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

1. 1
   The expression $(1+2x)^5$ is expanded in ascending powers of $x$.

   (a)

   Find the coefficient of $x^2$ in the expansion.

   [1 mark]

   • A$10$
   • B$20$
   • C$40$
   • D$80$

   (b)

   Find the coefficient of $x^3$ in the expansion.

   [1 mark]

   • A$40$
   • B$80$
   • C$160$
   • D$10$

   (c)

   Use the first three terms of the expansion, with a suitable value of $x$, to estimate $1.02^5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The random variable $X$ is the number of heads obtained when a fair coin is tossed 6 times, so $X\sim B\left(6,\frac12\right)$.

   (a)

   Find the value of $\binom62$.

   [1 mark]

   • A$12$
   • B$30$
   • C$36$
   • D$15$

   (b)

   Find $P(X=2)$.

   [1 mark]

   • A$\frac{15}{64}$
   • B$\frac{15}{32}$
   • C$\frac{1}{64}$
   • D$\frac{15}{16}$

   (c)

   Without evaluating either probability, explain why $P(X=4)=P(X=2)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The expression $(2+kx)^5$, where $k$ is a positive constant, is expanded in ascending powers of $x$. The coefficient of $x^2$ in the expansion is $720$.

   (a)

   Find the value of $k$.

   [3 marks]

   (b)

   Hence find the coefficient of $x$ and the coefficient of $x^3$ in the expansion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $f(x)=(1+3x)(2-x)^5$

   (a)

   (i) Find the first three terms of the expansion of $(2-x)^5$ in ascending powers of $x$.
   (ii) Hence find the coefficient of $x^2$ in the expansion of $f(x)$.

   [6 marks]

   (b)

   (i) Find the coefficient of $x^3$ in the expansion of $f(x)$.
   (ii) Use the first four terms of the expansion of $f(x)$, with a suitable value of $x$, to estimate $1.03\times1.99^5$, giving your answer to 4 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions