Questions — CAIE (7276 questions)

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CAIE P3 2020 Specimen Q2
4 marks Moderate -0.8
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
4 marks Easy -1.3
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
9 marks Standard +0.3
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
7 marks Standard +0.8
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
8 marks Moderate -0.5
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
10 marks Standard +0.3
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 \(6 M\).
    [0pt] [С
  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
11 marks Standard +0.5
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 S1 2020 Specimen Q1
5 marks Easy -1.2
1 Th fb low ing b ck te b ck stem-ad leaf il ag am sw stb a lsalaries \(\mathbf { 6 }\) agp \(\mathbf { 6 } \mathbf { 9 }\) females adgn ales.
FemalesMales
(4)520003(1
(9)9887640002007( \(\mathcal { B }\)
(8875331002004566( \(\varnothing\)
( )6421003002335677(9)
( ( )75400040112556889(1)
(4)950083457789\(( \gamma\)
(2)508046(3
Key 4 Q 3 m eas ( st \(\mathbf { o }\) females an of \(\mathbf { o }\) males.
  1. Fid b med ara d b ɛ rtiles \(\mathbf { 6 }\) th females' salaries. Yo are gie \(n\)th \(t\) th med an salary \(\mathbf { 6 }\) th males is \(\boldsymbol { \otimes } \rho\) th lw er \(\mathbf { q }\) rtile is \(\boldsymbol { \\) } \boldsymbol { \theta }\( ad th \)\mathbf { p }\( r e rtile is \\)50
  2. Drawap ir d ad wh sk rpos in a sig ed ag amo to g id b lw to rep esen th d ta. [β
    \includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-02_997_1589_1736_310}
CAIE S1 2020 Specimen Q2
4 marks Easy -1.2
2 A sm mary \(\mathbf { 6 }\) th sp esl, \(x \mathrm { k }\) lm etres \(\boldsymbol { \rho } \mathbf { r b }\), \(\mathbf { 0 } 2\) cars \(\boldsymbol { \rho }\) ssig a certain \(\dot { \mathrm { p } } \mathrm { ng }\) th fb low ig if o matin $$\Sigma ( x \oplus ) = 3 \mathrm { a } \quad \mathrm {~d} \quad \Sigma ( x \oplus ) ^ { 2 } = \mathbb { t }$$ Fid b riance 6 th sp ed ad \(n\) e fid b vale \(6 \Sigma x ^ { 2 }\). [4]
CAIE S1 2020 Specimen Q3
7 marks Moderate -0.5
3 A b clb sed 6 p p rb ck ad 2 h r ck b to Mrs Ho . Sb cb es 4 6 tb se b at rach to take with b r o b id y. Th rach \& riable \(X\) rep esen s tb m br \(\mathbf { b }\) p \(\mathbf { p }\) rb ck b sh cb es.
  1. Sth that th p b b lityt \(\mathbf { h }\) tsb cb es extlye perb clb is \(\frac { 3 } { 14 }\). [R
  2. Draw up b pb b lityd strib in tab e fo \(X\).
  3. Yu reg it h t \(\mathrm { E } ( X ) = 3\) Fid \(\operatorname { Var } ( X )\).
CAIE S1 2020 Specimen Q4
10 marks Moderate -0.5
4 A \(\boldsymbol { \rho }\) trb station fid th tits \(\mathbf { d }\) ily sales，in litres，are \(\mathbf { n }\) mally \(\dot { \mathbf { d } }\) strib ed with mean ad stad rd d \(\dot { \mathbf { v } }\) atin \(\quad 0\)
（a）Fid 0 may dy 6 th \(\mathbf { y }\) ar（B d \(\mathbf { y }\) ）th d ily sales can b eq cted to e区 eed \(\boldsymbol { \theta }\) litres． Th d ily sales at an \(\mathbf { b } r \mathbf { p }\) trb station are \(X\) litres，we re \(X\) is \(\mathbf { n }\) mally \(\dot { \mathbf { d } }\) strib ed with mean \(m\) ad stad rd iv atird \(\quad t\) is g it h \(\mathrm { t } \mathrm { P } ( X > 0 = \mathbb { 0 }\)
（b）Fid by le \(6 m\) ．
(c) Fid th p b b lity th t d ily sales at th s p trb station ex eed \(\theta\) litres \(\mathbf { n }\) fewer th n 266 rach lyc \(b\) end \(y\).
