Questions — Edexcel FP3 (160 questions)

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Edexcel FP3 2011 June Q5
9 marks Standard +0.8
The curve \(C_1\) has equation \(y = 3\sinh 2x\), and the curve \(C_2\) has equation \(y = 13 - 3e^{2x}\).
  1. Sketch the graph of the curves \(C_1\) and \(C_2\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. [4]
  2. Solve the equation \(3\sinh 2x = 13 - 3e^{2x}\), giving your answer in the form \(\frac{1}{2}\ln k\), where \(k\) is an integer. [5]
Edexcel FP3 2011 June Q6
10 marks Standard +0.3
The plane \(P\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 2 \\ 2 \end{pmatrix}$$
  1. Find a vector perpendicular to the plane \(P\). [2] The line \(l\) passes through the point \(A(1, 3, 3)\) and meets \(P\) at \((3, 1, 2)\). The acute angle between the plane \(P\) and the line \(l\) is \(\alpha\).
  2. Find \(\alpha\) to the nearest degree. [4]
  3. Find the perpendicular distance from \(A\) to the plane \(P\). [4]
Edexcel FP3 2011 June Q7
12 marks Challenging +1.2
The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} k & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}, \quad k \neq 1$$
  1. Show that \(\det \mathbf{M} = 2 - 2k\). [2]
  2. Find \(\mathbf{M}^{-1}\), in terms of \(k\). [5] The straight line \(l_1\) is mapped onto the straight line \(l_2\) by the transformation represented by the matrix \(\begin{pmatrix} 2 & -1 & 1 \\ 1 & 0 & -1 \\ 3 & -2 & 1 \end{pmatrix}\). The equation of \(l_2\) is \((\mathbf{r} - \mathbf{a}) \times \mathbf{b} = \mathbf{0}\), where \(\mathbf{a} = 4\mathbf{i} + \mathbf{j} + 7\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + \mathbf{j} + 3\mathbf{k}\).
  3. Find a vector equation for the line \(l_1\). [5]
Edexcel FP3 2011 June Q8
14 marks Challenging +1.3
The hyperbola \(H\) has equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
  1. Use calculus to show that the equation of the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\) may be written in the form $$xb\cosh\theta - ya\sinh\theta = ab$$ [4] The line \(l_1\) is the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\), \(\theta \neq 0\). Given that \(l_1\) meets the \(x\)-axis at the point \(P\),
  2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). [2] The line \(l_2\) is the tangent to \(H\) at the point \((a, 0)\). Given that \(l_1\) and \(l_2\) meet at the point \(Q\),
  3. find, in terms of \(a\), \(b\) and \(\theta\), the coordinates of \(Q\). [2]
  4. Show that, as \(\theta\) varies, the locus of the mid-point of \(PQ\) has equation $$x(4y^2 + b^2) = ab^2$$ [6]
Edexcel FP3 2014 June Q1
8 marks Standard +0.3
The line \(l\) passes through the point \(P(2, 1, 3)\) and is perpendicular to the plane \(\Pi\) whose vector equation is $$\mathbf{r} \cdot (\mathbf{i} - 2\mathbf{j} - \mathbf{k}) = 3$$ Find
  1. a vector equation of the line \(l\), [2]
  2. the position vector of the point where \(l\) meets \(\Pi\). [4]
  3. Hence find the perpendicular distance of \(P\) from \(\Pi\). [2]
Edexcel FP3 2014 June Q2
13 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$
  1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
  2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
  3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]
The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).
  1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]
Edexcel FP3 2014 June Q3
8 marks Standard +0.8
Using calculus, find the exact value of
  1. \(\int_1^2 \frac{1}{\sqrt{x^2 - 2x + 3}} \, dx\) [4]
  2. \(\int_0^1 e^{-x} \sinh x \, dx\) [4]
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]