2009 SAJC P2 Q3 [Modified] Given that ${{u}_{r}}=frac{1}{{{r}^{3}}}$, (i) Show that ${{u}_{r-1}}-{{u}_{r+1}}=frac{6{{r}^{2}}+2}{{{left( {{r}^{2}}-1 right)}^{3}}}$. (ii) Hence find $sumlimits_{r=2}^{n}{frac{3{{r}^{2}}+1}{{{left( {{r}^{2}}-1 right)}^{3}}}}$ in terms of $n$. (iii) Use your answer to part
2016 NJC P2 Q1 (a) Kenny took a loan of $$9600$ from a friend, and arranged to pay his loan fully in a period of exactly $48$ months. To fulfil
2021 NYJC P1 Q5 (b) The function $text{g}$ is given by $text{g}:xmapsto ln left( frac{{{e}^{x}}+5}{{{e}^{x}}-1} right)$, for $xin mathbb{R}$, $x>0$. (i) Find ${{text{g}}^{-1}}left( x right)$ and states its domain. The
2020 ACJC Promo Q7 The functions $text{f}$ and $text{g}$ are defined as follows: $text{f}:xmapsto 7{{left( x-a right)}^{2}}-1$, $xin mathbb{R}$, $x<3$ $text{g}:xmapsto 1+b-{{e}^{-x}}$, $xge 0$, where $a$ and $b$ are real
2022 EJC J2 MCT Q8 Planes $p$ and $q$ have equations $x+y+z=5$ and $3x+y-z=9$ respectively, and meet in the line $l$. (a) Find a vector equation for $l$. [2] (b)
2020 RVHS P1 Q10 Sales agent $A$ started work on 1st June 2020. He plans to acquire $2$ clients in his first month of work and thereafter increase his clientele
2020 SAJC P2 Q3 The variables $x$ and $y$ are related by the differential equation $frac{text{d}y}{text{d}x}=frac{x-y}{x-y+1}$. (i) Solve the differential equation, using the substitution $u=x-y+1$, and show that the general
2020 TMJC P1 Q8 (i) Show that the differential equation $frac{text{d}y}{text{d}x}=5left( x-y right)$ may be reduced by the substitution $w=x-y$ to $frac{text{d}w}{text{d}x}=1-5w$. Hence find the general solution for $~y$ in
2020 RVHS P2 Q2 A particle moving in a liquid is such that after $t$ seconds, its velocity is $v$ ms$^{-1}$, $vne 0$. $v$ satisfies the differential equation $vfrac{text{d}v}{text{d}t}+2{{v}^{2}}=t{{text{e}}^{-2t}}$. (i)
2020 RI P1 Q3 (i) Prove that for $x>0$, the substitution $y=ux$ reduces the differential equation $left( y-x right)left( frac{text{d}y}{text{d}x}-frac{y}{x} right)={{y}^{2}}+2{{x}^{2}}$ to $left( frac{u}{{{u}^{2}}+2}-frac{1}{{{u}^{2}}+2} right)left( frac{text{d}u}{text{d}x} right)=1$. [2] (ii) Hence