Questions FP3 (539 questions)

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Edexcel FP3 2014 June Q4
7 marks Standard +0.3
Using the definitions of hyperbolic functions in terms of exponentials,
  1. show that $$\operatorname{sech}^2 x = 1 - \tanh^2 x$$ [3]
  2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$ [4]
Edexcel FP3 2014 June Q5
4 marks Standard +0.8
Given that \(y = \arctan \frac{x}{\sqrt{1 + x^2}}\) show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2}}\) [4]
Edexcel FP3 2014 June Q6
10 marks Challenging +1.3
[In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.] The points \(P(3\cos \alpha, 2\sin \alpha)\) and \(Q(3\cos \beta, 2\sin \beta)\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$
  1. Show the equation of the chord \(PQ\) is $$\frac{x}{3}\cos\frac{(\alpha + \beta)}{2} + \frac{y}{2}\sin\frac{(\alpha + \beta)}{2} = \cos\frac{(\alpha - \beta)}{2}$$ [4]
  2. Write down the coordinates of the mid-point of \(PQ\). [1]
Given that the gradient, \(m\), of the chord \(PQ\) is a constant,
  1. show that the centre of the chord lies on a line $$y = -kx$$ expressing \(k\) in terms of \(m\). [5]
Edexcel FP3 2014 June Q7
9 marks Standard +0.8
A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x^2 + y^2 = r^2\) where \(r\) is a constant.
  1. Show that \(1 + \left(\frac{dy}{dx}\right)^2 = \frac{r^2}{r^2 - x^2}\) [3]
  2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4\pi r^2\). [5]
  3. Write down the length of the arc of the curve \(y = \sqrt{1 - x^2}\) from \(x = 0\) to \(x = 1\) [1]
Edexcel FP3 2014 June Q8
8 marks Standard +0.3
The position vectors of the points \(A\), \(B\) and \(C\) from a fixed origin \(O\) are $$\mathbf{a} = \mathbf{i} - \mathbf{j}, \quad \mathbf{b} = \mathbf{i} + \mathbf{j} + \mathbf{k}, \quad \mathbf{c} = 2\mathbf{j} + \mathbf{k}$$ respectively.
  1. Using vector products, find the area of the triangle \(ABC\). [4]
  2. Show that \(\frac{1}{6}\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 0\) [3]
  3. Hence or otherwise, state what can be deduced about the vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\). [1]
Edexcel FP3 2014 June Q9
8 marks Challenging +1.8
$$I_n = \int (x^2 + 1)^{-n} dx, \quad n > 0$$
  1. Show that, for \(n > 0\) $$I_{n+1} = \frac{x(x^2 + 1)^{-n}}{2n} + \frac{2n - 1}{2n}I_n$$ [5]
  2. Find \(I_2\) [3]
Edexcel FP3 Q1
5 marks Moderate -0.8
An ellipse has equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\).
