# Timothy Gan

#### Timothy Gan

This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.

Hope you find our articles enjoyable and have a great time learning math! 😁

### 2016 RI P1 Q11 [Modified]

2016 RI P1 Q11 [Modified] The line ${{l}_{1}}$ passes through the point $A$, whose position vector is $-mathbf{i}+2mathbf{j}$, and is parallel to the vector $mathbf{i}+mathbf{k}$. The line ${{l}_{2}}$ passes through

### 2023 SAJC P2 Q5

2023 SAJC P2 Q5 (i) Show that $frac{1}{r!}-frac{1}{left( r+1 right)!}=frac{r}{left( r+1 right)!}$. [1] (ii) Using the result in (i), evaluate the sum ${{S}_{n}}=frac{1}{2!}+frac{2}{3!}+frac{3}{4!}+…+frac{n}{left( n+1 right)!}$ in terms of $n$. [3]

### 2022 SAJC Promo Q8

2022 SAJC Promo Q8 [Volume of a cylinder, $V=pi {{r}^{2}}h$, where $r$ is the radius of the cylinder and $h$ is the vertical height]A manufacturer wants to produce a container

### 2023 JPJC Promo Q4

2023 JPJC Promo Q4 Water leaks out at a rate of $2$cm$^{3}$ per second from a container in the form of an open cone. The semi-vertical angle of the cone

### 2022 ACJC Promo Q1

2022 ACJC Promo Q1 (i) If $y=ln left( frac{{{text{e}}^{sqrt{x}}}}{{{cos }^{3}}x} right)$, find $frac{text{d}y}{text{d}x}$ in terms of $x$, where $0<x<frac{pi }{2}$. [2] (ii) Given that ${{y}^{frac{1}{x}}}={{x}^{ln x}}$, find $frac{text{d}y}{text{d}x}$ in terms

### 2022 SAJC Promo Q3

2022 SAJC Promo Q3 (a) Given that $y=frac{1}{sqrt{2{{text{e}}^{x}}-1}}$, show that (i) $frac{text{d}y}{text{d}x}=-{{y}^{3}}{{text{e}}^{x}}$, [2] (ii) $yfrac{{{text{d}}^{2}}y}{text{d}{{x}^{2}}}=yfrac{text{d}y}{text{d}x}+3{{left( frac{text{d}y}{text{d}x} right)}^{2}}$. [2] (b) Find $frac{text{d}}{text{d}x}left( ln left( frac{2x}{sqrt{{{x}^{2}}+1}} right) right)$. [2] Suggested Video Solutions

### 2022 EJC Promo Q11

2022 EJC Promo Q11 It is given that $sumlimits_{r=,,1}^{n}{{{r}^{2}}=frac{1}{6}nleft( n+1 right)left( 2n+1 right)}$. (a) Show that $sumlimits_{r=,,1}^{n}{left( 2r-7 right)left( r+1 right)}=frac{1}{6}nleft( 4{{n}^{2}}-9n-55 right)$. [4] (b) Find $sumlimits_{r=,,1}^{n}{{{3}^{-,r}}}$ in terms of

### 2022 SAJC Promo Q2

2022 SAJC Promo Q2 (a) (i) Using the method of difference, show that $sumlimits_{r=,,1}^{n}{left[ {{left( r+1 right)}^{3}}-{{r}^{3}} right]}={{left( n+1 right)}^{3}}-1$. [2] (ii) Expand and simplify ${{left( r+1 right)}^{3}}-{{r}^{3}}$. [1] (iii)

### HCI Tutorial Applications of Differentiation Q8

HCI Tutorial Applications of Differentiation Q8 In optics, the focal length of a thin lens (negligible thickness) is the distance between the centre of the lens and the point at

### HCI Applications of Differentiation Tutorial Q6

HCI Applications of Differentiation Tutorial Q6 The diagram shows a rectangle $ABCD$ inscribed in a semi-circle with fixed radius $r$cm. If $AD=x$cm, find an expression for the perimeter $P$ and