CAIE P3 (Pure Mathematics 3) 2020 Specimen

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. [ \(\beta\)

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

3

1. Sk tcht b g a \(\phi \quad y = | 2 x - 3 |\).
2. Sb the in a \(\operatorname { litg } x \rightarrow \quad | 2 x - 3 |\).

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.

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 \(\int _ { 0 } ^ { \sqrt { 3 } } x \tan ^ { - 1 } x \mathrm {~d} x = \frac { 2 } { 3 } \pi - \frac { 1 } { 2 } \sqrt { 3 }\).

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.

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 \(0 ^ { \circ } < x < \boldsymbol { \theta }\)

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 \(\frac { 3 } { 5 } \sqrt { 5 }\).

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.

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