Questions P3 (1203 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE P3 2015 June Q4
4 The equation of a curve is $$y = 3 \cos 2 x + 7 \sin x + 2$$ Find the \(x\)-coordinates of the stationary points in the interval \(0 \leqslant x \leqslant \pi\). Give each answer correct to 3 significant figures.
CAIE P3 2015 June Q5
5
  1. Find \(\int \left( 4 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \sin \left( x + \frac { 1 } { 6 } \pi \right) } { \sin x } \mathrm {~d} x\).
CAIE P3 2015 June Q6
6 The straight line \(l _ { 1 }\) passes through the points \(( 0,1,5 )\) and \(( 2 , - 2,1 )\). The straight line \(l _ { 2 }\) has equation \(\mathbf { r } = 7 \mathbf { i } + \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\).
  1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
  2. Find the acute angle between the direction of the line \(l _ { 2 }\) and the direction of the \(x\)-axis.
CAIE P3 2015 June Q7
7 Given that \(y = 1\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x \left( 3 y ^ { 2 } + 10 y + 3 \right)$$ obtaining an expression for \(y\) in terms of \(x\).
CAIE P3 2015 June Q8
8 The complex number \(w\) is defined by \(w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }\).
  1. Without using a calculator, show that \(w = 2 + 4 \mathrm { i }\).
  2. It is given that \(p\) is a real number such that \(\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi\). Find the set of possible values of \(p\).
  3. The complex conjugate of \(w\) is denoted by \(w ^ { * }\). The complex numbers \(w\) and \(w ^ { * }\) are represented in an Argand diagram by the points \(S\) and \(T\) respectively. Find, in the form \(| z - a | = k\), the equation of the circle passing through \(S , T\) and the origin.
CAIE P3 2015 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_399_696_255_721} The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { 2 - x }\) and its maximum point \(M\).
  1. Show that the \(x\)-coordinate of \(M\) is 2 .
  2. Find the exact value of \(\int _ { 0 } ^ { 2 } x ^ { 2 } \mathrm { e } ^ { 2 - x } \mathrm {~d} x\).
CAIE P3 2015 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_515_508_1105_815} The diagram shows part of the curve with parametric equations $$x = 2 \ln ( t + 2 ) , \quad y = t ^ { 3 } + 2 t + 3$$
  1. Find the gradient of the curve at the origin.
  2. At the point \(P\) on the curve, the value of the parameter is \(p\). It is given that the gradient of the curve at \(P\) is \(\frac { 1 } { 2 }\).
    (a) Show that \(p = \frac { 1 } { 3 p ^ { 2 } + 2 } - 2\).
    (b) By first using an iterative formula based on the equation in part (a), determine the coordinates of the point \(P\). Give the result of each iteration to 5 decimal places and each coordinate of \(P\) correct to 2 decimal places.
CAIE P3 2015 June Q1
1 Use the trapezium rule with three intervals to estimate the value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \ln ( 1 + \sin x ) \mathrm { d } x$$ giving your answer correct to 2 decimal places.
CAIE P3 2015 June Q2
2 Using the substitution \(u = 4 ^ { x }\), solve the equation \(4 ^ { x } + 4 ^ { 2 } = 4 ^ { x + 2 }\), giving your answer correct to 3 significant figures.
CAIE P3 2015 June Q3
3 A curve has equation \(y = \cos x \cos 2 x\). Find the \(x\)-coordinate of the stationary point on the curve in the interval \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P3 2015 June Q4
4
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \sin \theta + 2 \cos \theta = 1$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2015 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{d1377d66-73c8-4d97-9cae-d784b41fb0a8-2_519_800_1359_669} 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 the 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 perimeter of the shaded region is equal to the circumference 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 1.3.
  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 2015 June Q6
6 Let \(I = \int _ { 0 } ^ { 1 } \frac { \sqrt { } x } { 2 - \sqrt { } x } \mathrm {~d} x\).
  1. Using the substitution \(u = 2 - \sqrt { } x\), show that \(I = \int _ { 1 } ^ { 2 } \frac { 2 ( 2 - u ) ^ { 2 } } { u } \mathrm {~d} u\).
