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 2014 November Q5
7 marks Moderate -0.3
  1. Show that \(\sin^2 \theta - \cos^4 \theta = 2 \sin^2 \theta - 1\). [3]
  2. Hence solve the equation \(\sin^2 \theta - \cos^4 \theta = \frac{1}{2}\) for \(0° \leq \theta \leq 360°\). [4]
CAIE P1 2014 November Q6
7 marks Moderate -0.3
\(A\) is the point \((a, 2a - 1)\) and \(B\) is the point \((2a + 4, 3a + 9)\), where \(a\) is a constant.
  1. Find, in terms of \(a\), the gradient of a line perpendicular to \(AB\). [3]
  2. Given that the distance \(AB\) is \(\sqrt{260}\), find the possible values of \(a\). [4]
CAIE P1 2014 November Q7
8 marks Moderate -0.3
Three points, \(O\), \(A\) and \(B\), are such that \(\overrightarrow{OA} = \mathbf{i} + 3\mathbf{j} + p\mathbf{k}\) and \(\overrightarrow{OB} = -7\mathbf{i} + (1 - p)\mathbf{j} + p\mathbf{k}\), where \(p\) is a constant.
  1. Find the values of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [3]
  2. The magnitudes of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) are \(a\) and \(b\) respectively. Find the value of \(p\) for which \(b^2 = 2a^2\). [2]
  3. Find the unit vector in the direction of \(\overrightarrow{AB}\) when \(p = -8\). [3]
CAIE P1 2014 November Q8
8 marks Moderate -0.3
A curve \(y = f(x)\) has a stationary point at \((3, 7)\) and is such that \(f''(x) = 36x^{-3}\).
  1. State, with a reason, whether this stationary point is a maximum or a minimum. [1]
  2. Find \(f'(x)\) and \(f(x)\). [7]
CAIE P1 2014 November Q9
10 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3\sqrt{x}\) intersecting at points \(A\) and \(B\).
  1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\). [4]
  2. Find by integration the area of the shaded region. [6]
CAIE P1 2014 November Q10
13 marks Standard +0.3
  1. The functions \(f\) and \(g\) are defined for \(x \geq 0\) by $$f : x \mapsto (ax + b)^{\frac{1}{3}}, \text{ where } a \text{ and } b \text{ are positive constants,}$$ $$g : x \mapsto x^2.$$ Given that \(fg(1) = 2\) and \(gf(9) = 16\),
    1. calculate the values of \(a\) and \(b\), [4]
    2. obtain an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\). [4]
  2. A point \(P\) travels along the curve \(y = (7x^2 + 1)^{\frac{1}{3}}\) in such a way that the \(x\)-coordinate of \(P\) at time \(t\) minutes is increasing at a constant rate of 8 units per minute. Find the rate of increase of the \(y\)-coordinate of \(P\) at the instant when \(P\) is at the point \((3, 4)\). [5]
CAIE P1 2016 November Q1
3 marks Moderate -0.3
Find the set of values of \(k\) for which the curve \(y = kx^2 - 3x\) and the line \(y = x - k\) do not meet. [3]
CAIE P1 2016 November Q2
4 marks Moderate -0.3
The coefficient of \(x^3\) in the expansion of \((1 - 3x)^6 + (1 + ax)^5\) is 100. Find the value of the constant \(a\). [4]
CAIE P1 2016 November Q3
4 marks Moderate -0.8
Showing all necessary working, solve the equation \(6\sin^2 x - 5\cos^2 x = 2\sin^2 x + \cos^2 x\) for \(0° \leq x \leq 360°\). [4]
CAIE P1 2016 November Q4
4 marks Standard +0.3
The function \(f\) is such that \(f(x) = x^3 - 3x^2 - 9x + 2\) for \(x > n\), where \(n\) is an integer. It is given that \(f\) is an increasing function. Find the least possible value of \(n\). [4]
CAIE P1 2016 November Q5
6 marks Standard +0.3
\includegraphics{figure_1} The diagram shows a major arc \(AB\) of a circle with centre \(O\) and radius 6 cm. Points \(C\) and \(D\) on \(OA\) and \(OB\) respectively are such that the line \(AB\) is a tangent at \(E\) to the arc \(CED\) of a smaller circle also with centre \(O\). Angle \(COD = 1.8\) radians.
  1. Show that the radius of the arc \(CED\) is 3.73 cm, correct to 3 significant figures. [2]
  2. Find the area of the shaded region. [4]
CAIE P1 2016 November Q6
7 marks Moderate -0.3
Three points, \(A\), \(B\) and \(C\), are such that \(B\) is the mid-point of \(AC\). The coordinates of \(A\) are \((2, m)\) and the coordinates of \(B\) are \((n, -6)\), where \(m\) and \(n\) are constants.
  1. Find the coordinates of \(C\) in terms of \(m\) and \(n\). [2]
The line \(y = x + 1\) passes through \(C\) and is perpendicular to \(AB\).
