Pure Mathematics 1 (for Paper 1): 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 1Oct/Nov 2025, paper 11Structured4 marks

  Find the set of values of the constant $k$ for which the quadratic equation $$3k{x}^2+(k+8)x+3=0$$ has two distinct real roots.

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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 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 2Oct/Nov 2024, paper 11Structured4 marks

  The curve $y=x^2-\frac{a}{x}$ has a stationary point at $(-3,b)$. Find the values of the constants $a$ and $b$.

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• Question 2Oct/Nov 2023, paper 11Structured4 marks

  A line has equation $y=2cx+3$ and a curve has equation $y=c{x}^2+3x-c$, where $c$ is a constant. Showing all necessary working, determine which of the following statements is correct.

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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 3Oct/Nov 2024, paper 11Structured5 marksHas a diagram

  The diagram shows a sector of a circle, centre $O$, where $OB=OC=15\text{ cm}$. The size of angle $BOC$ is $\frac{2}{5}\pi \text{ radians}$. Points $A$ and $D$ on the lines $OB$ and $OC$ respectively are joined by an arc $AD$ of a circle with centre $O$.

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• Question 3Oct/Nov 2023, paper 11Structured3 marksHas a diagram

  The diagram shows a cubical closed container made of a thin elastic material which is filled with water and frozen. During the freezing process the length, $x\text{ cm}$, of each edge of the container increases at the constant rate of $0.01\text{ cm per minute}$.

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• Question 4Oct/Nov 2025, paper 11Structured6 marks

  Express $1-6x-x^2$ in the form $a-(x+b{)}^2$, where $a$ and $b$ are constants.

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• Question 4Oct/Nov 2024, paper 11Structured5 marks

  Show that the curve with equation $x^2-3xy-40=0$ and the line with equation $3x+y+k=0$ meet for all values of the constant $k$.

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• Question 5May/June 2026, paper 11Structured5 marks

  The equation of a curve is $$y=-\frac{1}{2}x+\sqrt{x^3-3x-2}\mathrm{.}$$ Find an equation of the normal to the curve at the point $(6,11)$.

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• Question 5Oct/Nov 2024, paper 11Structured7 marks

  The equation of a curve is such that $\frac{\mathrm{d}y}{\mathrm{d}x}=4x-3\sqrt{x}+1$. Find the $x$-coordinate of the point on the curve at which the gradient is $\frac{11}{2}$.

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• Question 5Oct/Nov 2023, paper 11Structured6 marks

  Show that the equation $$4\sin x+\frac{5}{\tan x}+\frac{2}{\sin x}=0$$ may be expressed in the form $a{\cos }^{2}x+b\cos x+c=0$, where $a$, $b$ and $c$ are integers to be found.

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• Question 6Oct/Nov 2023, paper 11Structured7 marksHas a diagram

  The diagram shows a motif formed by the major arc $AB$ of a circle with radius $r$ and centre $O$, and the minor arc $AOB$ of a circle, also with radius $r$ but with centre $C$. The point $C$ lies on the circle with centre $O$.

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• Question 7Oct/Nov 2025, paper 11Structured8 marksHas a diagram

  The diagram shows a sector of a circle with centre $O$ and radius $r\text{ cm}$. The shaded region is bounded by the chord $AB$ and the arc $AB$. The size of angle $AOB$ is $\frac{2}{3}\pi \text{ radians}$.

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• Question 7Oct/Nov 2024, paper 11Structured8 marks

  The diagram shows part of the curve with equation $y=\frac{12}{\sqrt[3]{2x+1}}$. The point $A$ on the curve has coordinates $(\frac{7}{2},6)$. Find the equation of the tangent to the curve at $A$.

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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 8May/June 2026, paper 11Structured8 marksHas a diagram

  The diagram shows part of the curve with equation $y=\frac{5}{2x-1}$ and the line $y=7-2x$. Find the exact volume formed when the shaded region is rotated through ${360}^{\circ }$ about the $x$-axis.

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• Question 8Oct/Nov 2025, paper 11Structured7 marksHas a diagram

  The diagram shows the curve with equation $y=\frac{1}{2}\sqrt{x}$ and the point $P$ with coordinates $(9,\frac{3}{2})$. The shaded region is bounded by the curve and the lines $x=0$ and $y=\frac{3}{2}$. Find the area of the shaded region.

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• Question 8Oct/Nov 2024, paper 11Structured8 marks

  It is given that $\beta$ is an angle between ${90}^{\circ }$ and ${180}^{\circ }$ such that $\sin \beta =a$. Express ${\tan }^2\beta -3\sin \beta \cos \beta$ in terms of $a$.

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• Question 8Oct/Nov 2023, paper 11Structured9 marksHas a diagram

  The diagram shows the curves with equations $y=2(2x-3{)}^4$ and $y=(2x-3{)}^2+1$ meeting at points $A$ and $B$. By using the substitution $u=2x-3$ find, by calculation, the coordinates of $A$ and $B$.

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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 9Oct/Nov 2024, paper 11Structured8 marks

  The equation of a curve is $y=4+5x+6x^2-3x^3$. Find the set of values of $x$ for which $y$ decreases as $x$ increases.

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• Question 9Oct/Nov 2023, paper 11Structured9 marks

  Express $4x^2-12x+13$ in the form $(2x+a{)}^2+b$, where $a$ and $b$ are constants.

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• Question 10May/June 2026, paper 11Structured11 marksHas a diagram

  The equation of a curve is such that $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=3x^2-6$. The point $P(2,3)$ is a stationary point on the curve. By evaluating $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$ at $P$, determine the nature of this stationary point.

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• Question 10Oct/Nov 2025, paper 11Structured7 marks

  A circle has equation $x^2+y^2+4y-21=0$ and a straight line has equation $2x+y-8=0$. The line intersects the circle at two points. Find the coordinates of these two points of intersection.

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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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