Binomial expansion for positive integer powersEdexcel International A Level Maths: Subtopic test

1. 1
   Consider the binomial expansion of $(2+x)^6$ in ascending powers of $x$.

   (a)

   Find the coefficient of $x^2$.

   [1 mark]

   • A$60$
   • B$240$
   • C$480$
   • D$15$

   (b)

   Which gives the first three terms of the expansion?

   [1 mark]

   • A$1+12x+60x^2$
   • B$64+32x+16x^2$
   • C$64+192x+240x^2$
   • D$64+192x+480x^2$

   (c)

   Find the coefficient of $x^3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For a positive integer $n$ and $0\le r\le n$, the coefficient of $x^r$ in the expansion of $(1+x)^n$ is $\binom{n}{r}=\frac{n!}{r!\,(n-r)!}$.

   (a)

   Find the value of $\binom83$.

   [1 mark]

   • A$56$
   • B$24$
   • C$336$
   • D$28$

   (b)

   Given that the coefficient of $x^2$ in the expansion of $(1+x)^n$ is $45$, find $n$.

   [1 mark]

   • A$9$
   • B$11$
   • C$90$
   • D$10$

   (c)

   Show that the coefficient of $x^3$ in the expansion of $(1+x)^n$ is $\frac{n(n-1)(n-2)}{6}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The coefficient of $x^2$ in the expansion of $(1+kx)^8$ is $252$, where $k>0$.

   (a)

   Find the value of $k$.

   [3 marks]

   (b)

   Find the coefficient of $x^3$ in the expansion of $(2-x)(1+kx)^8$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $f(x)=(3-2x)^4$.

   (a)

   Expand $f(x)$ in ascending powers of $x$, giving each coefficient as an integer.

   [6 marks]

   (b)

   (i) Use the first four terms of your expansion with a suitable value of $x$ to estimate $2.98^4$, giving your answer to 4 decimal places.
   (ii) The coefficient of $x^2$ in the expansion of $(1+px)f(x)$ is $36$. Find the value of $p$.

   [6 marks]

   Total for question 4: 12 marks

End of questions