Questions Pre-U 9795/1 (179 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
Pre-U Pre-U 9795/1 2013 November Q11
14 marks Standard +0.8
  1. Given that \(y = -4\) when \(x = 0\) and that $$\frac{dy}{dx} - y = e^{2x} + 3,$$ find the value of \(x\) for which \(y = 0\). [7]
  2. Find the general solution of $$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = e^{2x} + 3,$$ given that \(y = cx^2e^{2x} + d\) is a suitable form of particular integral. [7]
Pre-U Pre-U 9795/1 2013 November Q12
10 marks Challenging +1.3
    1. Use the method of differences to prove that $$\sum_{n=k}^N \frac{1}{n(n+1)} = \frac{1}{k} - \frac{1}{N+1}.$$ [4]
    2. Deduce the value of \(\sum_{n=k}^{\infty} \frac{1}{n(n+1)}\) and show that \(\sum_{n=k}^{\infty} \frac{1}{(n+1)^2} < \frac{1}{k}\). [3]
  2. Let \(S = \sum_{n=1}^{\infty} \frac{1}{n^2}\). Show that \(\frac{205}{144} < S < \frac{241}{144}\). [3]
Pre-U Pre-U 9795/1 2013 November Q13
24 marks Hard +2.3
  1. Let \(I_n = \int_0^{\alpha} \cosh^n x \, dx\) for integers \(n \geqslant 0\), where \(\alpha = \ln 2\).
    1. Prove that, for \(n \geqslant 2\), \(nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}\). [5]
    2. A curve has parametric equations \(x = 12 \sinh t + 4 \sinh^3 t\), \(y = 3 \cosh^4 t\), \(0 \leqslant t \leqslant \ln 2\). Find the length of the arc of this curve, giving your answer in the form \(a + b \ln 2\) for rational numbers \(a\) and \(b\). [8]
  2. The circle with equation \(x^2 + (y - R)^2 = r^2\), where \(r < R\), is rotated through one revolution about the \(x\)-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of \(\pi\), \(R\) and \(r\), the surface area of this torus. [11]
Pre-U Pre-U 9795/1 2015 June Q1
3 marks Moderate -0.5
Determine the volume of tetrahedron \(OABC\), where \(O\) is the origin and \(A\), \(B\) and \(C\) are, respectively, the points \((2, 3, -2)\), \((2, 0, 4)\) and \((6, 1, 7)\). [3]
Pre-U Pre-U 9795/1 2015 June Q2
3 marks Standard +0.8
The Taylor series expansion, about \(x = 1\), of the function \(y\) is $$y = 1 + \sum_{n=1}^{\infty} \frac{(-2)^{n-1}(x-1)^n}{1 \times 3 \times 5 \times \ldots \times (2n-1)}.$$ Find the value of \(\frac{\text{d}^4 y}{\text{d}x^4}\) when \(x = 1\). [3]
Pre-U Pre-U 9795/1 2015 June Q3
6 marks Challenging +1.2
\(\mathbf{M}\) is the matrix \(\begin{pmatrix} 1 & -2 & 2 \\ 2 & -1 & 2 \\ 2 & -2 & 3 \end{pmatrix}\). Use induction to prove that, for all positive integers \(n\), $$\mathbf{M}^n \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2n + 1 \\ 2n^2 + 2n \\ 2n^2 + 2n + 1 \end{pmatrix}.$$ [6]
Pre-U Pre-U 9795/1 2015 June Q4
7 marks Challenging +1.2
A curve has polar equation \(r = \sin \frac{1}{4}\theta\) for \(0 \leqslant \theta < 2\pi\).
  1. Sketch the curve. [3]
  2. Determine the area of the region enclosed by the curve. [4]
Pre-U Pre-U 9795/1 2015 June Q5
11 marks Standard +0.8
A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).
