Questions P1 (1401 questions)

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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]
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]