CAIE Further Paper 1 (Further Paper 1) 2020 Specimen

1

1. Gie it h \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) } , \mathrm { s }\) th t $$\mathrm { f } \left( r - 1 \quad \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) } \right.$$
2. Hen e fid \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
3. Ded e th le \(6 \sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

2 It is g \(n\)th \(\mathrm { t } \phi ( n ) = 5 ^ { n } ( 4 n + 1 )\), ff \(\quad \mathbf { o } \quad n = , \mathbb { B } .\).
Pro tyn ath matical id tin th \(t \phi ( n )\) is \(\dot { \mathbf { d } } \dot { \mathbf { v } }\) sib eff \(\quad \mathbf { o }\) eyp itie in eg \(r n\).

3 Th cn \(C \mathbf { h }\) sp areq tin \(r = 2 + 2\) co \(\theta\), fo \(0 \leqslant \theta \leqslant \pi\).

1. Sk tch \(C\).
2. Fid \(b\) area 6 th reg œ \(n\) lo edy \(C\) ad \(b\) in tial lie .
3. Sh the the Cartesiare q tim \(C\) carb essed \(\mathrm { s } \left( 4 x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\). \(\quad [ \beta\)

4 Th cb c ę tin $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ h s ro \(\mathrm { s } \alpha , \beta\) ad \(\gamma\).

1. Sth th t th le \(6 \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\) is 9
2. Fid he le \(6 \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }\).
3. Fird cb ceq tin \(N\) ith o s \(\alpha + 1 \beta + 1 \mathrm {~d} \gamma + \underset { \text { g } } { \text { vg } } \quad\) as wer in th fo m $$p x ^ { 3 } + q x ^ { 2 } + r x + s = 0$$ we re \(p , q , r\) ad \(s\) are co tan s to \(\mathbf { b } \quad \mathbf { d }\) termin d

5 Th matrix \(\mathbf { A }\) is g it $$\mathbf { A } = \left( \begin{array} { r r } 5 & k
- 3 & - 4 \end{array} \right)$$

1. Fid b le \(6 k\) fo wh ch \(\mathbf { A }\) is sig ar. It is \(\mathbf { M } \quad \mathbf { g }\) vert h \(\mathrm { t } k = 6\) d \(\mathbf { h } \mathrm { t } \mathbf { A } = \left( \begin{array} { r r } 5 & 6
   - 3 & - 4 \end{array} \right)\).
2. Fid th eq tim 6 th in rian lies s, th g th o ign 6 th tras fo matin in th \(x - y \mathrm { p }\) ae rep esen edy \(\mathbf { A }\).
3. Th triag e \(D E F\) in b \(x - y\) p ae is tras fo medy An a riag e \(P Q R\).
   1. Gie it \(\mathbf { h }\) th area 6 triag e \(D E F\) is \(\mathbb { I } \mathrm { cm } ^ { 2 }\), f id \(\mathbf { b }\) area 6 triag e \(P Q R\).
   2. Find b matrixw h cht ras fo ms triag e \(P Q R\) b d riag e \(D E F\).

6 Th p itim ctosg th \(\dot { \mathrm { p } }\) ns \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$ resp ctively, wh re \(m\) is an in eg r. It is g n th the sh test d stan e b tween th lie th g \(\quad A\) ad \(B\) ad lie th g \(\quad C\) ad \(D\) is 3

1. Shat that to b sib ey le \(6 m\) is 2
2. Fid b sh test il stan e \(6 D\) frm th lie thrg \(\mathrm { h } A\) ad \(C\).
3. Swat the actu eas eb tweert b p an \(\mathrm { s } A C D\) ad \(B C D\) is co \({ } ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).

7 Th cn \(C\) h s ę tin \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\).

1. State th eq tin 6 th asm po es \(6 C\).
2. Shat \(y \leqslant \frac { 25 } { 12 }\) at all \(\dot { p }\) nso C.
3. Fid b co dia tesb aws tatio ryip ns of \(C\).
4. Sk tch \(C\), statig th co dia tes 6 ay in ersectio \(6 C\) with th co \(\dot { \mathbf { d } } \mathbf { a }\) te aes ad th asm poes.
5. Sk tch th cn with eq tin \(y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|\) ad fid th set 6 les \(6 x\) fo wh ch \(\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2\)
   [0pt] [4] 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