Questions — CAIE P3 (1070 questions)

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CAIE P3 2024 November Q11
11 Let \(\mathrm { f } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } } { \mathrm { e } ^ { 2 x } - 3 \mathrm { e } ^ { x } + 2 }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-16_2718_35_107_2012}
  2. Use the substitution \(u = e ^ { x }\) and partial fractions to find the exact value of \(\int _ { \ln 3 } ^ { \ln 5 } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}, center]{9da6c2ac-31aa-4063-88b9-e15e38bedd8a-18_2718_42_107_2007}
CAIE P3 2024 November Q1
1 The complex number \(z\) satisfies \(| z | = 2\) and \(0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi\).
  1. On the Argand diagram below, sketch the locus of the points representing \(z\).
  2. On the same diagram, sketch the locus of the points representing \(z ^ { 2 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}
CAIE P3 2024 November Q2
2 Let \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + 4\).
  1. Show that if a sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { \frac { 4 } { 5 - 2 x _ { n } } }$$ converges, then it converges to a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. The equation has a root close to 1.2 . Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
CAIE P3 2024 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-04_527_634_255_717} The number of bacteria in a population, \(P\), at time \(t\) hours is modelled by the equation \(P = a \mathrm { e } ^ { k t }\), where \(a\) and \(k\) are constants. The graph of \(\ln P\) against \(t\), shown in the diagram, has gradient \(\frac { 1 } { 20 }\) and intersects the vertical axis at \(( 0,3 )\).
  1. State the value of \(k\) and find the value of \(a\) correct to 2 significant figures.
  2. Find the time taken for \(P\) to double. Give your answer correct to the nearest hour.
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-05_2723_33_99_22}
CAIE P3 2024 November Q4
4 Find the complex number \(z\) satisfying the equation $$\frac { z - 3 \mathrm { i } } { z + 3 \mathrm { i } } = \frac { 2 - 9 \mathrm { i } } { 5 }$$ Give your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2024 November Q5
5
  1. Show that \(\cos ^ { 4 } \theta - \sin ^ { 4 } \theta - 4 \sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \cos ^ { 2 } 2 \theta + \cos 2 \theta - 1\).
  2. Solve the equation \(\cos ^ { 4 } \alpha - \sin ^ { 4 } \alpha = 4 \sin ^ { 2 } \alpha \cos ^ { 2 } \alpha\) for \(0 ^ { \circ } \leqslant \alpha \leqslant 180 ^ { \circ }\).
CAIE P3 2024 November Q6
6 The lines \(l\) and \(m\) have vector equations $$l : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { k } ) \quad \text { and } \quad m : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) .$$ Lines \(l\) and \(m\) intersect at the point \(P\).
  1. State the coordinates of \(P\).
  2. Find the exact value of the cosine of the acute angle between \(l\) and \(m\).
  3. The point \(A\) on line \(I\) has coordinates ( \(0,1,1\) ). The point \(B\) on line \(m\) has coordinates ( \(0,2 , - 8\) ). Find the exact area of triangle \(A P B\).
CAIE P3 2024 November Q7
7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
  2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.
CAIE P3 2024 November Q8
4 marks
8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
  2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
  3. State the set of values of \(x\) for which the expansion in part (b) is valid.
CAIE P3 2024 November Q9
9
  1. Find the quotient and remainder when \(x ^ { 4 } + 16\) is divided by \(x ^ { 2 } + 4\).
  2. Hence show that \(\int _ { 2 } ^ { 2 \sqrt { 3 } } \frac { x ^ { 4 } + 16 } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 4 } { 3 } ( \pi + 4 )\).
CAIE P3 2024 November Q10
10 A water tank is in the shape of a cuboid with base area \(40000 \mathrm {~cm} ^ { 2 }\). At time \(t\) minutes the depth of water in the tank is \(h \mathrm {~cm}\). Water is pumped into the tank at a rate of \(50000 \mathrm {~cm} ^ { 3 }\) per minute. Water is leaking out of the tank through a hole in the bottom at a rate of \(600 \mathrm {~cm} ^ { 3 }\) per minute.
