Statistics related articles

JC Mathematics

Here is a compilation of all the Statistics articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

1. What does unbiased in Math mean?
2. What is r-value?
3. Why is the r-value independent of translation and scaling? #1
4. Why is the r-value independent of translation and scaling? #2
5. Why can’t we extrapolate regression lines?
6. What is a regression line?
7. Simulation for Hypothesis Testing
8. Type I vs Type II Errors
9. A little about Central Limit Theorem
10. What is a good estimate?
11. Using symmetrical properties of Normal Curve to solve questions
12. Bayesian vs Frequentist Statistics

Integration related articles

JC Mathematics

Here is a compilation of all the Integration articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

1. Integrating Trigonometric Functions (1)
2. Integrating Trigonometric Functions (2)
3. Integrating Trigonometric Functions (3)
4. Integrating Trigonometric Functions (4)
5. Integrating Trigonometric Functions (5)
6. Leibniz’s Formula for \pi
7. Simpson’s Rule
8. Evaluating Integrals with Modulus

Trigonometric related articles

JC Mathematics

Here is a compilation of all the trigonometry articles KS has done. Students should read them when they are free to improve their mathematics skills. They will come in handy! 🙂

1. How to derive the sum to produce formula
2. Integrating Trigonometric Functions (1)
3. Integrating Trigonometric Functions (2)
4. Integrating Trigonometric Functions (3)
5. Integrating Trigonometric Functions (4)
6. Integrating Trigonometric Functions (5)
7. Easiest way to remember cosec, sec, and cot
8.

2010 A-level H2 Mathematics (9740) Paper 1 Question 10 Suggested Solutions

JC Mathematics

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

Direction vector of l \text{~is~} \begin{pmatrix}{-3}\\6\\9\end{pmatrix} = -3 \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}

Normal vector of p \text{~is~} \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}

Since l is parallel to the normal vector of p, l is perpendicular to p.

(ii)
l: r = \begin{pmatrix}10\\{-1}\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}

[\begin{pmatrix}10\\{-1}\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix}] \bullet \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix} = 0

10 + \lambda + 2 4 \lambda + 9 + 9 \lambda = 0

\lambda = - \frac{3}{2}

Point of intersection = (\frac{17}{2}, 2, \frac{3}{2}).

(iii)
\begin{pmatrix}10\\{-1}\\{-3}\end{pmatrix} + \lambda \begin{pmatrix}1\\{-2}\\{-3}\end{pmatrix} = \begin{pmatrix}{-2}\\23\\33\end{pmatrix}

10 + \lambda = - 2 — (1)

-1 - 2 \lambda = 23 — (2)

-3 - 3 \lambda = 33 — (3)

Since \lambda = -12 satisfies all 3 equations, A lies on l.

Since = (\frac{17}{2}, 2, \frac{3}{2}) is the midpoint of A & B, by ratio theorem,

\begin{pmatrix}{\frac{17}{2}}\\2\\{\frac{3}{2}}\end{pmatrix} = \frac{\vec{OA} + \vec{OB}}{2}

\vec{OB} = 2 \begin{pmatrix}{\frac{17}{2}}\\2\\{\frac{3}{2}}\end{pmatrix} - \begin{pmatrix}{-2}\\23\\33\end{pmatrix} = \begin{pmatrix}19\\{-19}\\{-30}\end{pmatrix}

B(19, -19, -30)

(iv)
Area

= \frac{1}{2} |\vec{OA} \times \vec{OB}|

= \frac{1}{2} |\begin{pmatrix}{-2}\\23\\33\end{pmatrix} \times \begin{pmatrix}19\\{-19}\\{-30}\end{pmatrix}|

= \frac{1}{2} |\begin{pmatrix}{-63}\\{567}\\{-399}\end{pmatrix}|

= \frac{1}{2} \sqrt{63^2 +567^2 + 399^2}

= 348 to the nearest whole number.

KS Comments:

Students must give answers in coordinates, rather than as a position vector.