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

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

   (a)

   Find the coefficient of $x^2$.

   [1 mark]

   • A$240$
   • B$60$
   • C$15$
   • D$192$

   (b)

   Find the constant term.

   [1 mark]

   • A$1$
   • B$12$
   • C$32$
   • D$64$

   (c)

   Hence find the coefficient of $x^2$ in the expansion of $(1+x)(2+x)^6$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In the expansion of $(1+3x)^n$, where $n$ is a positive integer, the term in $x$ is $24x$.

   (a)

   Find the value of $n$.

   [1 mark]

   • A$24$
   • B$21$
   • C$8$
   • D$72$

   (b)

   Find the coefficient of $x^2$.

   [1 mark]

   • A$28$
   • B$252$
   • C$84$
   • D$756$

   (c)

   Use the first three terms of the expansion to estimate the value of $1.03^8$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the expansion of $(3-2x)^7$ in ascending powers of $x$.

   (a)

   Find the first three terms in the expansion, in ascending powers of $x$.

   [3 marks]

   (b)

   Find the coefficient of $x^3$ in the expansion of $(2+x)(3-2x)^7$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A quiz has 5 multiple-choice questions. A student guesses every answer, and each guess is correct independently with probability $\frac14$. Let $X$ be the number of correct answers.

   (a)

   (i) Show that $P(X=2)=\frac{135}{512}$.
   (ii) Find the exact value of $P(X\ge4)$.

   [6 marks]

   (b)

   (i) Explain, by referring to the expansion of $\left(\frac34+\frac14\right)^5$, why the probabilities $P(X=0),\ldots,P(X=5)$ add up to $1$.
   (ii) To pass the quiz a student needs at least $3$ correct answers. Find the probability that a guessing student passes.
   (iii) A student claims that guessing gives a probability of passing greater than $10\%$. Is this claim correct? Justify your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions