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Edexcel C4 2014 June Q7
15 marks Challenging +1.2
\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3\tan\theta, \quad y = 4\cos^2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$ The point \(P\) lies on \(C\) and has coordinates \((3, 2)\). The line \(l\) is the normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
  1. [(a)] Find the \(x\) coordinate of the point \(Q\). \hfill [6]
The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This shaded region is rotated \(2\pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. [(b)] Find the exact value of the volume of the solid of revolution, giving your answer in the form \(p\pi + q\pi^2\), where \(p\) and \(q\) are rational numbers to be determined. [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.] \hfill [9] \end{enumerate} \end{enumerate}
Edexcel C4 2014 June Q8
15 marks Standard +0.3
Relative to a fixed origin \(O\), the point \(A\) has position vector \(\begin{pmatrix} -2 \\ 4 \\ 7 \end{pmatrix}\) and the point \(B\) has position vector \(\begin{pmatrix} -1 \\ 3 \\ 8 \end{pmatrix}\) The line \(l_1\) passes through the points \(A\) and \(B\).
  1. [(a)] Find the vector \(\overrightarrow{AB}\). \hfill [2]
  2. [(b)] Hence find a vector equation for the line \(l_1\) \hfill [1]
The point \(P\) has position vector \(\begin{pmatrix} 0 \\ 2 \\ 3 \end{pmatrix}\) Given that angle \(PBA\) is \(\theta\),
  1. [(c)] show that \(\cos\theta = \frac{1}{3}\) \hfill [3]
The line \(l_2\) passes through the point \(P\) and is parallel to the line \(l_1\)
  1. [(d)] Find a vector equation for the line \(l_2\) \hfill [2]
The points \(C\) and \(D\) both lie on the line \(l_2\) Given that \(AB = PC = DP\) and the \(x\) coordinate of \(C\) is positive,
  1. [(e)] find the coordinates of \(C\) and the coordinates of \(D\). \hfill [3]
  2. [(f)] find the exact area of the trapezium \(ABCD\), giving your answer as a simplified surd. \hfill [4] \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate}
CAIE P1 2023 June Q1
3 marks Moderate -0.8
The equation of a curve is such that \(\frac{dy}{dx} = \frac{4}{(x-3)^2}\) for \(x > 3\). The curve passes through the point \((4, 5)\). Find the equation of the curve. [3]
CAIE P1 2023 June Q2
4 marks Standard +0.3
The coefficient of \(x^4\) in the expansion of \((x + a)^6\) is \(p\) and the coefficient of \(x^2\) in the expansion of \((ax + 3)^4\) is \(q\). It is given that \(p + q = 276\). Find the possible values of the constant \(a\). [4]
CAIE P1 2023 June Q3
3 marks Moderate -0.8
  1. Express \(4x^2 - 24x + p\) in the form \(a(x + b)^2 + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\). [2]
  2. Hence or otherwise find the set of values of \(p\) for which the equation \(4x^2 - 24x + p = 0\) has no real roots. [1]
CAIE P1 2023 June Q4
3 marks Standard +0.3
Solve the equation \(8x^6 + 215x^3 - 27 = 0\). [3]
CAIE P1 2023 June Q5
4 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows the curve with equation \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) for \(x > 0\). The curve meets the \(x\)-axis at the points \((0, 0)\) and \((4, 0)\). Find the area of the shaded region. [4]
CAIE P1 2023 June Q6
6 marks Standard +0.3
\includegraphics{figure_6} The diagram shows a sector \(OAB\) of a circle with centre \(O\). Angle \(AOB = \theta\) radians and \(OP = AP = x\).
  1. Show that the arc length \(AB\) is \(2x\theta \cos \theta\). [2]
  2. Find the area of the shaded region \(APB\) in terms of \(x\) and \(\theta\). [4]
CAIE P1 2023 June Q7
11 marks Standard +0.3
    1. By first expanding \((\cos \theta + \sin \theta)^2\), find the three solutions of the equation $$(\cos \theta + \sin \theta)^2 = 1$$ for \(0 \leqslant \theta \leqslant \pi\). [3]
    2. Hence verify that the only solutions of the equation \(\cos \theta + \sin \theta = 1\) for \(0 < \theta < \pi\) are \(0\) and \(\frac{1}{2}\pi\). [2]
  2. Prove the identity \(\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} \equiv \frac{\cos \theta + \sin \theta - 1}{1 - 2\sin^2 \theta}\). [3]
  3. Using the results of (a)(ii) and (b), solve the equation $$\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} = 2(\cos \theta + \sin \theta - 1)$$ for \(0 \leqslant \theta \leqslant \pi\). [3]
CAIE P1 2023 June Q8
12 marks Moderate -0.3
\includegraphics{figure_8} The diagram shows the graph of \(y = f(x)\) where the function \(f\) is defined by $$f(x) = 3 + 2\sin \frac{1}{4}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$
  1. On the diagram above, sketch the graph of \(y = f^{-1}(x)\). [2]
  2. Find an expression for \(f^{-1}(x)\). [2]
  3. \includegraphics{figure_8c} The diagram above shows part of the graph of the function \(g(x) = 3 + 2\sin \frac{1}{4}x\) for \(-2\pi \leqslant x \leqslant 2\pi\). Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]
  4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\) to the graph of \(y = f(x)\), making clear the order in which the transformations are applied. [6]
CAIE P1 2023 June Q9
8 marks Standard +0.3
The second term of a geometric progression is 16 and the sum to infinity is 100.
  1. Find the two possible values of the first term. [4]
  2. Show that the \(n\)th term of one of the two possible geometric progressions is equal to \(4^{n-2}\) multiplied by the \(n\)th term of the other geometric progression. [4]
CAIE P1 2023 June Q10
13 marks Standard +0.3
The equation of a circle is \((x - a)^2 + (y - 3)^2 = 20\). The line \(y = \frac{1}{2}x + 6\) is a tangent to the circle at the point \(P\).
  1. Show that one possible value of \(a\) is 4 and find the other possible value. [5]
  2. For \(a = 4\), find the equation of the normal to the circle at \(P\). [4]
  3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b). [4]
CAIE P1 2023 June Q11
8 marks Moderate -0.3
The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.
  1. Find \(\frac{dy}{dx}\). [2]
  2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
  3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan^{-1}(2)\) with the positive \(x\)-axis. [4]
CAIE P1 2024 June Q1
5 marks Moderate -0.8
  1. Express \(3y^2 - 12y - 15\) in the form \(3(y + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
  2. Hence find the exact solutions of the equation \(3x^4 - 12x^2 - 15 = 0\). [3]
CAIE P1 2024 June Q2
6 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = \text{f}(x)\).
  1. In order to transform the curve \(y = \sin x\) to the curve \(y = \text{f}(x)\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required. [4]
  2. Find f\((x)\) in terms of \(\sin x\). [2]
CAIE P1 2024 June Q3
5 marks Moderate -0.8
The coefficient of \(x^3\) in the expansion of \((3 + ax)^6\) is 160.
  1. Find the value of the constant \(a\). [2]
  2. Hence find the coefficient of \(x^5\) in the expansion of \((3 + ax)^6(1 - 2x)\). [3]
CAIE P1 2024 June Q4
3 marks Easy -1.8
The equation of a curve is \(y = \text{f}(x)\), where f\((x) = (2x - 1)\sqrt{3x - 2} - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. \(A(2, 4)\), \(B(2.0001, k)\), \(C(2.001, 4.00625)\), \(D(2.01, 4.06261)\), \(E(2.1, 4.63566)\), \(F(3, 11.22876)\)
  1. Find the value of \(k\). Give your answer correct to 5 decimal places. [1]
The table shows the gradients of the chords \(AB\), \(AC\), \(AD\) and \(AF\).
Chord\(AB\)\(AC\)\(AD\)\(AE\)\(AF\)
Gradient of chord6.25016.25116.26087.2288
  1. Find the gradient of the chord \(AE\). Give your answer correct to 4 decimal places. [1]
  2. Deduce the value of f\('(2)\) using the values in the table. [1]
CAIE P1 2024 June Q5
6 marks Moderate -0.8
  1. Prove the identity \(\frac{\sin^2 x - \cos x - 1}{1 + \cos x} \equiv -\cos x\). [3]
  2. Hence solve the equation \(\frac{\sin^2 x - \cos x - 1}{2 + 2\cos x} = \frac{1}{4}\) for \(0° \leq x \leq 360°\). [3]
CAIE P1 2024 June Q6
7 marks Moderate -0.3
\includegraphics{figure_6} The function f is defined by f\((x) = \frac{2}{x^2} + 4\) for \(x < 0\). The diagram shows the graph of \(y = \text{f}(x)\).
  1. On this diagram, sketch the graph of \(y = \text{f}^{-1}(x)\). Show any relevant mirror line. [2]
  2. Find an expression for f\(^{-1}(x)\). [3]
  3. Solve the equation f\((x) = 4.5\). [1]
  4. Explain why the equation f\(^{-1}(x) = \text{f}(x)\) has no solution. [1]
CAIE P1 2024 June Q7
8 marks Standard +0.3
\includegraphics{figure_7} In the diagram, \(AOD\) and \(BC\) are two parallel straight lines. Arc \(AB\) is part of a circle with centre \(O\) and radius \(15\text{cm}\). Angle \(BOA = \theta\) radians. Arc \(CD\) is part of a circle with centre \(O\) and radius \(10\text{cm}\). Angle \(COD = \frac{1}{3}\pi\) radians.
  1. Show that \(\theta = 0.7297\), correct to 4 decimal places. [1]
  2. Find the perimeter and the area of the shape \(ABCD\). Give your answers correct to 3 significant figures. [7]
CAIE P1 2024 June Q8
8 marks Moderate -0.3
  1. The first three terms of an arithmetic progression are \(25\), \(4p - 1\) and \(13 - p\), where \(p\) is a constant. Find the value of the tenth term of the progression. [4]
  2. The first three terms of a geometric progression are \(25\), \(4q - 1\) and \(13 - q\), where \(q\) is a positive constant. Find the sum to infinity of the progression. [4]
CAIE P1 2024 June Q9
6 marks Standard +0.3
\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \frac{1}{(5x - 4)^3}\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \((1, 1)\). Find the exact volume of the solid generated when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]
CAIE P1 2024 June Q10
8 marks Standard +0.8
The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]
CAIE P1 2024 June Q11
13 marks Standard +0.8
\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.
  1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
  2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]
CAIE P1 2024 June Q1
3 marks Moderate -0.5
The coefficient of \(x^2\) in the expansion of \((1-4x)^6\) is 12 times the coefficient of \(x^2\) in the expansion of \((2+ax)^5\). Find the value of the positive constant \(a\). [3]