Coefficient zero after multiplying binomial - Binomial Theorem (positive integer n)

6 marks Standard +0.3

3

Give the complete expansion of $\left( x + \frac { 2 } { x } \right) ^ { 5 }$.
In the expansion of $\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }$, the coefficient of $x$ is zero and the coefficient of $\frac { 1 } { x }$ is 80 . Find the values of the constants $a$ and $b$.

5 marks Moderate -0.3

4

Find the first 3 terms in the expansion of $( 2 - x ) ^ { 6 }$ in ascending powers of $x$.
Find the value of $k$ for which there is no term in $x ^ { 2 }$ in the expansion of $( 1 + k x ) ( 2 - x ) ^ { 6 }$.

5 marks Moderate -0.3

3

Find the first 3 terms in the expansion of $( 2 + 3 x ) ^ { 5 }$ in ascending powers of $x$.
Hence find the value of the constant $a$ for which there is no term in $x ^ { 2 }$ in the expansion of $( 1 + a x ) ( 2 + 3 x ) ^ { 5 }$.

5 marks Moderate -0.3

2

In the expression $( 1 - p x ) ^ { 6 } , p$ is a non-zero constant. Find the first three terms when $( 1 - p x ) ^ { 6 }$ is expanded in ascending powers of $x$.
It is given that the coefficient of $x ^ { 2 }$ in the expansion of $( 1 - x ) ( 1 - p x ) ^ { 6 }$ is zero. Find the value of $p$.

4 marks Moderate -0.8

1

Find the coefficients of $x ^ { 4 }$ and $x ^ { 5 }$ in the expansion of $( 1 - 2 x ) ^ { 5 }$.
It is given that, when $( 1 + p x ) ( 1 - 2 x ) ^ { 5 }$ is expanded, there is no term in $x ^ { 5 }$. Find the value of the constant $p$.

5 marks Moderate -0.8

1

Find the first three terms when $( 2 + 3 x ) ^ { 6 }$ is expanded in ascending powers of $x$.
In the expansion of $( 1 + a x ) ( 2 + 3 x ) ^ { 6 }$, the coefficient of $x ^ { 2 }$ is zero. Find the value of $a$.

7 marks Standard +0.3

11. $\mathrm { f } ( x ) = ( a - x ) ( 3 + a x ) ^ { 5 }$, where $a$ is a positive constant

Find the first 3 terms, in ascending powers of $x$, in the binomial expansion of $$( 3 + a x ) ^ { 5 }$$ Give each term in its simplest form. Given that in the expansion of $\mathrm { f } ( x )$ the coefficient of $x$ is zero,
find the exact value of $a$.

7 marks Moderate -0.8

Find the first 3 terms in the expansion of $(1 + ax)^4$ in ascending powers of $x$. [2]
Given that there is no term in $x$ in the expansion of $(1 - 2x)(1 + ax)^5$, find the value of the constant $a$. [2]
For this value of $a$, find the coefficient of $x^2$ in the expansion of $(1 - 2x)(1 + ax)^5$. [3]

5 marks Moderate -0.8

Find the terms in $x^2$ and $x^3$ in the expansion of $\left(1 - \frac{3}{2}x\right)^6$. [3]
Given that there is no term in $x^3$ in the expansion of $(k + 2x)\left(1 - \frac{3}{2}x\right)^6$, find the value of the constant $k$. [2]

6 marks Moderate -0.3

Find and simplify the first three terms in the expansion, in ascending powers of $x$, of $\left(2 + \frac{1}{3}kx\right)^6$, where $k$ is a constant. [3]
In the expansion of $(3 - 4x)\left(2 + \frac{1}{3}kx\right)^6$, the constant term is equal to the coefficient of $x^2$. Determine the exact value of $k$, given that $k$ is positive. [3]