Series: past paper questions

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• Question 1May/June 2026, paper 11Structured4 marks

  In a geometric progression, the third term is 50 and the fifth term is 12.5. Given that the common ratio of the progression is positive, find the sum to infinity of the progression.

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• Question 2Oct/Nov 2025, paper 11Structured5 marks

  A geometric progression has first term $a$ and common ratio $\cos \theta$, where $0<\theta <\frac{1}{2}\pi$. It is given that the second term is 8 and the fifth term is $\frac{1}{8}$. Find the value of $\theta$.

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• Question 1Oct/Nov 2024, paper 11Structured4 marks

  In the expansion of ${(kx+\frac{2}{x})}^4$, where $k$ is a positive constant, the term independent of $x$ is equal to 150. Find the value of $k$ and hence determine the coefficient of $x^2$ in the expansion.

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• Question 1Oct/Nov 2023, paper 11Structured4 marks

  Expand $(1+3x{)}^6$ in ascending powers of $x$ up to, and including, the term in $x^2$.

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• Question 2May/June 2026, paper 11Structured5 marks

  Find the expansion of each of the following expressions, in ascending powers of $x$, up to and including the term in $x^2$.

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• Question 3Oct/Nov 2025, paper 11Structured5 marks

  In the expansion of $$(px+3{)}^5-{(x^3+\frac{p}{x})}^4,$$ the coefficient of $x^4$ is 216. Find the value of the positive constant $p$.

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• Question 10Oct/Nov 2024, paper 11Structured8 marks

  An arithmetic progression has first term 5 and common difference $d$, where $d>0$. The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression. Find the value of $d$.

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• Question 7Oct/Nov 2023, paper 11Structured7 marks

  The sum of the first two terms of a geometric progression is 15 and the sum to infinity is $\frac{125}{7}$. The common ratio of the progression is negative. Find the third term of the progression.

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• Question 9Oct/Nov 2025, paper 11Structured8 marks

  An arithmetic progression has first term 2 and common difference $d$. The sum of the first $n$ terms is denoted by $S_n$. It is given that $(S_2-1)$, $S_4$, $S_9$ are the first three terms of a second arithmetic progression.

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• Question 2Oct/Nov 2024, paper 12Structured5 marks

  The first term of an arithmetic progression is $-20$ and the common difference is 5. Find the sum of the first 20 terms of the progression.

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• Question 1Oct/Nov 2023, paper 12Structured4 marks

  The coefficient of $x^3$ in the expansion of $(3+2ax{)}^5$ is six times the coefficient of $x^2$ in the expansion of $(2+ax{)}^6$. Find the value of the constant $a$.

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• Question 4May/June 2026, paper 12Structured7 marks

  Find the first three terms, in ascending powers of $x$, in the expansion of ${(3-\frac{1}{2}x)}^4$.

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• Question 8Oct/Nov 2025, paper 12Structured11 marks

  The first three terms of a geometric progression are $a$, $b$ and $c$ respectively, where $a$, $b$ and $c$ are positive constants. The first three terms of an arithmetic progression are $a$, $b$ and $-3c$ respectively. Show that $a^2-10ac+9c^2=0$.

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• Question 4Oct/Nov 2024, paper 12Structured6 marks

  Find the term independent of $x$ in the expansion of each of the following: ${(x+\frac{3}{x^2})}^6$

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• Question 5Oct/Nov 2023, paper 12Structured6 marks

  The first, second and third terms of a geometric progression are $\sin \theta$, $\cos \theta$ and $2-\sin \theta$ respectively, where $\theta$ radians is an acute angle. Find the value of $\theta$.

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• Question 7May/June 2026, paper 12Structured10 marks

  The second, fourth and last terms of an arithmetic progression are 60, 52 and $-28$ respectively. Find the sum of all of the terms in the progression.

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• Question 1Oct/Nov 2025, paper 13Structured5 marks

  Expand ${(2-\frac{1}{2}x)}^6$ in ascending powers of $x$ up to and including the term in $x^3$.

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• Question 1Oct/Nov 2024, paper 13Structured3 marks

  An arithmetic progression has fourth term 15 and eighth term 25. Find the 30th term of the progression.

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• Question 4Oct/Nov 2023, paper 13Structured7 marks

  Expand the following in ascending powers of $x$ up to and including the term in $x^2$.

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• Question 5May/June 2026, paper 13Structured8 marks

  Find the first four terms, in ascending powers of $x$, in the expansion of $(2+3x{)}^5$.

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• Question 4Oct/Nov 2025, paper 13Structured6 marks

  The first, second and third terms of a progression are 20, $k$ and $k-5$ respectively. Given that the progression is arithmetic, find the 30th term.

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• Question 3Oct/Nov 2024, paper 13Structured6 marks

  Find the coefficients of $x^3$ and $x^4$ in the expansion of $(3-ax{)}^5$, where $a$ is a constant.

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• Question 10May/June 2026, paper 13Structured7 marks

  The first, second and third terms of a geometric progression are equal to the first, fourth and sixth terms of an arithmetic progression respectively. The first term of both progressions is non-zero. The geometric progression has common ratio $r$.

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• Question 3May/June 2025, paper 11Structured7 marks

  The third term of a geometric progression is 18 and the sum of the first three terms is 26. It is given that the common ratio is negative. Find the tenth term of the progression.

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• Question 6Oct/Nov 2024, paper 13Structured5 marks

  The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is $p$ and the sum of the first 8 terms of the progression is $q$. It is given that $\frac{q}{p}=\frac{17}{16}$.

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• Question 2May/June 2023, paper 11Structured6 marks

  Find the first three terms in the expansion, in ascending powers of $x$, of $(2+3x{)}^4$.

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• Question 5May/June 2025, paper 11Structured7 marks

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

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• Question 3May/June 2024, paper 11Structured5 marks

  The coefficient of $x^3$ in the expansion of $(3+ax{)}^6$ is 160. Find the value of the constant $a$.

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• Question 2May/June 2023, paper 12Structured4 marks

  The coefficient of $x^4$ in the expansion of $(x+a{)}^6$ is $p$ and the coefficient of $x^2$ in the expansion of $(ax+3{)}^4$ is $q$. It is given that $p+q=276$. Find the possible values of the constant $a$.

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• Question 1May/June 2024, paper 12Structured3 marks

  The coefficient of $x^2$ in the expansion of $(1-4x{)}^6$ is 12 times the coefficient of $x^2$ in the expansion of $(2+ax{)}^5$. Find the value of the positive constant $a$.

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