  1. Sketch the ellipse. [1]
  2. Find the value of the eccentricity \(e\). [2]
  3. State the coordinates of the foci of the ellipse. [2]
Edexcel FP3 Q3
7 marks Standard +0.8
Solve the equation $$10 \cosh x + 2 \sinh x = 11.$$ Give each answer in the form \(\ln a\) where \(a\) is a rational number. [7]
Edexcel FP3 Q4
9 marks Challenging +1.8
$$I_n = \int_0^{\frac{\pi}{2}} x^n \cos x \, dx, \quad n \geq 0.$$
  1. Prove that \(I_n = \left(\frac{\pi}{2}\right)^n - n(n-1)I_{n-2}\), \(n \geq 2\). [5]
  2. Find an exact expression for \(I_6\). [4]
Edexcel FP3 Q5
10 marks Standard +0.8
  1. Given that \(y = \arctan 3x\), and assuming the derivative of \(\tan x\), prove that $$\frac{dy}{dx} = \frac{3}{1 + 9x^2}.$$ [4]
  2. Show that $$\int_0^{\frac{\sqrt{3}}{3}} 6x \arctan 3x \, dx = \frac{1}{3}(4\pi - 3\sqrt{3}).$$ [6]
Edexcel FP3 Q6
16 marks Challenging +1.2
\includegraphics{figure_6} The curve \(C\) shown in Fig. 1 has equation \(y^2 = 4x\), \(0 \leq x \leq 1\). The part of the curve in the first quadrant is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Show that the surface area of the solid generated is given by $$4\pi \int_0^1 \sqrt{1+x} \, dx.$$ [4]
  2. Find the exact value of this surface area. [3]
  3. Show also that the length of the curve \(C\), between the points \((1, -2)\) and \((1, 2)\), is given by $$2 \int_0^1 \sqrt{\frac{x+1}{x}} \, dx.$$ [3]
  4. Use the substitution \(x = \sinh^2 \theta\) to show that the exact value of this length is $$2[\sqrt{2} + \ln(1 + \sqrt{2})].$$ [6]
Edexcel FP3 Q7
4 marks Challenging +1.2
Prove that \(\sinh(i\pi - \theta) = \sinh \theta\). [4]
Edexcel FP3 Q8
13 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 1 & 0 & 4 \\ 0 & 5 & 4 \\ 4 & 4 & 3 \end{pmatrix}.$$
  1. Verify that \(\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\) and find the corresponding eigenvalue. [3]
  2. Show that \(9\) is another eigenvalue of \(\mathbf{A}\) and find the corresponding eigenvector. [5]
  3. Given that the third eigenvector of \(\mathbf{A}\) is \(\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\), write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{D}.$$ [5]
Edexcel FP3 Q9
18 marks Standard +0.8
The plane \(\Pi\) passes through the points $$A(-1, -1, 1), B(4, 2, 1) \text{ and } C(2, 1, 0).$$
  1. Find a vector equation of the line perpendicular to \(\Pi\) which passes through the point \(D(1, 2, 3)\). [3]
  2. Find the volume of the tetrahedron \(ABCD\). [3]
  3. Obtain the equation of \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [3]
The perpendicular from \(D\) to the plane \(\Pi\) meets \(\Pi\) at the point \(E\).
  1. Find the coordinates of \(E\). [4]
  2. Show that \(DE = \frac{11\sqrt{35}}{35}\). [2]
The point \(D'\) is the reflection of \(D\) in \(\Pi\).
  1. Find the coordinates of \(D'\). [3]
Edexcel FP3 Q10
6 marks Standard +0.3
Find the values of \(x\) for which $$4 \cosh x + \sinh x = 8,$$ giving your answer as natural logarithms. [6]
Edexcel FP3 Q11
7 marks Challenging +1.2
  1. Prove that the derivative of \(\operatorname{artanh} x\), \(-1 < x < 1\), is \(\frac{1}{1-x^2}\). [3]
  2. Find \(\int \operatorname{artanh} x \, dx\). [4]
Edexcel FP3 Q12
7 marks Challenging +1.2
\includegraphics{figure_12} Figure 1 shows the cross-section \(R\) of an artificial ski slope. The slope is modelled by the curve with equation $$y = \frac{10}{\sqrt{4x^2 + 9}}, \quad 0 \leq x \leq 5.$$ Given that 1 unit on each axis represents 10 metres, use integration to calculate the area \(R\). Show your method clearly and give your answer to 2 significant figures. [7]
Edexcel FP3 Q13
9 marks Standard +0.8
\includegraphics{figure_13} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh\left(\frac{x}{a}\right), \quad -ka \leq x \leq ka,$$ where \(a\) and \(k\) are positive constants.
  1. Prove that the length of the rope is \(2a \sinh k\). [5]
Given that the length of the rope is \(8a\),
  1. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate. [4]
Edexcel FP3 Q14
11 marks Challenging +1.2
The curve \(C\) has equation $$y = \operatorname{arcsec} e^x, \quad x > 0, \quad 0 < y < \frac{1}{2}\pi.$$
  1. Prove that \(\frac{dy}{dx} = \frac{1}{\sqrt{e^{2x} - 1}}\). [5]
  2. Sketch the graph of \(C\). [2]
The point \(A\) on \(C\) has \(x\)-coordinate \(\ln 2\). The tangent to \(C\) at \(A\) intersects the \(y\)-axis at the point \(B\).