  2. Hence show that \(I = 8 \ln 2 - 5\).
CAIE P3 2015 June Q7
5 marks
7 The complex number \(u\) is given by \(u = - 1 + ( 4 \sqrt { } 3 ) \mathrm { i }\).
  1. Without using a calculator and showing all your working, find the two square roots of \(u\). Give your answers in the form \(a + \mathrm { i } b\), where the real numbers \(a\) and \(b\) are exact.
  2. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the relation \(| z - u | = 1\). Determine the greatest value of \(\arg z\) for points on this locus.
    \(8 \quad\) Let \(f ( x ) = \frac { 5 x ^ { 2 } + x + 6 } { ( 3 - 2 x ) \left( x ^ { 2 } + 4 \right) }\).
CAIE P3 2015 June Q9
9 The number of organisms in a population at time \(t\) is denoted by \(x\). Treating \(x\) as a continuous variable, the differential equation satisfied by \(x\) and \(t\) is $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x \mathrm { e } ^ { - t } } { k + \mathrm { e } ^ { - t } }$$ where \(k\) is a positive constant.
  1. Given that \(x = 10\) when \(t = 0\), solve the differential equation, obtaining a relation between \(x , k\) and \(t\).
  2. Given also that \(x = 20\) when \(t = 1\), show that \(k = 1 - \frac { 2 } { \mathrm { e } }\).
  3. Show that the number of organisms never reaches 48, however large \(t\) becomes.
CAIE P3 2015 June Q10
10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O B } = \mathbf { i } + \mathbf { j } + 5 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )\).
  1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
  2. Find the equation of the plane containing the line \(l\) and the point \(A\). Give your answer in the form \(a x + b y + c z = d\).
CAIE P3 2015 June Q1
1 Solve the equation \(\ln ( x + 4 ) = 2 \ln x + \ln 4\), giving your answer correct to 3 significant figures.
CAIE P3 2015 June Q2
2 Solve the inequality \(| x - 2 | > 2 x - 3\).
CAIE P3 2015 June Q3
3 Solve the equation \(\cot 2 x + \cot x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2015 June Q4
4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }\) has one stationary point. Find the exact values of the coordinates of this point.
CAIE P3 2015 June Q5
5 The parametric equations of a curve are $$x = a \cos ^ { 4 } t , \quad y = a \sin ^ { 4 } t$$ where \(a\) is a positive constant.
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin ^ { 2 } t + y \cos ^ { 2 } t = a \sin ^ { 2 } t \cos ^ { 2 } t$$
  3. Hence show that if the tangent meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\), then $$O P + O Q = a$$ where \(O\) is the origin.
CAIE P3 2015 June Q6
6 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).
  1. Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
  2. Verify by calculation that \(a\) is greater than 1 .
  3. Use the iterative formula $$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.
CAIE P3 2015 June Q7
7 The number of micro-organisms in a population at time \(t\) is denoted by \(M\). At any time the variation in \(M\) is assumed to satisfy the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = k ( \sqrt { } M ) \cos ( 0.02 t )$$ where \(k\) is a constant and \(M\) is taken to be a continuous variable. It is given that when \(t = 0 , M = 100\).
  1. Solve the differential equation, obtaining a relation between \(M , k\) and \(t\).
  2. Given also that \(M = 196\) when \(t = 50\), find the value of \(k\).
  3. Obtain an expression for \(M\) in terms of \(t\) and find the least possible number of micro-organisms.
CAIE P3 2015 June Q8
8 The complex number 1 - i is denoted by \(u\).
  1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
  2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
  3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).
CAIE P3 2015 June Q9
9 Two planes have equations \(x + 3 y - 2 z = 4\) and \(2 x + y + 3 z = 5\). The planes intersect in the straight line \(l\).
  1. Calculate the acute angle between the two planes.
  2. Find a vector equation for the line \(l\).