  1. Show that \(200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 250 - 3 h\).
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-17_2723_33_99_22}
  2. It is given that when \(t = 0 , h = 50\). Find the time taken for the depth of water in the tank to reach 80 cm . Give your answer correct to 2 significant figures.
CAIE P3 2024 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_565_634_260_717} The diagram shows the curve \(y = 2 \sin x \sqrt { 2 + \cos x }\), for \(0 \leqslant x \leqslant 2 \pi\), and its minimum point \(M\), where \(x = a\).
  1. Find the value of \(a\) correct to 2 decimal places.
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-18_2716_38_109_2012}
    \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-19_2726_33_97_22}
  2. Use the substitution \(u = 2 + \cos x\) to find the exact area of the shaded region \(R\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE P3 2012 June Q9
9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.
  1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
  2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
  3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).
CAIE P3 2021 November Q9
9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.
  1. Show that \(l\) and \(m\) are perpendicular.
  2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
  3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q1
1 Fird bet \(\mathbf { 6 }\) le \(\mathrm { s } \mathbf { 6 } x\) fo wh clB \(\left. 2 ^ { 3 x + 1 } \right) < \mathscr { G }\) in an wer ira simp ified \& ct fo m. [ \(\beta\)
CAIE P3 2020 Specimen Q2
2
  1. Ed \(( 1 + 3 x ) ^ { - \frac { 1 } { 3 } }\) in asced g N ers \(6 x\), p to ad in lid g th term in \(x ^ { 2 }\), simp ify g th co fficien s.
  2. State the set \(\mathbf { 6 }\) \& le s \(\mathbf { 6 }\) x fo wh cht b e nsin s valid
CAIE P3 2020 Specimen Q3
3
  1. Sk tcht b g a \(\phi \quad y = | 2 x - 3 |\).
  2. Sb the in a \(\operatorname { litg } x \rightarrow \quad | 2 x - 3 |\).
CAIE P3 2020 Specimen Q4
4 Th \(\mathbf { p }\) rametric eq tion \(\mathbf { 6 }\) a cn \(\mathbf { E }\) are $$x = \mathrm { e } ^ { 2 t - 3 } , \quad y = 4 \ln t$$ wh re \(t > 0\) Wh \(\mathrm { n } t = a\) th \(\mathbf { g }\) ad en 6 th cn ⊕ is 2
  1. Sba that \(a\) satisfies th eq tin \(a = \frac { 1 } { 2 } ( 3 \quad \mathrm { n } a )\).
  2. Verifyb \(y c\) alch atin \(\mathbf { h }\) tth s eq tim sarb \(\mathbf { b }\) tween \(\mathbf { l } \mathbf { d }\)
  3. Use th iterati fo mlu a \(a _ { n + 1 } = \frac { 1 } { 2 } \left( 3 - \ln a _ { n } \right)\) to calch ate \(a\) correct to 2 d cimal p aces, sh ig th resh to each teratin od cimal \(p\) aces.
CAIE P3 2020 Specimen Q5
5
  1. Sb that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x - \tan ^ { - 1 } x \right) = \frac { x ^ { 2 } } { 1 + x ^ { 2 } }\).
  2. Sth the \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).
CAIE P3 2020 Specimen Q6
6 Th cm plexm b rs \(1 + B\) ad \(4 + \quad 2\) are \(d \mathbf { b }\) ed \(\forall u\) ad \(v\) resp ctie ly.
  1. Fid \(\frac { \mathrm { u } } { \mathrm { V } }\) irt b fo \(\mathrm { m } x + \mathrm { i } y , \mathrm { w } \mathbf { b }\) re \(x\) ad \(y\) are real.
  2. State th argn en \(6 \frac { u } { v }\). In an Arg nd id ag am, with o ign \(O\), th \(\dot { \mathrm { p } } \mathrm { ns } A , B\) ad \(C\) represen th cm p ex m b rs \(u , v\) ad \(u - v\) resp ctie ly.
