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CAIE Further Paper 1 2020 Specimen Q2
7 marks Standard +0.3
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 \(r n\).
CAIE Further Paper 1 2020 Specimen Q3
10 marks Standard +0.8
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 \(\mathrm { s } \left( 4 x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\). \(\quad [ \beta\)
CAIE Further Paper 1 2020 Specimen Q4
9 marks Standard +0.8
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
CAIE Further Paper 1 2020 Specimen Q5
12 marks Standard +0.3
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 \(D E F\).
CAIE Further Paper 1 2020 Specimen Q6
14 marks Challenging +1.8
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 \({ } ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).
CAIE Further Paper 1 2020 Specimen Q7
17 marks Challenging +1.2
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
CAIE P2 2019 June Q1
3 marks Moderate -0.8
1 Show that \(\ln \left( x ^ { 3 } - 4 x \right) - \ln \left( x ^ { 2 } - 2 x \right) \equiv \ln ( x + 2 )\).
CAIE P2 2019 June Q2
6 marks Standard +0.3
2
  1. Solve the inequality \(| 3 x - 5 | < | x + 3 |\).
  2. Hence find the greatest integer \(n\) satisfying the inequality \(\left| 3 ^ { 0.1 n + 1 } - 5 \right| < \left| 3 ^ { 0.1 n } + 3 \right|\).
CAIE P2 2019 June Q3
7 marks Standard +0.3
3 Find the equation of the normal to the curve $$x ^ { 2 } \ln y + 2 x + 5 y = 11$$ at the point \(( 3,1 )\).
CAIE P2 2019 June Q4
7 marks Moderate -0.3
4
  1. Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 3 x } + 4 } { \mathrm { e } ^ { x } } \mathrm {~d} x\). Show all necessary working.
CAIE P2 2019 June Q5
8 marks Moderate -0.8
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 5 x ^ { 3 } + a x ^ { 2 } + b x - 16$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 27 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2019 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{6694ccc1-c8b1-42a7-8b21-829a89af74c9-08_732_807_258_667} The diagram shows the curve with equation \(y = \frac { 8 + x ^ { 3 } } { 2 - 5 x }\). The maximum point is denoted by \(M\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine the gradient of the curve at the point where the curve crosses the \(x\)-axis.
  2. Show that the \(x\)-coordinate of the point \(M\) satisfies the equation \(x = \sqrt { } \left( 0.6 x + 4 x ^ { - 1 } \right)\).
  3. Use an iterative formula, based on the equation in part (ii), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2019 June Q7
10 marks Standard +0.8
7
  1. Show that \(2 \operatorname { cosec } 2 \theta \cot \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
  2. Hence show that \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } \tan 15 ^ { \circ } = 4\).
  3. Solve the equation \(2 \operatorname { cosec } \phi \cot \frac { 1 } { 2 } \phi + \operatorname { cosec } \frac { 1 } { 2 } \phi = 12\) for \(- 360 ^ { \circ } < \phi < 360 ^ { \circ }\). Show all necessary working.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2019 June Q1
3 marks Moderate -0.8
1 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + ( k + 1 ) x ^ { 2 } - m x + 3 k$$ where \(k\) and \(m\) are constants. Given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\), express \(m\) in terms of \(k\).
CAIE P2 2019 June Q2
5 marks Moderate -0.3
2
  1. Solve the equation \(| 4 + 2 x | = | 3 - 5 x |\).
  2. Hence solve the equation \(\left| 4 + 2 e ^ { 3 y } \right| = \left| 3 - 5 e ^ { 3 y } \right|\), giving the answer correct to 3 significant figures.
CAIE P2 2019 June Q3
5 marks Standard +0.8
3 Find the exact coordinates of the stationary point of the curve with equation \(y = \frac { 3 x } { \ln x }\).
CAIE P2 2019 June Q4
8 marks Moderate -0.3
4
  1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( 4 \sin 2 x + 2 \cos ^ { 2 } x \right) \mathrm { d } x\). Show all necessary working.
  2. Use the trapezium rule with two intervals to find an approximation to \(\int _ { 2 } ^ { 8 } \sqrt { } ( \ln ( 1 + x ) ) \mathrm { d } x\)
CAIE P2 2019 June Q5
8 marks Standard +0.3
5
  1. Find the quotient and remainder when \(2 x ^ { 3 } + x ^ { 2 } - 8 x\) is divided by ( \(2 x + 1\) ).
  2. Hence find the exact value of \(\int _ { 0 } ^ { 3 } \frac { 2 x ^ { 3 } + x ^ { 2 } - 8 x } { 2 x + 1 } \mathrm {~d} x\), giving the answer in the form \(\ln \left( k \mathrm { e } ^ { a } \right)\) where \(k\) and \(a\) are constants.
CAIE P2 2019 June Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{f5e0b088-73db-405b-a832-aa01d9fcba64-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .
  1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
  2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
  3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.
CAIE P2 2019 June Q7
11 marks Standard +0.8
7
    1. Express \(4 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
    2. Hence find the smallest positive value of \(\theta\) satisfying the equation \(4 \sin \theta + 4 \cos \theta = 5\).
  2. Solve the equation $$4 \cot 2 x = 5 + \tan x$$ for \(0 < x < \pi\), showing all necessary working and giving the answers correct to 2 decimal places.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2019 June Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{0d15e5a1-d05f-48bc-8613-198804ff605c-08_396_716_260_712} The diagram shows the curve with parametric equations $$x = 3 t - 6 \mathrm { e } ^ { - 2 t } , \quad y = 4 t ^ { 2 } \mathrm { e } ^ { - t }$$ for \(0 \leqslant t \leqslant 2\). At the point \(P\) on the curve, the \(y\)-coordinate is 1 .
  1. Show that the value of \(t\) at the point \(P\) satisfies the equation \(t = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t }\).
  2. Use the iterative formula \(t _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } t _ { n } }\) with \(t _ { 1 } = 0.7\) to find the value of \(t\) at \(P\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
  3. Find the gradient of the curve at \(P\), giving the answer correct to 2 significant figures.
CAIE P2 2016 March Q1
3 marks Easy -1.2
1 Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 10\) is divided by \(( x + 2 )\).
CAIE P2 2016 March Q2
4 marks Standard +0.3
2 Solve the inequality \(| x - 5 | < | 2 x + 3 |\).
CAIE P2 2016 March Q3
5 marks Standard +0.3
3 It is given that \(k\) is a positive constant. Solve the equation \(2 \ln x = \ln ( 3 k + x ) + \ln ( 2 k - x )\), expressing \(x\) in terms of \(k\).
CAIE P2 2016 March Q4
5 marks Standard +0.3
4 The sequence of values given by the iterative formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } + 4 x _ { n } ^ { - 3 } \right)$$ with initial value \(x _ { 1 } = 1.5\), converges to \(\alpha\).
  1. Use this iterative formula to find \(\alpha\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
  2. State an equation that is satisfied by \(\alpha\) and hence find the exact value of \(\alpha\).