 # Exam-Style Questions on Binomial Theorem

## Problems 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 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 $$x^6$$.

### 4.

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 $$j$$.

### 5.

IB Standard

The constant term in the expansion of $$x^4(2x^2+\frac{m}{x})^7$$ is 896

Find $$m$$.

### 6.

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.

### 7.

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 $$b \lt c$$.

### 8.

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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