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + \mathrm { i } b\), where the numbers \(a\) and \(b\) are real and exact.
  2. On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(| z - 2 - i | \leqslant 2\) and \(0 \leqslant \arg ( z - i ) \leqslant \frac { 1 } { 4 } \pi\).
    [0pt] [4]
    \(8 \quad\) Let \(\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 9 x - 8 } { ( x + 2 ) ( 2 x - 1 ) }\).
    1. Express \(\mathrm { f } ( x )\) in the form \(A + \frac { B } { x + 2 } + \frac { C } { 2 x - 1 }\).
    2. Hence show that \(\int _ { 1 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 6 + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 7 } \right)\).
CAIE P3 2017 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{be0fb208-2ef1-4fae-84ff-ad2e8bf2dcc5-16_446_956_260_593} The diagram shows the curve \(y = \left( 1 + x ^ { 2 } \right) \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
  1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve.
  2. Show that the exact value of the area of \(R\) is \(18 - \frac { 42 } { \mathrm { e } }\).
CAIE P3 2017 November Q10
10 The equations of two lines \(l\) and \(m\) are \(\mathbf { r } = 3 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\) respectively.
  1. Show that the lines do not intersect.
  2. Calculate the acute angle between the directions of the lines.
  3. Find the equation of the plane which passes through the point \(( 3 , - 2 , - 1 )\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2017 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{746d2c39-7d78-4478-bc36-15ea5e3ba72a-02_460_807_258_667} The diagram shows a sketch of the curve \(y = \frac { 3 } { \sqrt { } \left( 9 - x ^ { 3 } \right) }\) for values of \(x\) from - 1.2 to 1.2 .
  1. Use the trapezium rule, with two intervals, to estimate the value of $$\int _ { - 1.2 } ^ { 1.2 } \frac { 3 } { \sqrt { \left( 9 - x ^ { 3 } \right) } } \mathrm { d } x$$ giving your answer correct to 2 decimal places.
  2. Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.
CAIE P3 2017 November Q2
2 Showing all necessary working, solve the equation \(2 \log _ { 2 } x = 3 + \log _ { 2 } ( x + 1 )\), giving your answer correct to 3 significant figures.
CAIE P3 2017 November Q3
3 By expressing the equation \(\tan \left( \theta + 60 ^ { \circ } \right) + \tan \left( \theta - 60 ^ { \circ } \right) = \cot \theta\) in terms of \(\tan \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2017 November Q4
4 The curve with equation \(y = \frac { 2 - \sin x } { \cos x }\) has one stationary point in the interval \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
  1. Find the exact coordinates of this point.
  2. Determine whether this point is a maximum or a minimum point.
CAIE P3 2017 November Q5
5 The variables \(x\) and \(y\) satisfy the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( x + 2 )$$ and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).
CAIE P3 2017 November Q6
6 The equation of a curve is \(x ^ { 3 } y - 3 x y ^ { 3 } = 2 a ^ { 4 }\), where \(a\) is a non-zero constant.
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 x ^ { 2 } y - 3 y ^ { 3 } } { 9 x y ^ { 2 } - x ^ { 3 } }\).
  2. Hence show that there are only two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points.