Questions — CAIE P1 (1228 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 Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a sector of a circle, centre \(O\), where \(OB = OC = 15\) cm. The size of angle \(BOC\) is \(\frac{2}{5}\pi\) radians. Points \(A\) and \(D\) on the lines \(OB\) and \(OC\) respectively are joined by an arc \(AD\) of a circle with centre \(O\). The shaded region is bounded by the arcs \(AD\) and \(BC\) and by the straight lines \(AB\) and \(DC\). It is given that the area of the shaded region is \(\frac{90}{7}\pi\) cm\(^2\). Find the perimeter of the shaded region. Give your answer in terms of \(\pi\). [5]
CAIE P1 2024 November Q4
5 marks Standard +0.3
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 \(k\). [5]
CAIE P1 2024 November Q5
7 marks Moderate -0.8
The equation of a curve is such that \(\frac{dy}{dx} = 4x - 3\sqrt{x} + 1\).
  1. Find the \(x\)-coordinate of the point on the curve at which the gradient is \(\frac{11}{2}\). [3]
  2. Given that the curve passes through the point \((4, 11)\), find the equation of the curve. [4]
CAIE P1 2024 November Q6
7 marks Moderate -0.3
Circles \(C_1\) and \(C_2\) have equations $$x^2 + y^2 + 6x - 10y + 18 = 0 \text{ and } (x-9)^2 + (y+4)^2 - 64 = 0$$ respectively.
  1. Find the distance between the centres of the circles. [4] \(P\) and \(Q\) are points on \(C_1\) and \(C_2\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
  2. Find the greatest and least possible values of \(d\). [3]
CAIE P1 2024 November Q7
8 marks Moderate -0.3
\includegraphics{figure_7} The diagram shows part of the curve with equation \(y = \frac{12}{\sqrt{2x+1}}\). The point \(A\) on the curve has coordinates \(\left(\frac{7}{2}, 6\right)\).
  1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = mx + c\). [4]
  2. Find the area of the region bounded by the curve and the lines \(x = 0\), \(x = \frac{7}{2}\) and \(y = 0\). [4]
CAIE P1 2024 November Q8
8 marks Moderate -0.3
  1. It is given that \(\beta\) is an angle between \(90°\) and \(180°\) such that \(\sin \beta = a\). Express \(\tan^2 \beta - 3 \sin \beta \cos \beta\) in terms of \(a\). [3]
  2. Solve the equation \(\sin^2 \theta + 2 \cos^2 \theta = 4 \sin \theta + 3\) for \(0° < \theta < 360°\). [5]
CAIE P1 2024 November Q9
8 marks Standard +0.3
The equation of a curve is \(y = 4 + 5x + 6x^2 - 3x^3\).
  1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases. [4]
  2. It is given that \(y = 9x + k\) is a tangent to the curve. Find the value of the constant \(k\). [4]
CAIE P1 2024 November Q10
8 marks Standard +0.3
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.
  1. Find the value of \(d\). [3]
  2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S_{77}\). The sum of the first 10 terms of the geometric progression is denoted by \(G_{10}\). Find the value of \(S_{77} - G_{10}\). [5]
CAIE P1 2024 November Q11
11 marks Moderate -0.3
The function f is defined by f\((x) = 3 + 6x - 2x^2\) for \(x \in \mathbb{R}\).
  1. Express f\((x)\) in the form \(a - b(x - c)^2\), where \(a\), \(b\) and \(c\) are constants, and state the range of f. [3]
  2. The graph of \(y = \)f\((x)\) is transformed to the graph of \(y = \)h\((x)\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \)h\((x)\) has a minimum point at the origin. Give details of the reflection and translation involved. [2] The function g is defined by g\((x) = 3 + 6x - 2x^2\) for \(x \leqslant 0\).
  3. Sketch the graph of \(y = \)g\((x)\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]
  4. Sketch the graph of \(y = \)g\(^{-1}(x)\) on your diagram in (c), and find an expression for g\(^{-1}(x)\). You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]
CAIE P1 2024 November Q1
5 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the curve with equation \(y = a\sin(bx) + c\) for \(0 \leqslant x \leqslant 2\pi\), where \(a\), \(b\) and \(c\) are positive constants.
  1. State the values of \(a\), \(b\) and \(c\). [3]
  2. For these values of \(a\), \(b\) and \(c\), determine the number of solutions in the interval \(0 \leqslant x \leqslant 2\pi\) for each of the following equations:
    1. \(a\sin(bx) + c = 7 - x\) [1]
    2. \(a\sin(bx) + c = 2\pi(x - 1)\). [1]
CAIE P1 2024 November Q2
5 marks Moderate -0.8
The first term of an arithmetic progression is \(-20\) and the common difference is \(5\).
  1. Find the sum of the first 20 terms of the progression. [2]
It is given that the sum of the first \(2k\) terms is 10 times the sum of the first \(k\) terms.
  1. Find the value of \(k\). [3]
CAIE P1 2024 November Q3
5 marks Moderate -0.8
The equation of a curve is \(y = 2x^2 - 3\). Two points \(A\) and \(B\) with \(x\)-coordinates 2 and \((2 + h)\) respectively lie on the curve.
  1. Find and simplify an expression for the gradient of the chord \(AB\) in terms of \(h\). [3]
  2. Explain how the gradient of the curve at the point \(A\) can be deduced from the answer to part (a), and state the value of this gradient. [2]
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]