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 2011 November Q1
3 marks Moderate -0.5
The coefficient of \(x^2\) in the expansion of \(\left(k + \frac{1}{x}\right)^5\) is 30. Find the value of the constant \(k\). [3]
CAIE P1 2011 November Q2
4 marks Easy -1.2
The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is
  1. an arithmetic progression, [2]
  2. a geometric progression. [2]
CAIE P1 2011 November Q3
5 marks Moderate -0.8
\includegraphics{figure_3} The diagram shows the curve \(y = 2x^5 + 3x^3\) and the line \(y = 2x\) intersecting at points \(A\), \(O\) and \(B\).
  1. Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2x^4 + 3x^2 - 2 = 0\). [2]
  2. Solve the equation \(2x^4 + 3x^2 - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form. [3]
CAIE P1 2011 November Q4
6 marks Moderate -0.3
\includegraphics{figure_4} In the diagram, \(ABCD\) is a parallelogram with \(AB = BD = DC = 10\) cm and angle \(ABD = 0.8\) radians. \(APD\) and \(BQC\) are arcs of circles with centres \(B\) and \(D\) respectively.
  1. Find the area of the parallelogram \(ABCD\). [2]
  2. Find the area of the complete figure \(ABQCDP\). [2]
  3. Find the perimeter of the complete figure \(ABQCDP\). [2]
CAIE P1 2011 November Q5
7 marks Moderate -0.3
  1. Given that $$3\sin^2 x - 8\cos x - 7 = 0,$$ show that, for real values of \(x\), $$\cos x = -\frac{2}{3}.$$ [3]
  2. Hence solve the equation $$3\sin^2(\theta + 70°) - 8\cos(\theta + 70°) - 7 = 0$$ for \(0° \leqslant \theta \leqslant 180°\). [4]
CAIE P1 2011 November Q6
8 marks Moderate -0.8
Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{3i} + 4\mathbf{j} - \mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) respectively.
  1. Use a scalar product to find angle \(BOA\). [4]
The point \(C\) is the mid-point of \(AB\). The point \(D\) is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).
  1. Find \(\overrightarrow{DC}\). [4]
CAIE P1 2011 November Q7
9 marks Moderate -0.3
  1. A straight line passes through the point \((2, 0)\) and has gradient \(m\). Write down the equation of the line. [1]
  2. Find the two values of \(m\) for which the line is a tangent to the curve \(y = x^2 - 4x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve. [6]
  3. Express \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve. [2]
CAIE P1 2011 November Q8
10 marks Moderate -0.3
A curve \(y = \mathrm{f}(x)\) has a stationary point at \(P(3, -10)\). It is given that \(\mathrm{f}'(x) = 2x^2 + kx - 12\), where \(k\) is a constant.
  1. Show that \(k = -2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\). [4]
  2. Find \(\mathrm{f}''(x)\) and determine the nature of each of the stationary points \(P\) and \(Q\). [2]
  3. Find \(\mathrm{f}(x)\). [4]
CAIE P1 2011 November Q9
11 marks Standard +0.3
Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \begin{align} \mathrm{f} : x \mapsto 2x + 3 \quad &\text{for } x \leqslant 0,
\mathrm{g} : x \mapsto x^2 - 6x \quad &\text{for } x \leqslant 3. \end{align}
  1. Express \(\mathrm{f}^{-1}(x)\) in terms of \(x\) and solve the equation \(\mathrm{f}(x) = \mathrm{f}^{-1}(x)\). [3]
  2. On the same diagram sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\), showing the coordinates of their point of intersection and the relationship between the graphs. [3]
  3. Find the set of values of \(x\) which satisfy \(\mathrm{gf}(x) \leqslant 16\). [5]
CAIE P1 2011 November Q10
12 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt{(x + 1)}\), meeting at \((-1, 0)\) and \((0, 1)\).
  1. Find the area of the shaded region. [5]
  2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [7]
CAIE P1 2014 November Q1
4 marks Standard +0.3
\includegraphics{figure_1} The diagram shows part of the curve \(y = x^2 + 1\). Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [4]
CAIE P1 2014 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} The diagram shows a triangle \(AOB\) in which \(OA\) is 12 cm, \(OB\) is 5 cm and angle \(AOB\) is a right angle. Point \(P\) lies on \(AB\) and \(OP\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(AB\) and \(OQ\) is an arc of a circle with centre \(B\).