  1. Determine the equations of the asymptotes. [3]
  2. Find the coordinates of the turning points. [5]
  3. Sketch the curve. [3]
Pre-U Pre-U 9795/1 2015 June Q6
9 marks Challenging +1.2
  1. Given the complex number \(z = \cos \theta + \text{i} \sin \theta\), show that \(z^n + \frac{1}{z^n} = 2 \cos n\theta\). [1]
  2. Deduce the identity \(16 \cos^5 \theta \equiv \cos 5\theta + 5 \cos 3\theta + 10 \cos \theta\). [4]
  3. For \(0 < \theta < 2\pi\), solve the equation \(\cos 5\theta + 5 \cos 3\theta + 9 \cos \theta = 0\). [4]
Pre-U Pre-U 9795/1 2015 June Q7
7 marks Standard +0.8
  1. On an Argand diagram, shade the region whose points satisfy $$|z - 20 + 15\text{i}| \leqslant 7.$$ [3]
  2. The complex number \(z_1\) represents that value of \(z\) in the region described in part (i) for which \(\arg(z)\) is least. Mark \(z_1\) on your Argand diagram and determine \(\arg(z_1)\) correct to 3 decimal places. [4]
Pre-U Pre-U 9795/1 2015 June Q8
9 marks Challenging +1.8
The group \(G\), of order 8, consists of the elements \(\{e, a, b, c, ab, bc, ca, abc\}\), together with a multiplicative binary operation, where \(e\) is the identity and $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb \quad \text{and} \quad ca = ac.$$
  1. Construct the group table of \(G\). [You are not required to show how individual elements of the table are determined.] [4]
  2. List all the proper subgroups of \(G\). [5]
Pre-U Pre-U 9795/1 2015 June Q9
9 marks Challenging +1.2
The differential equation \((\star)\) is $$\frac{\text{d}^2 u}{\text{d}x^2} + 4u = 8x + 1.$$
  1. Find the general solution of \((\star)\). [5]
  2. The differential equation \((\star \star)\) is $$x \frac{\text{d}^2 v}{\text{d}x^2} + 2 \frac{\text{d}v}{\text{d}x} + 4xv = 8x + 1.$$ By using the substitution \(u = xv\), show that \((\star)\) becomes \((\star \star)\) and deduce the general solution of \((\star \star)\). [4]
Pre-U Pre-U 9795/1 2015 June Q10
11 marks Standard +0.8
  1. Find a vector equation for the line of intersection of the planes with cartesian equations $$x + 7y - 6z = -10 \quad \text{and} \quad 3x - 5y + 8z = 48.$$ [5]
  2. Determine the value of \(k\) for which the system of equations \begin{align} x + 7y - 6z &= -10
    3x - 5y + 8z &= 48
    kx + 2y + 3z &= 16 \end{align} does not have a unique solution and show that, for this value of \(k\), the system of equations is inconsistent. [6]
Pre-U Pre-U 9795/1 2015 June Q11
13 marks Challenging +1.3
  1. The cubic equation \(x^3 + 2x^2 + 3x - 4 = 0\) has roots \(p\), \(q\) and \(r\). A second cubic equation has roots \(qr\), \(rp\) and \(pq\). Show how the substitution \(y = \frac{4}{x}\) can be used to determine this second equation. Hence, or otherwise, find this equation in the form \(y^3 + ay^2 + by + c = 0\). [6]
  2. The cubic equation \(x^3 - 4x^2 + 5x - 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha\) is real and positive, and that \(\beta\) and \(\gamma\) are complex.
    1. Describe the relationship between \(\beta\) and \(\gamma\). [1]
    2. Explain why \(|\beta| = \frac{2}{\sqrt{\alpha}}\). [2]
    3. Verify that \(\alpha = 2.70\) correct to 3 significant figures, and deduce that \(\text{Re}(\beta) = 0.65\) correct to 2 significant figures. [4]
Pre-U Pre-U 9795/1 2015 June Q12
22 marks Challenging +1.8
Let \(I_n = \int_0^2 x^n \sqrt{1 + 2x^2} \, \text{d}x\) for \(n = 0, 1, 2, 3, \ldots\).
