CAIE P1 (Pure Mathematics 1) 2020 Specimen

1 Th fb lowiற் ns $$A ( \Omega , B ( \mathbb { \phi } , C ( \mathbb { \phi } , D ( \mathbb { \phi } ) \quad \text { ad } \quad E ( \mathbb { \phi } )$$ lie t b cure \(y = \mathrm { f } ( x )\). Th tabeb lw sho s th ad ens 6 th \(\mathrm { ch } \mathrm { s } A E\) ad \(B E\). \begin{center} \begin{tabular}{ | c | c | c | c | c | } \hline Ch d & \(A E\) & \(B E\) & \(C E\) & \(D E\)
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2 Fn tin f adg re d fie dy $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + 2 \quad x \in \mathbb { R }
& \operatorname { g } \quad x \mapsto 4 x - 2 \quad x \in \mathbb { R } \end{aligned}$$ Sb th equ tiff \({ } ^ { - 1 } ( x ) = g ( x )\).

3 Ara rith etic p og essich s first term 7 Th \(n\)th erm is \& d (3n)ttl erm is \% Fid b lue \(6 n\).

4 A cu h s eq tin \(y = \mathrm { f } ( x )\).I t is g it h \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }\) ad h \(\mathrm { f } ( \mathcal { \beta } = 1\)
Fif ( \(x\) ).

5

1. Th cn \(y = x ^ { 2 } + 3 x + 4\) s tras latedy \(\binom { 2 } { 0 }\).
   Find imp ify \(\mathbf { b }\) eq tim the tras lated \(\mathrm { n } \mathbb { E }\).
2. Th g ad \(y = \mathrm { f } ( x )\) is tras fo med \& b g ap \(6 y = \mathrm { B } ( - x )\). Describ fly ly th two sig le tras fo matin wh ch hav b en cm be d to ge th resh tig tras fo matio
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6

1. Fid \(\mathbf { b }\) co fficien \(\mathrm { s } 6 x ^ { 2 }\) ad \(x ^ { 3 }\) irt \(\mathbf { b }\) e in ( \(\left. 2 x \right) ^ { 6 }\). [
2. Hen e fid b co fficien \(6 x ^ { 3 }\) in b \(\mathbf { b }\) in \(\left( 3 x + 1 ( 2 \quad x ) ^ { 6 } \right.\).

7

1. Sw that the equr tin \(\sin x \tan x = 5 \mathrm { co } x\) carb esseds s $$6 \text { св } ^ { 2 } x \in \text { в } x \neq 0$$
2. Hen e sb e the tinl \(\sin x \tan x = 5 c o x\) fo \(\theta \leqslant x \leqslant \theta\)

8 A cu t h seq tin \(y = \frac { 12 } { 3 - 2 x }\).

1. Fid \(\frac { \mathrm { dy } } { \mathrm { dx } }\). A \(\dot { p } \cap \mathrm { n } \in \mathrm { s }\) alg thscue. Astb \(\dot { p } \cap \mathbf { p }\) sses th \(\mathbf { g } \quad A\), th \(x\)-co \(\dot { \mathbf { d } } \mathbf { a }\) te is in reasig at a rate \(\boldsymbol { 6 }\) ts p r sech d b \(y\)-co d a te is in reasign ta rate \(\mathbf { 6 }\) th ts p r secd
2. Fid b s sib e \(x\)-co du tes \(6 A\).

9
\includegraphics[max width=\textwidth, alt={}, center]{3986478a-062a-4bc6-bce2-85408b51a0b2-14_716_912_258_571} Th id ag am sh s a circle with cen re \(A\) ad rad s r. Diameters \(C A D\) ad \(B A E\) are \(\mathbf { p }\) re \(\dot { \text { d } }\) ch ar to each b r. A larg r circle \(\mathbf { h }\) s cen re \(B\) a¢ sses th \(\mathbf { g } \quad C\) ad \(D\).

1. Sth that th rad s 6 th larg r circle is \(r \sqrt { 2 }\).
2. Fid b area \(\mathbf { 6 }\) th sh d d eg i r it erms \(\mathbf { 6 } r\).

10 Th circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - \quad\) Ch s cen re \(C\) a¢ sses the id s \(A\) ad \(B\).

1. State th co ida tesg \(C\). It is \(\dot { \mathbf { g } } \dot { \mathrm { n } }\) the t th mid oin, \(D , 6 \quad A B \mathbf { h }\) s co \(\dot { \mathrm { d } } \mathbf { a }\) tes \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
2. Fid eq tin \(A B , \dot { \mathrm {~g} } \dot { \mathrm { v } }\) as wer in th fo \(\mathrm { m } y = m x + c\).
3. Fidy calch atitil b \(x\)-co dia tes \(6 A\) ad \(B\).

11 Th fo tin f is d fin f \(\mathbf { o } x \in \mathbb { R } , \quad x \mapsto x ^ { 2 } + a x + b , \mathrm { w } \mathbf { b }\) re \(a\) ad \(b\) are \(\mathrm { c } \mathbf { B }\) tan s .

1. It is g it h t \(a = 6\) d \(b = 8\) Fid b rag 6 f .
2. It is g ven in tead th \(\mathrm { t } a = 5\) ad th t th ro s \(\mathbf { 6 }\) the eq tin \(\mathrm { f } ( x ) = 0\) are \(k\) ad \(z k\), wh re \(k\) is a CB tan. Fid b le s \(6 b\) ad \(k\).
3. Sha th tif the equ tif \(( x + a ) = a \mathbf { h }\) so eal ro sth \(\mathrm { n } a ^ { 2 } < ( 4 b - a )\).

12
\includegraphics[max width=\textwidth, alt={}, center]{3986478a-062a-4bc6-bce2-85408b51a0b2-20_542_1003_260_539} Th d ag am sw s th cn \& with equ tin \(y = x \left( x - \mathcal { P } ^ { 2 } \right.\). Th min mm \(\dot { \mathbf { p } } n\) n th cn \(\mathbf { t }\) s co dia tes \(( a , \emptyset )\) ad l \(x\)-co id a te of th max \(\mathrm { mm } \dot { \mathrm { p } } \quad \mathrm { n }\) is \(b , \mathrm { w } \mathbf { b }\) re \(a\) ad \(b\) are \(\mathrm { c } \mathbf { n }\) tan s .

1. State the le \(6 a\).
2. Calch ate th \& le 6 b.
3. Fid b area 6 th sh d d eg n
4. Th g ad en, \(\frac { \mathrm { dy } } { \mathrm { dx } } , 6\) th cn a sa min mm \& le \(m\). Calch ate th le \(6 m\). 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