|
|
Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Analysis and Approaches |
(a) Write down the first three terms of the binomial expansion of \( (1 - a)^{-1} \) in ascending powers of \( a \).
(b) By using the Maclaurin series for \( \dfrac{\sin x}{x} \) and the result from part (a), find the Maclaurin series for \( x\text{cosec} x \) up to and including the term in \( x^4 \).
(c) By using the Maclaurin series for \( \arctan x \) and the result from part (b) find:
$$ \lim_{{x \to 0}} \left( \frac{x \arctan 2x}{x\text{cosec}x - 1} \right) $$
2. | IB Analysis and Approaches |
(a) Find the first two non-zero terms in the Maclaurin series of
(i) \( \cos(x^2) \);
(ii) \( \cos^2(x^2) \).
(b) Hence, or otherwise, find the first two non-zero terms in the Maclaurin series of
$$6x \cos(x^2) \sin(x^2)$$
3. | IB Analysis and Approaches |
The function \(f\) is defined by \(f(x) = e^x \sin{x} \), where \(x \in \mathbb{R}\).
(a) Find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term.
(b) Hence, find an approximate value for \( \int_0^1 e^{x^2}\sin(x^2)dx\).
The function \(g\) is defined by \(g(x) = e^x\cos{x}\), where \(x \in \mathbb{R}\).
(c) Show that \(g(x)\) satisfies the equation \(g''(x) = 2(g'(x) — g(x)) \).
(d) Show that \(g(x)\) also satisfies the equation \(g^{(4)}(x) = 2(g'''(x) — g''(x)) \).
(e) Using these results, find the Maclaurin series for \(g(x)\) up to and including the \(x^4 \) term.
(f) Hence determine the value of \( \lim \limits_{x \to 0} \frac{e^x \cos{x} -1-x}{x^3} \)
If you would like space on the right of the question to write out the solution try this Thinning Feature. It will collapse the text into the left half of your screen but large diagrams will remain unchanged.
The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
The solutions to the questions on this website are only available to those who have a Transum Subscription.
Exam-Style Questions Main Page
To search the entire Transum website use the search box in the grey area below.
Do you have any comments about these exam-style questions? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.
