Questions P1 (1401 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
CAIE P1 2024 November Q4
6 marks Moderate -0.8
Find the term independent of \(x\) in the expansion of each of the following:
  1. \(\left(x + \frac{3}{x^2}\right)^6\) [2]
  2. \((4x^3 - 5)\left(x + \frac{3}{x^2}\right)^6\) [4]
CAIE P1 2024 November Q5
10 marks Moderate -0.3
The function f is defined by \(\mathrm{f}(x) = \frac{2x + 1}{2x - 1}\) for \(x < \frac{1}{2}\).
    1. State the value of f\((-1)\). [1]
    2. \includegraphics{figure_5} The diagram shows the graph of \(y = \mathrm{f}(x)\). Sketch the graph of \(y = \mathrm{f}^{-1}(x)\) on this diagram. Show any relevant mirror line. [2]
    3. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of the function \(\mathrm{f}^{-1}\). [4]
The function g is defined by \(\mathrm{g}(x) = 3x + 2\) for \(x \in \mathbb{R}\).
  1. Solve the equation \(\mathrm{f}(x) = \mathrm{gf}\left(\frac{1}{4}\right)\). [3]
CAIE P1 2024 November Q6
6 marks Standard +0.8
\includegraphics{figure_6} The diagram shows a metal plate \(OABCDEF\) consisting of sectors of two circles, each with centre \(O\). The radii of sectors \(AOB\) and \(EOF\) are \(r\) cm and the radius of sector \(COD\) is \(2r\) cm. Angle \(AOB =\) angle \(EOF = \theta\) radians and angle \(COD = 2\theta\) radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is 10 cm\(^2\). Given that \(r \geqslant \frac{3}{2}\) and \(\theta < \frac{3}{4}\), find the values of \(r\) and \(\theta\). [6]
CAIE P1 2024 November Q7
8 marks Standard +0.3
  1. By expressing \(-2x^2 + 8x + 11\) in the form \(-a(x - h)^2 + c\), where \(a\), \(b\) and \(c\) are positive integers, find the coordinates of the vertex of the graph with equation \(y = -2x^2 + 8x + 11\). [3]
  2. \includegraphics{figure_7} The diagram shows part of the curve with equation \(y = -2x^2 + 8x + 11\) and the line with equation \(y = 8x + 9\). Find the area of the shaded region. [5]
CAIE P1 2024 November Q8
10 marks Moderate -0.3
The equation of a circle is \(x^2 + y^2 + px + 2y + q = 0\), where \(p\) and \(q\) are constants.
  1. Express the equation in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\) is to be given in terms of \(p\) and \(r^2\) is to be given in terms of \(p\) and \(q\). [2]
The line with equation \(x + 2y = 10\) is the tangent to the circle at the point \(A(4, 3)\).
    1. Find the equation of the normal to the circle at the point \(A\). [3]
    2. Find the values of \(p\) and \(q\). [5]
CAIE P1 2024 November Q9
10 marks Standard +0.3
The equation of a curve is \(y = \frac{1}{3}k^2x^2 - 2kx + 2\) and the equation of a line is \(y = kx + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).
  1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left(\frac{6}{5}, \frac{3}{5}\right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection. [7]
  2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\). [3]
CAIE P1 2024 November Q10
10 marks Standard +0.3
A function f with domain \(x > 0\) is such that \(\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}\). It is given that the curve with equation \(y = \mathrm{f}(x)\) passes through the point \((1, 0)\).
  1. Find the equation of the normal to the curve at the point \((1, 0)\). [3]
  2. Find f\((x)\). [4]
It is given that the equation \(\mathrm{f}'(x) = 0\) can be expressed in the form $$125x^2 - 128x + 192 = 0.$$
  1. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]
CAIE P1 2024 November Q1
3 marks Easy -1.2
An arithmetic progression has fourth term 15 and eighth term 25. Find the 30th term of the progression. [3]
CAIE P1 2024 November Q2
2 marks Challenging +1.2
Find the exact solution of the equation $$\cos\frac{x}{6} + \tan 2x + \frac{\sqrt{3}}{2} = 0 \text{ for } -\frac{1}{4}\pi < x < \frac{1}{4}\pi.$$ [2]
CAIE P1 2024 November Q3
6 marks Moderate -0.8
  1. Find the coefficients of \(x^3\) and \(x^4\) in the expansion of \((3 - ax)^5\), where \(a\) is a constant. Give your answers in terms of \(a\). [3]
  2. Given that the coefficient of \(x^4\) in the expansion of \((ax + 7)(3 - ax)^5\) is 240, find the positive value of \(a\). [3]
CAIE P1 2024 November Q4
4 marks Moderate -0.3
Solve the equation \(4\sin^4\theta + 12\sin^2\theta - 7 = 0\) for \(0° \leqslant \theta \leqslant 360°\). [4]
CAIE P1 2024 November Q5
8 marks Standard +0.3
\includegraphics{figure_5} In the diagram, the graph with equation \(y = \text{f}(x)\) is shown with solid lines and the graph with equation \(y = \text{g}(x)\) is shown with broken lines.