[0pt] [ \(\beta\)
CAIE S1 2020 Specimen Q5
7 marks Moderate -0.5
5 A fair six sid dl e,w itlf aces mark dress s ther im imes.
  1. Use ara \(p\) in matin of id b pb b lity b ta 3 s ob ain of ewer th rㅇs imes. [4]
  2. Js tifys se 6 th ap ox matin pe rt (a). Ora \(\mathbf { h }\) b roccasity he same \(\dot { \mathbf { d } }\) e is th \(\boldsymbol { w }\) ep ated y il a \(\mathbf { 3 } \mathrm { sb }\) aie d
  3. Fid b pb b lity b tb ain g ʒ eq res fewer th \(n\) st \(h\) s.
CAIE S1 2020 Specimen Q6
7 marks Standard +0.3
6 Ag \(\mathbf { \Phi }\) of ries trac ls to b airp t irt wd axis, \(P\) ad Q.E acht ax cart ak \(\boldsymbol { \mathcal { C } }\) sseg rs.
  1. Th 8 fried dive th msele s in o two gp 6,4 日 gp fo tax \(P\) ad o gp fo tax \(Q\),w ithlo il aralt rae llig it te same tax. Fid b m brd dl fferen way inw hick his carb de .
    \includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-11_298_492_226_447}
    \includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-11_301_478_223_1142} Each tax can tak \(1 \boldsymbol { \rho }\) sseg r in th fro ad \(3 \boldsymbol { \rho }\) sseg rs in th \(\mathbf { b }\) ck (see \(\dot { \mathbf { d } }\) ag am). Mark sits in th
    \includegraphics[max width=\textwidth, alt={}, center]{1fef5f2c-b375-4be2-b8a1-c30136bd0063-11_51_1227_598_242}
  2. Fid b m brd d fferen seatig rrag men s th tare \(\mathbf { w }\) sibefo th of ried . [4]
CAIE S1 2020 Specimen Q7
10 marks Standard +0.3
7 Bag \(A\) ch ais \(4 \mathbf { b }\) lls \(\mathrm { m } \quad \mathbf { b }\) red \(2,4,58\) Bag \(B\) ch ais \(5 \mathbf { b }\) lls \(\mathrm { m } \quad \mathbf { b }\) red 1,3688 Bag \(C\) co ais 7 b lls m b redram a b \(l l\) is selected \(t\) rach frm eaclb \(g\)
  • Ed \(n X\) is 'ed ctlyt wo th selecteb lls \(\mathbf { h }\) th same m br'.
  • Ed n \(Y\) is 'tb b ll selected rm bag \(A \mathbf { h }\) sm br4.
    1. FidP (X).
    2. Fid ( \(X \cap Y\) ) aid \(\mathbf { n }\) ed termin wh ther or \(\mathbf { n }\) even \(\mathrm { s } X\) ad \(Y\) are id \(\mathbf { p } \mathbf { d } \quad \mathrm { h }\). [B
    3. Fid the p b b lity th t two \(\mathbf { b }\) lls are \(\mathrm { m } \quad \mathbf { b }\) red \(2 \dot { \mathrm {~g} }\) n th t ex ctly two \(\mathbf { 6 }\) th selected \(\mathbf { b }\) lls h \& th same m br.