  3. State fullyt bg m etrical relatio h申 tween \(O C\) ad \(B A\).
  4. Sth the tag e \(A O B = \frac { 1 } { 4 } \pi\) rad as.
CAIE P3 2020 Specimen Q7
7
  1. By first d g co \(\left( x + \Omega ^ { \circ } \right)\), ev ess co \(\left( x + \Omega ^ { \circ } \right) - \sqrt { 2 } \sin x\) in th fo \(\mathrm { m } R \mathrm { co } ( x + \alpha )\), wh re \(R > 0\) ad \(0 ^ { \circ } < \alpha < \theta { } ^ { \circ }\). Gie th le \(6 R\) co rect to 4 sig fican fig res ad th le \(6 \alpha\) co rect tod cimal p aces. [ $\$$
  2. Hen e sb th teq tin $$\text { CB } \left( x + 3 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ fo \(0 ^ { \circ } < x < \boldsymbol { \theta }\)
CAIE P3 2020 Specimen Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{258f9a6f-9339-49c3-8118-6ae9e934f1bb-14_503_727_251_669} In th id ag am, \(O A B C\) is a ply amid in wh ch \(O A = 2\) in ts, \(O B = 4\) in ts ad \(O C = 2\) in ts. Th ed \(O C\) is rtical, th \(\mathbf { b }\) se \(O A B\) is \(\mathbf { b }\) izd al ad ag e \(A O B = \theta ^ { \circ }\). Un t cto s \(\mathbf { i } , \mathbf { j }\) ad \(\mathbf { k }\) are \(\mathbf { p }\) rallel to \(O A\), \(O B\) ad \(O C\) resp ctie ly. Th mij nsg \(A B\) ad \(B C\) are \(M\) ad \(N\) resp ctie ly.
  1. Eq ess th cto s \(\overrightarrow { \mathrm { ON } }\) ad \(\overrightarrow { \mathrm { CM } }\) irt erms \(\boldsymbol { 6 } \mathbf { i } , \mathbf { j }\) ad \(\mathbf { k }\).
  2. Calch ate th ab eb tweert b di rectis \(6 \overrightarrow { \mathrm { ON } }\) ad \(\overrightarrow { \mathrm { CM } }\).
  3. Sth the leg lo th p rp d ich ar from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).
CAIE P3 2020 Specimen Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{258f9a6f-9339-49c3-8118-6ae9e934f1bb-16_321_602_260_735} Th d ag am sto \(\mathrm { su } y = \sin ^ { 2 } 2 x \mathrm { co } x\) fo \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), ad ts max mm \(\dot { \mathrm { p } }\) n \(M\).
  1. Fid b \(x\)-co dia te \(6 M\).
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  2. Usig th stb tittu in \(u = \sin x\), find th area 6 th sh d d regn \(\mathbf { d } \quad \mathrm { d }\) y th cn e ad th \(x\)-ax s.
CAIE P3 2020 Specimen Q10
10 Ira ch mical reactio a cm \(\mathbf { p } \quad X\) is fo med rm twœ \(\mathbf { m } \mathbf { p } \quad Y\) ad \(Z\).
Tb masses in g ams \(\varnothing \quad X , Y\) ad \(Z \mathrm { p }\) esen at time \(t\) secd s after th start \(\varnothing\) th reactin are \(x , \mathbb { Q } - x\) ad \(0 - x\) resp ctie ly. At ay time th rate 6 fo matin \(6 X\) is p p tio l to to pd t \(\mathbf { 6 }\) th masses \(6 Y\) ad \(Z \mathrm { p }\) esen at th time. Wh \(\mathrm { n } t = \rho x = 0 \mathrm {~d} \frac { \mathrm { dx } } { \mathrm { dt } } = 2\).
  1. Sh the t \(x\) ad \(t\) satisfyt b d fferen ial eq tin $$\frac { \mathrm { dx } } { \mathrm { dt } } = \left( \frac { 1 } { 1 } \quad x \quad x \right) \left( \begin{array} { l l } 1 & x \end{array} \right) .$$
  2. Sb this d fferen ial eq tin \(\mathbf { C }\) aira ressift \(\mathbf { D }\) irt erms \(\boldsymbol { 6 } t\).
  3. State wh th p в to b \& le \(6 x\) wd \(n t\) b cm es larg If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin \(\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )\) ms tb clearlys n n
CAIE P3 2020 June Q7
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Using your answer to part (a), show that $$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
  3. Hence show that \(\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)\).