## 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 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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