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 Further Paper 4 2020 Specimen Q1
7 marks Moderate -0.5
1
  1. State b iefly tb circm stan es d r wh cha a \(\quad\) p rametric test \(\mathbf { 6 }\) sig fican e sh lid be \(\mathbf { b }\) ed rath r thara \(\mathbf { P }\) rametric test. Th le l6 p ltu in ira rie r was measu ed at \(\mathbb { r }\) ach lyc \(\mathbf { b }\) serl o atin . Tb resli ts, iı ù tab e i ts,a re sh wib lo , w b reh g r le s rep esen g eater p ltu in B 56 B 158 58 日 9
  2. Use a Wilc sig d rah test to test wh th r th a rag p ltu in le l in th rie r is mo e th iU se a \(\%\) sig fican e le 1 .
    [0pt] [6
CAIE Further Paper 4 2020 Specimen Q2
7 marks Challenging +1.2
2 Each \(\mathbf { 6 }\) Q id \(n\) ically \(\mathbf { b }\) ased d ce is th \(n\) rep ated \(\mathrm { y } n\) il an eq \(n\) b \(\mathbf { r }\) is \(\mathbf { b }\) ain d Th \(\mathbf { m }\) b \(\mathbf { r }\) 6 th se ed d s reco d d d b resu ts are sm marised it b fb low ig ab e.
Numb r \(\mathbf { 6 }\) th s123456\(\geqslant 7\)
Freq n y\(\boldsymbol { 6 }\)\(\mathbf { 3 }\)23510
Carry \(\mathbf { a }\) a ss \(\mathbf { 6 }\) fit test, at th \(\% _ { 0 }\) sig fican e lev l, to test wh th r Ge( Đ is a satisfacto y md lfo th d ta.
[0pt] []
CAIE Further Paper 4 2020 Specimen Q3
8 marks Standard +0.3
3 Empø es at ap rtich ar comp y \(\mathbf { h }\) \textbackslash ＆ b en wo kg seqnb s each \(\mathbf { d y }\) ，frm 9 am to 4 p ．To try to red e ab en e，th cm \(\mathbf { p } \mathrm { y }\) d cid s to in rd e ＇flex－time＇ad all emp 甲 es to wo k th ir see nb s each d y at ay time b tween 7 am ad 9 p ．Fo a rach sampe \(\mathbf { 6 } \mathbf { 0 }\) emp \(\boldsymbol { \varphi }\) es，th m b rs 6 b s of ab en e in th \(y\) arb fo e ad th \(y\) ar after th in rd tin 6 flex－time are g t rit t fb low ig ab e．
Emp \(\varphi\) e\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Befo e2364058360
After3302326300
Test，at th \textbackslash ％sig ficance le l，wh th r th \(\mathbf { p }\) atin mean m br \(\mathbf { 6 }\) b s \(\mathbf { 6 }\) ab en e \(\mathbf { s }\) d creased b lw ig b in rd tim flex－time，s tatig pr ssm p it b tm ak．［ア
CAIE Further Paper 4 2020 Specimen Q4
7 marks Standard +0.8
4 Th m b r, \(x , 6\) a certain ty sea sh ll was co ed at \(\theta\) rach ly cb en sites, each \(\mathbf { a }\) metre seq re, alg th co stlie in co ry \(A\). Th \(m \quad \mathbf { b } , y , \boldsymbol { 6 }\) th same \(\quad \mathbf { 6 }\) sea sh ll was co ed at \(\theta\) rach ly cb en sites, each \(\mathbf {} { } _ { \text {t } }\) metre sq re, alog th co stlie in co ry \(B\). Tb results are sm marised as fb lw s,w b re \(\bar { x }\) ad \(\bar { y } \mathbf { d } \mathbf { h }\) e th samp e meas \(\mathbf { 6 } x\) ad \(y\) resp ctiv ly. $$\bar { x } = 9 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = \mathbf { 3 } \quad \bar { y } = \mathbf { 4 } \quad \Sigma ( y - \bar { y } ) ^ { 2 } = \text { 日 }$$ \includegraphics[max width=\textwidth, alt={}]{0df58f9d-6700-46cc-bcf0-903e94cccc02-06_58_1667_539_239} metre,it b co stlin sirc \(\mathbf { b }\) ry \(A\) ad inc \(\mathbf { b }\) ry \(B\).