  1. Describe fully a sequence of three transformations which transforms the graph of \(y = \text{f}(x)\) to the graph of \(y = \text{g}(x)\). [6]
  2. Find an expression for g(x) in the form \(af(bx + c)\), where \(a\), \(b\) and \(c\) are integers. [2]
CAIE P1 2024 November Q6
5 marks Standard +0.3
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 \(\frac{q}{p} = \frac{17}{16}\). Find the two possible values of the sum to infinity. [5]
CAIE P1 2024 November Q7
10 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a metal plate \(ABCDEF\) consisting of five parts. The parts \(BCD\) and \(DEF\) are semicircles. The part \(BAFO\) is a sector of a circle with centre \(O\) and radius 20 cm, and \(D\) lies on this circle. The parts \(OBD\) and \(ODF\) are triangles. Angles \(BOD\) and \(DOF\) are both \(\theta\) radians.
  1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures. [5]
  2. Given instead that the area of each semicircle is \(50\pi \text{ cm}^2\), find the exact perimeter of the metal plate. [5]
CAIE P1 2024 November Q8
9 marks Moderate -0.3
  1. Express \(3x^2 - 12x + 14\) in the form \(3(x + a)^2 + b\), where \(a\) and \(b\) are constants to be found. [2]
The function f(x) = \(3x^2 - 12x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.
  1. Find the least value of \(k\) for which the function \(\text{f}^{-1}\) exists. [1]
For the rest of this question, you should assume that \(k\) has the value found in part (b).
  1. Find an expression for \(\text{f}^{-1}(x)\). [3]
  2. Hence or otherwise solve the equation \(\text{f f}(x) = 29\). [3]
CAIE P1 2024 November Q9
7 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).
  1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
  2. Find the area of the shaded region. [4]
CAIE P1 2024 November Q10
9 marks Standard +0.3
Points \(A\) and \(B\) have coordinates \((4, 3)\) and \((8, -5)\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).
  1. Show that the centre of the circle lies on the line \(y = \frac{1}{2}x - 4\). [4]
  2. Find the two possible equations of the circle. [5]
CAIE P1 2024 November Q11
12 marks Standard +0.3
The equation of a curve is \(y = kx^{\frac{1}{2}} - 4x^2 + 2\), where \(k\) is a constant.
  1. Find \(\frac{\text{d}y}{\text{d}x}\) and \(\frac{\text{d}^2y}{\text{d}x^2}\) in terms of \(k\). [2]
  2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature. [4]
  3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6. Find this value of \(k\). [6]
CAIE P1 2010 June Q1
4 marks Moderate -0.8
  1. Show that the equation $$3(2\sin x - \cos x) = 2(\sin x - 3\cos x)$$ can be written in the form \(\tan x = -\frac{4}{5}\). [2]
  2. Solve the equation \(3(2\sin x - \cos x) = 2(\sin x - 3\cos x)\), for \(0° \leq x \leq 360°\). [2]
CAIE P1 2010 June Q2
4 marks Moderate -0.3
\includegraphics{figure_2} The diagram shows part of the curve \(y = \frac{a}{x}\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]
CAIE P1 2010 June Q3
5 marks Moderate -0.3
The functions f and g are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto 4x - 2x^2,$$ $$g : x \mapsto 5x + 3.$$
  1. Find the range of f. [2]
  2. Find the value of the constant \(k\) for which the equation \(gf(x) = k\) has equal roots. [3]
CAIE P1 2010 June Q4
6 marks Moderate -0.3
\includegraphics{figure_4} In the diagram, \(A\) is the point \((-1, 3)\) and \(B\) is the point \((3, 1)\). The line \(L_1\) passes through \(A\) and is parallel to \(OB\). The line \(L_2\) passes through \(B\) and is perpendicular to \(AB\). The lines \(L_1\) and \(L_2\) meet at \(C\). Find the coordinates of \(C\). [6]
CAIE P1 2010 June Q5
6 marks Moderate -0.8
Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix}$$
  1. Find the value of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [2]
  2. Find the values of \(p\) for which the magnitude of \(\overrightarrow{AB}\) is 7. [4]
CAIE P1 2010 June Q6
7 marks Moderate -0.8
  1. Find the first 3 terms in the expansion of \((1 + ax)^4\) in ascending powers of \(x\). [2]
  2. Given that there is no term in \(x\) in the expansion of \((1 - 2x)(1 + ax)^5\), find the value of the constant \(a\). [2]
  3. For this value of \(a\), find the coefficient of \(x^2\) in the expansion of \((1 - 2x)(1 + ax)^5\). [3]
CAIE P1 2010 June Q7
8 marks Moderate -0.8
  1. Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]
  2. A geometric progression has a common ratio of \(-\frac{2}{3}\) and the sum of the first 3 terms is 35. Find
    1. the first term of the progression, [3]
    2. the sum to infinity. [2]