2011 A-level H2 Mathematics (9740) Paper 1 Question 4 Suggested Solutions

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

cos ^6 x

= (cos x)^6

\approx (1 -\frac{x^2}{2} + \frac{x^4}{24})^6

= 1 + {6 \choose 1}(1)(-\frac{x^2}{2} + \frac{x^4}{24}) + {6 \choose 2}(1)(-\frac{x^2}{2} + \frac{x^4}{24})^2 + \ldots

= 1 + 6(-\frac{x^2}{2} + \frac{x^4}{24}) + 15(-\frac{x^2}{2} + \frac{x^4}{24})^2 + \ldots

\approx 1 - 3x^2 + 4x^4


\int_0^a g(x) dx

= \int_0^a 1 - 3x^2 + 4x^4 dx

= x - x^3 + \frac{4x^5}{5} \biggl|_0^a

= a - a^3 + \frac{4a^5}{5}

When a = \frac{\pi}{4}, ~ \int_0^a g(x) dx \approx 0.540

Using the Graphing Calculator, \int_0^{\frac{\pi}{4}} g(x) dx = 0.475

The approximation is not very good since \frac{\pi}{4} is not very close to zero.
We only considered up to x^4, thus the approximation can be improved by taking into consideration more terms.

KS Comments:

Just don’t be careless for (i)! Some students still can be unsure of how to evaluate a definite integral using Graphing Calculator. Lastly, maclaurin’s series expansion are most accurate when value is close to one since they are centred at x = 0.

    pingbacks / trackbacks

    Leave a Comment

    16 + twelve =

    Contact Us

    CONTACT US We would love to hear from you. Contact us, or simply hit our personal page for more contact information

    Not readable? Change text. captcha txt

    Start typing and press Enter to search