  1. Find the exact value of the \(y\)-coordinate of \(B\). [4]
Edexcel FP3 Q15
13 marks Standard +0.8
$$I_n = \int_0^1 x^n e^x \, dx \text{ and } J_n = \int_0^1 x^n e^{-x} \, dx, \quad n \geq 0.$$
  1. Show that, for \(n \geq 1\), $$I_n = e - nI_{n-1}.$$ [2]
  2. Find a similar reduction formula for \(J_n\). [3]
  3. Show that \(J_2 = 2 - \frac{5}{e}\). [3]
  4. Show that \(\int_0^1 x^n \cosh x \, dx = \frac{1}{2}(I_n + J_n)\). [1]
  5. Hence, or otherwise, evaluate \(\int_0^1 x^2 \cosh x \, dx\), giving your answer in terms of \(e\). [4]
Edexcel FP3 Q16
14 marks Challenging +1.3
The hyperbola \(C\) has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
  1. Show that an equation of the normal to \(C\) at the point \(P(a \sec t, b \tan t)\) is $$ax \sin t + by = (a^2 + b^2) \tan t.$$ [6]
The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and \(S\) is a focus of \(C\). Given that the eccentricity of \(C\) is \(\frac{3}{2}\), and that \(OA = 3OS\), where \(O\) is the origin,
  1. determine the possible values of \(t\), for \(0 \leq t < 2\pi\). [8]
Edexcel FP3 Q17
5 marks Challenging +1.2
Referred to a fixed origin \(O\), the position vectors of three non-collinear points \(A\), \(B\) and \(C\) are \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. By considering \(\overrightarrow{AB} \times \overrightarrow{AC}\), prove that the area of \(\triangle ABC\) can be expressed in the form \(\frac{1}{2}|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|\). [5]
Edexcel FP3 Q18
7 marks Standard +0.3
$$\mathbf{M} = \begin{pmatrix} 4 & -5 \\ 6 & -9 \end{pmatrix}$$
  1. Find the eigenvalues of \(\mathbf{M}\). [4]
A transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{M}\). There is a line through the origin for which every point on the line is mapped onto itself under \(T\).
  1. Find a cartesian equation of this line. [3]
Edexcel FP3 Q19
11 marks Standard +0.3
$$\mathbf{A} = \begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & u \end{pmatrix}, \quad u \neq 1.$$
  1. Show that \(\det \mathbf{A} = 2(u - 1)\). [2]
  2. Find the inverse of \(\mathbf{A}\). [6]
The image of the vector \(\begin{pmatrix} a \\ b \\ c \end{pmatrix}\) when transformed by the matrix \(\begin{pmatrix} 3 & 1 & -1 \\ 1 & 1 & 1 \\ 5 & 3 & 6 \end{pmatrix}\) is \(\begin{pmatrix} 3 \\ 1 \\ 6 \end{pmatrix}\).
  1. Find the values of \(a\), \(b\) and \(c\). [3]
Edexcel FP3 Q20
12 marks Standard +0.3
The plane \(\Pi_1\) passes through the \(P\), with position vector \(\mathbf{i} + 2\mathbf{j} - \mathbf{k}\), and is perpendicular to the line \(L\) with equation $$\mathbf{r} = 3\mathbf{i} - 2\mathbf{k} + \lambda(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}).$$
  1. Show that the Cartesian equation of \(\Pi_1\) is \(x - 5y - 3z = -6\). [4]
The plane \(\Pi_2\) contains the line \(L\) and passes through the point \(Q\), with position vector \(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\).
  1. Find the perpendicular distance of \(Q\) from \(\Pi_1\). [4]
  2. Find the equation of \(\Pi_2\) in the form \(\mathbf{r} = \mathbf{a} + s\mathbf{b} + t\mathbf{c}\). [4]