CAIE Further Paper 4 2020 Specimen Q5
8 marks Standard +0.3
5 Th co in rach \& riab e \(X \mathbf { h }\) s prb b lity e \(\mathbf { s }\) ityf \(\mathbf { n }\) tiff \(\dot { \mathrm { g } }\) \& $$f ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 6 } { 5 } x & 0 \leqslant x \leqslant 1 \\ \frac { 6 } { 5 } x ^ { - 4 } & x > 1 \end{cases}$$
  1. FidP \(( X > 1\).
  2. Fid b med arm le \(6 X\).
  3. Gie it \(\mathbf { h } \mathrm { t } \mathrm { E } ( X ) =\), fif id \(\mathbf { b }\) riance \(6 X\).
  4. Fide \(( \sqrt { X } )\).
CAIE Further Paper 4 2020 Specimen Q6
13 marks Standard +0.3
6 Aish h sab g co ainig 3 red b lls ad 3 wh te b lls. Sb selects a b ll at rach , b es its cb o ad return it to th b g th same p o ess is rep ated twice mo e. Tb m brd red b lls selected b Aish is d no edy \(X\).
  1. Fid \(\mathbf { b } \mathbf { p b }\) b lityg \(\mathbf { a }\) ratig \(\mathbf { a }\) ting \({ } _ { X } ( t ) \boldsymbol { 6 } X\). Basan also s sab g co ain g 3 red balls ad 3 wh te b lls. He selects th ee b lls at rach , with rep acemen, frm hsbg Th m brg red lls selectedB asan is d n edy \(Y\).
  2. Fid \(\mathbf { b } \mathbf { p } \mathbf { b }\) b litys \(\mathbf { a }\) ratif \(\mathbf { a }\) ting \({ } _ { Y } ( t ) \underset { \text { b } } { } Y\). Th rad \(m\) riab e \(Z\) is to to alm brø reb lls selected y Aish adB asan.
  3. Fid \(\mathbf { b } \mathbf { p } \mathbf { b }\) b lityg \(\mathbf { e }\) ratig \(\mathbf { u }\) tim \(Z\), essig as wer as \(\mathrm { p } \mathbf { p }\) ial. [β
  4. Use th p b b lityg \(\mathbf { B }\) ratig u tim \(Z\) tof idE ( \(Z\) ) ad \(\operatorname { Var } ( Z )\). [\$ 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 FP1 2008 June Q1
4 marks Standard +0.8
1 The finite region enclosed by the line \(y = k x\), where \(k\) is a positive constant, the \(x\)-axis for \(0 \leqslant x \leqslant h\), and the line \(x = h\) is rotated through 1 complete revolution about the \(x\)-axis. Prove by integration that the centroid of the resulting cone is at a distance \(\frac { 3 } { 4 } h\) from the origin \(O\).
[0pt] [The volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE FP1 2008 June Q2
5 marks Standard +0.8
2 Given that $$u _ { n } = \ln \left( \frac { 1 + x ^ { n + 1 } } { 1 + x ^ { n } } \right)$$ where \(x > - 1\), find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(N\) and \(x\). Find the sum to infinity of the series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ when
  1. \(- 1 < x < 1\),
  2. \(x = 1\).
CAIE FP1 2008 June Q3
6 marks Challenging +1.2
3 Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector, then \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector. The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 4 & - 6 & - 6 \\ 5 & 11 & 10 \end{array} \right)$$ has \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) as an eigenvector. Find the corresponding eigenvalue. The other two eigenvalues of \(\mathbf { A }\) are 1 and 2, with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. The matrix \(\mathbf { B }\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\), \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
[0pt] [You are not required to evaluate \(\mathbf { P } ^ { - 1 }\).]