    1. Evaluate \(I_1\). [3]
    2. Prove that, for \(n \geqslant 2\), $$(2n + 4)I_n = 27 \times 2^{n-1} - (n - 1)I_{n-2}.$$ [6]
    3. Using a suitable substitution, or otherwise, show that $$I_0 = 3 + \frac{1}{\sqrt{2}} \ln(1 + \sqrt{2}).$$ [8]
  2. The curve \(y = \frac{1}{\sqrt{2}} x^2\), between \(x = 0\) and \(x = 2\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface with area \(S\). Find the exact value of \(S\). [5]
Pre-U Pre-U 9795/1 2015 June Q13
10 marks Challenging +1.2
  1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
  2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
  3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]
Pre-U Pre-U 9795/1 2018 June Q1
5 marks Moderate -0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using the method of differences, prove that \(\sum_{r=1}^{n} \frac{3}{(3r-1)(3r+2)} = \frac{1}{2} - \frac{1}{3n+2}\). [2]
  3. Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(3r-1)(3r+2)}\). [1]
Pre-U Pre-U 9795/1 2018 June Q2
10 marks Standard +0.3
  1. Determine the asymptotes and turning points of the curve with equation \(y = \frac{x^2+3}{x+1}\). [7]
  2. Sketch the curve. [3]
Pre-U Pre-U 9795/1 2018 June Q3
7 marks Standard +0.3
The complex numbers \(z_1\) and \(z_2\) are such that \(|z_1| = 2\), \(\arg(z_1) = \frac{7}{12}\pi\), \(|z_2| = \sqrt{2}\) and \(\arg(z_2) = -\frac{1}{8}\pi\).
  1. Find, in exact form, the modulus and argument of \(\frac{z_1}{z_2}\). [3]
  2. Let \(z_3 = \left(\frac{z_1}{z_2}\right)^n\). It is given that \(n\) is the least positive integer for which \(z_3\) is a positive real number. Find this value of \(n\) and the exact value of \(z_3\). [4]
Pre-U Pre-U 9795/1 2018 June Q4
7 marks Challenging +1.2
A curve has polar equation \(r = \frac{3}{10}e^{3\theta}\) for \(\theta \geq 0\). The length of the arc of this curve between \(\theta = 0\) and \(\theta = \alpha\) is denoted by \(L(\alpha)\).
  1. Show that \(L(\alpha) = \frac{1}{3}(e^{3\alpha} - 1)\). [5]
  2. The point \(P\) on the curve corresponding to \(\theta = \beta\) is such that \(L(\beta) = OP\), where \(O\) is the pole. Find the value of \(\beta\). [2]
Pre-U Pre-U 9795/1 2018 June Q5
8 marks Standard +0.8
Find, in the form \(y = f(x)\), the solution of the differential equation \(\frac{dy}{dx} + y\tanh x = 2\cosh x\), given that \(y = \frac{3}{4}\) when \(x = \ln 2\). [8]
Pre-U Pre-U 9795/1 2018 June Q6
8 marks Challenging +1.8
The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).
  1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
  2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]
Pre-U Pre-U 9795/1 2018 June Q7
6 marks Challenging +1.2
The function \(y\) satisfies \(\frac{d^2y}{dx^2} + x^2y = x\), and is such that \(y = 1\) and \(\frac{dy}{dx} = 1\) when \(x = 1\).
  1. Using the given differential equation
    1. state the value of \(\frac{d^2y}{dx^2}\) when \(x = 1\), [1]
    2. find, by differentiation, the value of \(\frac{d^3y}{dx^3}\) when \(x = 1\). [2]
  2. Hence determine the Taylor series for \(y\) about \(x = 1\) up to and including the term in \((x-1)^3\) and deduce, correct to 4 decimal places, an approximation for \(y\) when \(x = 1.1\). [3]
Pre-U Pre-U 9795/1 2018 June Q8
8 marks Challenging +1.2
  1. Write down the values of the constants \(a\) and \(b\) for which \(m^3 = \frac{1}{6}m^3(am^2 + 2) - \frac{1}{12}m^2(bm)\). [1]
  2. Prove by induction that \(\sum_{r=1}^{n} r^5 = \frac{1}{6}n^3(n+1)^3 - \frac{1}{12}n^2(n+1)^2\) for all positive integers \(n\). [7]
Pre-U Pre-U 9795/1 2018 June Q9
8 marks Standard +0.3
  1. Use de Moivre's theorem to prove that \(\cos 3\theta = 4c^3 - 3c\), where \(c = \cos\theta\). [3]
  2. Solve the equation \(2\cos 3\theta - \sqrt{3} = 0\) for \(0 < \theta < \pi\), giving each answer in an exact form. [2]
  3. Deduce, in trigonometric form, the three roots of the equation \(x^3 - 3x - \sqrt{3} = 0\). [3]