Anderson Serangoon Junior College H2 Math papers, with video solutions

1. 1. Find \(\frac{\mathrm{d}}{\mathrm{d}x}\left( \frac{{{x}^{2}}}{1-{{x}^{2}}} \right)\).[1]
   2. Hence find \(\displaystyle\int \frac{2x\ln x}{{{\left( 1-{{x}^{2}} \right)}^{2}}}\,\mathrm{d}x\).[3]
2. Using the substitution \(x=\sin \theta \), where \( 0 < \theta < \frac{\pi }{2} \), show that
   \(8\displaystyle\int{{{x}^{2}}\sqrt{1-{{x}^{2}}}\,\mathrm{d}x={{\sin }^{-1}}x-\mathrm{p}(x)\sqrt{1-{{x}^{2}}}+C}\),
   where \(\mathrm{p}(x)\) is a cubic polynomial to be determined.[5]

1. Find $\displaystyle\int{\left( {{\cot }^{6}}2x+{{\cot }^{4}}2x-\sin \frac{x}{2}\sin \frac{3x}{2} \right)}\mathrm{ d}x$.

   [3]

2. The diagram shows the graphs of $y={{\mathrm{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.
   $R$ is the finite region bounded by the two curves $y={{\mathrm{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.
   Find the volume of the solid formed when $R$ is rotated through $2\pi $ radians about the $y$-axis, giving your answer correct to $4$ decimal places.

   [3]

   
3. A curve $C$ has parametric equations
   $x=2\left( \sin \frac{t}{2}-\cos \frac{t}{2} \right)$, $y=\sin \frac{t}{2}+\cos \frac{t}{2}$, for $-\frac{3\pi }{2}\le t\le -\frac{\pi }{2}$.
   The line $L$ with equation $y=-2x-5$ meets the curve $C$ at the point $\left( -2,\,-1 \right)$. $S$ is the region enclosed by line $L$, curve $C$and the $x$-axis as shown in the diagram below.
   
   1. Show that the x-intercept of curve $C$ is $-2\sqrt{2}$.

      [2]

   2. Find the exact area of $S$.

      [6]

For positive integer $n$, a complex number $z$ is such that ${{\left| z \right|}^{n}}=\frac{1}{\sqrt{2}}$. By considering the conjugate of $1+2{{z}^{2n}}$, show that $\frac{2{{z}^{n}}}{1+2{{z}^{2n}}}$ is a real number.

[4]

The diagram above shows an object with $O$ at the centre of its rectangular base $ABCD$ where $AB=8$cm and $BC=4$cm. The top side of the object, $EFGH$ is a square with side $2$cm long and is parallel to the base. The centre of the top side is vertically above $O$ at a height $h$ cm.

1. Show that the equation of the line $BG$ may be expressed as $r=\left( \begin{matrix}4 \\-2 \\ 0 \\\end{matrix} \right)+t\left( \begin{matrix}-3 \\ 1 \\ h \\\end{matrix} \right)$, where $t$ is a parameter.

   [1]

2. Find the sine of the angle between the line $BG$ and the rectangular base $ABCD$ in terms of $h$.

   [2]

It is given that $h=6$.

3. Find the cartesian equation of the plane $BCFG$.

   [3]

4. Find the shortest distance from the point $A$ to the plane $BCFG$.

   [2]

5. The line $l$, which passes through the point $A$, is parallel to the normal of plane $BCFG$. Given that, the line $l$ intersects the plane $BCFG$ at a point $M$, use your answer in part (iv) to find the shortest distance from point $M$ to the rectangular base $ABCD$.

   [2]

Birth weight can be used to predict short and long-term health complications for babies. Studies show that the birth weight of babies born to mothers who do not smoke in a certain hospital can be assumed to follow a normal distribution with mean $3.05$ kg and variance ${{\sigma }^{2}}$kg$^{2}$.

The hospital classifies babies based on their birth weight as shown as in the table below.

1. A sample showed that $20.2%$ of the babies born to mothers who do not smoke have low birth weight. If this is true for the entire population, find two possible values of $\sigma $, corrected to $2$ decimal places. Explain clearly why one of the values of $\sigma $ found should be rejected.

   [4]