Temasek Junior College H2 Math papers, with video solutions

1. Relative to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors.
   1. It is given that $\mathbf{a}\times \mathbf{b}=\mathbf{0}$. Find $\mathbf{a}\cdot \mathbf{b}$ in terms of $\left| \mathbf{b} \right|$.

      [2]

      It is now given that $\mathbf{a}\times \mathbf{b}\ne \mathbf{0}$. Point $P$ is the foot of the perpendicular from $A$ to the line $OB$ and the point $Q$ is the foot of the perpendicular from $B$ to the line $OA$. It is given that $AP=BQ$.
   2. Write down the lengths $AP$ and $BQ$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

      [2]

      Hence show that $\left| \mathbf{a} \right|=\left| \mathbf{b} \right|$.

      [1]

   3. Given also that the angle between $\mathbf{a}$ and $\mathbf{a}-\mathbf{b}$ is $\phi $ radians, show that $\mathbf{a}\cdot \mathbf{b}=-{{\left| \mathbf{a} \right|}^{2}}\cos \,2\phi $.

      [2]

2. The pyramid $ORST$ has a triangular base $ORS$ and height $OT$. The position vectors of $R$ and $S$ are $-4\mathbf{i}+\mathbf{j}+3\mathbf{k}$ and $5\mathbf{i}+\mathbf{j}$ respectively. Find the possible coordinates of $T$ if the volume of the pyramid $ORST$ is $35$ units$^{3}$.

   [5]

   [Volume of pyramid $=\frac{1}{3}\times $base area$\times $height]

1. Sketch the curve with equation \(y=\left| \frac{\alpha x}{x+1} \right|\), where \(\alpha \) is a positive constant, stating the equations of the asymptotes. On the same diagram, sketch the line with equation \(y=\alpha x-2\).[3]
2. Solve the inequality \(\left| \frac{\alpha x}{x+1} \right|\ge \alpha x-2\), giving your answers in term of \(\alpha \).[3]

A curve $C$ has parametric equations

$x=2{{e}^{t}}$, $y={{t}^{3}}-t$,

where $-1\le t\le \frac{3}{2}$.

1. Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$. Hence find the exact equations of the normals to the curve which are parallel to the $y$-axis.

   [3]

2. Sketch the curve $C$.

   [1]

3. Find the equation of the tangent to $C$ at the point$\left( 2{{e}^{a}},{{a}^{3}}-a \right)$. Find the possible exact values of $a$ such that the tangent cuts the $y$-axis at $y=1$.

   [4]

The curve $C$ has equation$y=k{{x}^{3}}$ . The tangent at the point $P$ on $C$ meets the curve again at point $Q$. The tangent at point $Q$ meets the curve again at point $R$. If the $x$ coordinates of $P$, $Q$ and $R$ are $p$ , $q$, and $r$ respectively where $p\ne 0$ .

1. Show that $p$ and $q$ satisfy the equation ${{\left( \frac{q}{p} \right)}^{2}}+\left( \frac{q}{p} \right)-2=0$ .

   [4]

2. Show that $p$ , $q$, and $r$ are three consecutive terms of a geometric progression. Hence determine if this geometric series is convergent.

   [4]

[You may use the identity ${{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)$ for $a,b\in \mathbb{R}$.]