Binomial expansionAQA A-Level Maths: Topic test

1. 1
   The binomial expansion of $(1+2x)^8$ is written in ascending powers of $x$.

   (a)

   What is the coefficient of $x^3$?

   [1 mark]

   • A$448$
   • B$56$
   • C$168$
   • D$1792$

   (b)

   What is the coefficient of $x^2$?

   [1 mark]

   • A$28$
   • B$56$
   • C$112$
   • D$224$

   (c)

   Write down the first three terms of the expansion.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A biased coin lands heads with probability $0.3$. It is tossed 6 times and the tosses are independent.

   (a)

   What is the probability that the coin lands heads exactly twice?

   [1 mark]

   • A$0.0216$
   • B$0.324$
   • C$0.662$
   • D$0.185$

   (b)

   What is the probability that the coin never lands heads?

   [1 mark]

   • A$0.0007$
   • B$0.3$
   • C$0.882$
   • D$0.118$

   (c)

   Find the probability that the coin lands heads exactly four times. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The binomial expansion of $(2+ax)^6$, where $a$ is a non-zero constant, is written in ascending powers of $x$.

   (a)

   Find the first three terms of the expansion, giving each coefficient in terms of $a$.

   [3 marks]

   (b)

   The coefficient of $x^2$ is equal to the coefficient of $x$. Find the value of $a$ and hence find the coefficient of $x^3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A random sample of 8 components is taken from a large batch. Each component is independently faulty with probability $0.2$. The number of faulty components in the sample is $X$.

   (a)

   Find $P(X=2)$ and $P(X\ge2)$, giving each answer to 3 significant figures.

   [6 marks]

   (b)

   Show that $P(X=3)=\dfrac{1}{2}P(X=2)$. Hence find $P(X=3)$ to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In the expansion of $\left(1-\dfrac{x}{2}\right)^{10}$ in ascending powers of $x$, the coefficients of $x$, $x^2$ and $x^3$ are considered.

   (a)

   What is the coefficient of $x$?

   [1 mark]

   • A$5$
   • B$-10$
   • C$10$
   • D$-5$

   (b)

   What is the coefficient of $x^2$?

   [1 mark]

   • A$45$
   • B$\dfrac{45}{4}$
   • C$-\dfrac{45}{4}$
   • D$\dfrac{45}{2}$

   (c)

   Find the coefficient of $x^3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A fair six-sided die is rolled 5 times. The number of sixes is counted.

   (a)

   What is the probability of exactly two sixes?

   [1 mark]

   • A$0.0161$
   • B$0.0804$
   • C$0.161$
   • D$0.402$

   (b)

   What is the probability of at least one six?

   [1 mark]

   • A$0.598$
   • B$0.402$
   • C$0.833$
   • D$0.9999$

   (c)

   Find the probability of exactly four sixes. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   In the expansion of $(2+x)^n$, where $n$ is a positive integer, the coefficient of $x$ is $80$.

   (a)

   Show that $n\times2^{n-1}=80$ and hence find the value of $n$.

   [3 marks]

   (b)

   Find the first three terms of the expansion in ascending powers of $x$ and use them to estimate $2.01^5$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The expression $(1+2x)(3-x)^5$ is expanded in ascending powers of $x$.

   (a)

   Find the first three terms of the expansion of $(3-x)^5$ and hence find the coefficient of $x^2$ in the expansion of $(1+2x)(3-x)^5$.

   [6 marks]

   (b)

   Find the coefficient of $x^3$ in the expansion of $(1+2x)(3-x)^5$. Find the value of $a$ for which the coefficient of $x^3$ in the expansion of $(1+ax)(3-x)^5$ is zero.

   [6 marks]

   Total for question 8: 12 marks

End of questions