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Exam-Style Questions on Binomial TheoremProblems on Binomial Theorem adapted from questions set in previous Mathematics exams. |
1. | IB Standard |
If \((x+5)^{10}\) is expanded
(a) how many terms would there be?
(b) what is the coefficient of the term containing \(x^4\)?
2. | IB Standard |
If \((2x+7)^{6}\) is expanded
(a) how many terms would there be?
(b) what is the coefficient of the term containing \(x^4\)?
3. | IB Analysis and Approaches |
Consider the expansion of:
$$ (3x^4+\frac{1}{5x})^n $$where \( n \in \mathbb{Z}^+\)
Determine all possible values of \(n\) for which the expansion has a non-zero constant term.
4. | IB Standard |
If you expanded \((2x-3)^{15}\), the term containing \(x^6\) can be written as \(\binom{15}{a}\times(2x)^b\times(-3)^c\)
(a) Write down the values of \(a\), of \(b\) and \(c\).
(b) Find the coefficient of the term containing \(x^6\).
5. | IB Analysis and Approaches |
In the expansion of \( (x+j)^{9}\) where \(j \in \mathbb{R}\), the coefficient of the term in \(x^7\) is 144.
Find the possible values of \(j\).
6. | IB Analysis and Approaches |
The expansion of \((x + g)^7\), where \(g \in \mathbb{Q}^+\), can be written as:
$$x^7 + ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + g^7$$where \(a, b, c, d, e, f \in \mathbb{R}\).
Given that the coefficients, \( a, c \text{ and } d\), are the first three terms of a geometric sequence, find the value of \(g\).
7. | IB Standard |
The constant term in the expansion of \(x^4(2x^2+\frac{m}{x})^7\) is 896
Find \(m\).
8. | IB Standard |
Consider the expansion of \( (3x+ \frac{c}{x})^8\) where \( c \gt 0 \).
The coefficient of the term in \(x^4\) is equal to the coefficient of the term in \(x^6\).
Find c.
9. | A-Level |
(a) Find the binomial expansion of \( (1-6x)^{\frac34} \) up to and including the term in \(x^2\).
(b) Find the binomial expansion of \( (16-6x)^{\frac34} \) up to and including the term in \(x^2\).
(c) Use your expansion from part (b) to find an estimate for \( 19^{\frac34} \) giving your answer in the form \(a + \frac{b}{c} \) where a, b and c are positive integers with \( b \lt c \).
10. | IB Analysis and Approaches |
Consider the expansion of \( (7-x^2)^{n-1}\) where \(n \in \mathbb{Z}^+\).
Given that the coefficient of \(x^6\) is -9882516, find the value of \(n\).
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