  1. Show that angle \(BAO\) is 0.3948 radians, correct to 4 decimal places. [1]
  2. Calculate the area of the shaded region. [5]
CAIE P1 2014 November Q3
5 marks Moderate -0.8
  1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \((1 + x)^5\). [2]
The coefficient of \(x^2\) in the expansion of \((1 + (px + x^2))^5\) is 95.
  1. Use the answer to part (i) to find the value of the positive constant \(p\). [3]
CAIE P1 2014 November Q4
6 marks Standard +0.3
A curve has equation \(y = \frac{12}{5 - 2x}\).
  1. Find \(\frac{dy}{dx}\). [2]
A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
  1. Find the possible \(x\)-coordinates of \(A\). [4]
CAIE P1 2014 November Q5
6 marks Moderate -0.3
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos^2 x - \cos x - 1 = 0.$$ [3]
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0° \leqslant x \leqslant 180°\). [3]
CAIE P1 2014 November Q6
6 marks Moderate -0.3
The equation of a curve is \(y = x^3 + ax^2 + bx\), where \(a\) and \(b\) are constants.
  1. In the case where the curve has no stationary point, show that \(a^2 < 3b\). [3]
  2. In the case where \(a = -6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\). [3]
CAIE P1 2014 November Q7
7 marks Moderate -0.8
\includegraphics{figure_7} The diagram shows a pyramid \(OABCX\). The horizontal square base \(OABC\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(XD = 10\) units. The mid-point of \(OX\) is \(M\). The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards.
  1. Express the vectors \(\overrightarrow{AM}\) and \(\overrightarrow{AC}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [3]
  2. Use a scalar product to find angle \(MAC\). [4]
CAIE P1 2014 November Q8
8 marks Moderate -0.3
  1. The sum, \(S_n\), of the first \(n\) terms of an arithmetic progression is given by \(S_n = 32n - n^2\). Find the first term and the common difference. [3]
  2. A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression. [5]
CAIE P1 2014 November Q9
8 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows a trapezium \(ABCD\) in which \(AB\) is parallel to \(DC\) and angle \(BAD\) is \(90°\). The coordinates of \(A\), \(B\) and \(C\) are \((2, 6)\), \((5, -3)\) and \((8, 3)\) respectively.
  1. Find the equation of \(AD\). [3]
  2. Find, by calculation, the coordinates of \(D\). [3]
The point \(E\) is such that \(ABCE\) is a parallelogram.
  1. Find the length of \(BE\). [2]
CAIE P1 2014 November Q10
9 marks Moderate -0.3
A curve is such that \(\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4\). The curve has a stationary point at \(P\) where \(x = 2\).
  1. State, with a reason, the nature of this stationary point. [1]
  2. Find an expression for \(\frac{dy}{dx}\). [4]
  3. Given that the curve passes through the point \((1, 13)\), find the coordinates of the stationary point \(P\). [4]
CAIE P1 2014 November Q11
10 marks Moderate -0.3
The function \(f : x \mapsto 6 - 4\cos(\frac{1}{2}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).
  1. Find the exact value of \(x\) for which \(f(x) = 4\). [3]
  2. State the range of \(f\). [2]
  3. Sketch the graph of \(y = f(x)\). [2]
  4. Find an expression for \(f^{-1}(x)\). [3]
CAIE P1 2014 November Q1
4 marks Moderate -0.3
In the expansion of \((2 + ax)^6\), the coefficient of \(x^2\) is equal to the coefficient of \(x^3\). Find the value of the non-zero constant \(a\). [4]
CAIE P1 2014 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} In the diagram, \(OADC\) is a sector of a circle with centre \(O\) and radius 3 cm. \(AB\) and \(CB\) are tangents to the circle and angle \(ABC = \frac{1}{4}\pi\) radians. Find, giving your answer in terms of \(\sqrt{3}\) and \(\pi\),
  1. the perimeter of the shaded region, [3]
  2. the area of the shaded region. [3]
CAIE P1 2014 November Q3
6 marks Moderate -0.8
  1. Express \(9x^2 - 12x + 5\) in the form \((ax + b)^2 + c\). [3]
  2. Determine whether \(3x^3 - 6x^2 + 5x - 12\) is an increasing function, a decreasing function or neither. [3]
CAIE P1 2014 November Q4
6 marks Standard +0.3
Three geometric progressions, \(P\), \(Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2, 1, \frac{1}{2}, \frac{1}{4}, \ldots\) Progression \(Q\) is \(3, 1, \frac{1}{3}, \frac{1}{9}, \ldots\)
  1. Find the sum to infinity of progression \(R\). [3]
  2. Given that the first term of \(R\) is 4, find the sum of the first three terms of \(R\). [3]