  1. Find the values of \(m\) and \(n\). [5]
CAIE P1 2016 November Q7
7 marks Moderate -0.3
\includegraphics{figure_2} The diagram shows a triangular pyramid \(ABCD\). It is given that $$\overrightarrow{AB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}, \quad \overrightarrow{AC} = \mathbf{i} - 2\mathbf{j} - \mathbf{k} \quad \text{and} \quad \overrightarrow{AD} = \mathbf{i} + 4\mathbf{j} - 7\mathbf{k}.$$
  1. Verify, showing all necessary working, that each of the angles \(DAB\), \(DAC\) and \(CAB\) is \(90°\). [3]
  2. Find the exact value of the area of the triangle \(ABC\), and hence find the exact value of the volume of the pyramid. [4]
[The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac{1}{3}Ah\).]
CAIE P1 2016 November Q8
8 marks Moderate -0.3
  1. Express \(4x^2 + 12x + 10\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Functions \(f\) and \(g\) are both defined for \(x > 0\). It is given that \(f(x) = x^2 + 1\) and \(fg(x) = 4x^2 + 12x + 10\). Find \(g(x)\). [1]
  3. Find \((fg)^{-1}(x)\) and give the domain of \((fg)^{-1}\). [4]
CAIE P1 2016 November Q9
8 marks Standard +0.3
  1. Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are \(6\) and \(6r\) respectively. The first and second terms of \(Q\) are \(12\) and \(-12r\) respectively. Find the value of the common sum to infinity. [3]
  2. The first term of an arithmetic progression is \(\cos\theta\) and the second term is \(\cos\theta + \sin^2\theta\), where \(0 \leq \theta \leq \pi\). The sum of the first \(13\) terms is \(52\). Find the possible values of \(\theta\). [5]
CAIE P1 2016 November Q10
12 marks Standard +0.3
A curve is such that \(\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-\frac{3}{2}}\), where \(a\) is a positive constant. The point \(A(a^2, 3)\) lies on the curve. Find, in terms of \(a\),
  1. the equation of the tangent to the curve at \(A\), simplifying your answer, [3]
  2. the equation of the curve. [4]
It is now given that \(B(16, 8)\) also lies on the curve.
  1. Find the value of \(a\) and, using this value, find the distance \(AB\). [5]
CAIE P1 2016 November Q11
12 marks Standard +0.3
A curve has equation \(y = (kx - 3)^{-1} + (kx - 3)\), where \(k\) is a non-zero constant.
  1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point. Justify your answers. [7]
  1. \includegraphics{figure_3} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360°\) about the \(x\)-axis. [5]
CAIE P1 2018 November Q1
4 marks Standard +0.3
Find the coefficient of \(\frac{1}{x^2}\) in the expansion of \(\left(3x + \frac{2}{3x^2}\right)^7\). [4]
CAIE P1 2018 November Q2
4 marks Moderate -0.8
Showing all necessary working, find \(\int_1^4 \left(\sqrt{x} + \frac{2}{\sqrt{x}}\right) \text{d}x\). [4]
CAIE P1 2018 November Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows part of the curve \(y = x(9 - x^2)\) and the line \(y = 5x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.
  1. Express the length of \(PQ\) in terms of \(t\), simplifying your answer. [2]
  2. Given that \(t\) can vary, find the maximum value of the length of \(PQ\). [3]
CAIE P1 2018 November Q4
6 marks Moderate -0.8
Functions f and g are defined by $$f : x \mapsto 2 - 3\cos x \text{ for } 0 \leqslant x \leqslant 2\pi,$$ $$g : x \mapsto \frac{1}{2}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$
  1. Solve the equation \(\text{fg}(x) = 1\). [3]
  2. Sketch the graph of \(y = \text{f}(x)\). [3]
CAIE P1 2018 November Q5
7 marks Standard +0.3
The first three terms of an arithmetic progression are \(4\), \(x\) and \(y\) respectively. The first three terms of a geometric progression are \(x\), \(y\) and \(18\) respectively. It is given that both \(x\) and \(y\) are positive.
  1. Find the value of \(x\) and the value of \(y\). [4]
  2. Find the fourth term of each progression. [3]
CAIE P1 2018 November Q6
7 marks Standard +0.3
\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).
  1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
  2. Hence, showing all necessary working, calculate \(\theta\). [3]
CAIE P1 2018 November Q7
7 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards. Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]
CAIE P1 2018 November Q8
7 marks Standard +0.3
\includegraphics{figure_8} The diagram shows an isosceles triangle \(ACB\) in which \(AB = BC = 8\) cm and \(AC = 12\) cm. The arc \(XC\) is part of a circle with centre \(A\) and radius \(12\) cm, and the arc \(YC\) is part of a circle with centre \(B\) and radius \(8\) cm. The points \(A\), \(B\), \(X\) and \(Y\) lie on a straight line.
  1. Show that angle \(CBY = 1.445\) radians, correct to \(4\) significant figures. [3]
  2. Find the perimeter of the shaded region. [4]