Questions — Pre-U Pre-U 9795 (27 questions)

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Pre-U Pre-U 9795 Specimen Q1
Standard +0.3
1 The \(n\)th triangular number, \(T _ { n }\), is given by the formula \(T _ { n } = \frac { 1 } { 2 } n ( n + 1 )\).
  1. Express \(\frac { 1 } { T _ { n } }\) in terms of partial fractions.
  2. Hence, using the method of differences, show that \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).
Pre-U Pre-U 9795 Specimen Q2
Moderate -0.3
2
  1. On a single Argand diagram, sketch and clearly label each of the following loci:
    1. \(| z | = 4\),
    2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
    3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$
Pre-U Pre-U 9795 Specimen Q3
Standard +0.3
3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.
Pre-U Pre-U 9795 Specimen Q4
Standard +0.8
4 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Use de Moivre's theorem to prove that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
  2. Deduce the identity \(\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )\).
Pre-U Pre-U 9795 Specimen Q5
Standard +0.3
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 72 \mathrm { e } ^ { 3 x }$$
Pre-U Pre-U 9795 Specimen Q6
Challenging +1.2
6
  1. Given that \(y = \cos ( \ln ( 1 + x ) )\), prove that
    1. \(\quad ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = - \sin ( \ln ( 1 + x ) )\),
    2. \(( 1 + x ) ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + y = 0\).
    3. Obtain an equation relating \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    4. Hence find the Maclaurin series for \(y\), up to and including the term in \(x ^ { 3 }\).
Pre-U Pre-U 9795 Specimen Q7
Standard +0.3
7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$
  1. Find the equations of the asymptotes of \(C\).
  2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.
Pre-U Pre-U 9795 Specimen Q8
Challenging +1.2
8 The equation \(8 x ^ { 3 } + 12 x ^ { 2 } + 4 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. By considering a suitable substitution, or otherwise, show that the equation whose roots are \(2 \alpha + 1,2 \beta + 1,2 \gamma + 1\) can be written in the form $$y ^ { 3 } - y - 1 = 0 .$$
  2. The sum \(( 2 \alpha + 1 ) ^ { n } + ( 2 \beta + 1 ) ^ { n } + ( 2 \gamma + 1 ) ^ { n }\) is denoted by \(S _ { n }\). Evaluate \(S _ { 3 }\) and \(S _ { - 2 }\).
Pre-U Pre-U 9795 Specimen Q9
Standard +0.8
9
  1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)\), and hence solve the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 10 \\ 3 x + 8 y + 2 z & = 8 \end{aligned}$$
  2. Find the value of \(k\) for which the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 15 \\ 3 x + 8 y + 3 z & = k \end{aligned}$$ is consistent. Find the solution in this case and interpret it geometrically.
Pre-U Pre-U 9795 Specimen Q10
Challenging +1.8
10 A group \(G\) has distinct elements \(e , a , b , c , \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation.
  1. Prove that if \(a \circ a = b\) and \(b \circ b = a\), then the set of elements \(\{ e , a , b \}\) forms a subgroup of \(G\).
  2. Prove that if \(a \circ a = b , b \circ b = c\) and \(c \circ c = a\), then the set of elements \(\{ e , a , b , c \}\) does not form a subgroup of \(G\).
Pre-U Pre-U 9795 Specimen Q11
Challenging +1.2
11 With respect to an origin \(O\), the points \(A , B , C\) and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad \mathbf { b } = \mathbf { i } - 2 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { d } = - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ,$$ respectively. Find
  1. a vector perpendicular to the plane \(O A B\),
  2. the acute angle between the planes \(O A B\) and \(O C D\), correct to the nearest \(0.1 ^ { \circ }\),
  3. the shortest distance between the line \(A B\) and the line \(C D\),
  4. the perpendicular distance from the point \(A\) to the line \(C D\).
Pre-U Pre-U 9795 Specimen Q12
Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{0f5edc87-cb14-4583-a54d-badec47741d1-08_414_659_804_744} The diagram shows a sketch of the curve \(C\) with polar equation \(r = 4 \cos ^ { 2 } \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\).
  2. The equation of \(C\) may be expressed in the form \(r = k ( 1 + \cos 2 \theta )\). State the value of \(k\) and use this form to show that the area of the region enclosed by \(C\) is given by $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } ( 3 + 4 \cos 2 \theta + \cos 4 \theta ) d \theta ,$$ and hence find this area.
  3. The length of \(C\) is denoted by \(L\). Show that $$L = 8 \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos \theta \sqrt { 1 + 3 \sin ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(\sinh x = \sqrt { 3 } \sin \theta\) to determine \(L\) in an exact form.
Pre-U Pre-U 9795 Specimen Q13
Challenging +1.8
13 Let \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\), where \(n\) is a positive integer.
  1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ( \ln x ) ^ { n } \right)\), or otherwise, show that \(I _ { n } = \mathrm { e } - n I _ { n - 1 }\).
  2. Let \(J _ { n } = \frac { I _ { n } } { n ! }\). Prove by induction that $$\sum _ { r = 2 } ^ { n } \frac { ( - 1 ) ^ { r } } { r ! } = \frac { 1 } { \mathrm { e } } \left( 1 + ( - 1 ) ^ { n } J _ { n } \right)$$ for all positive integers \(n \geqslant 2\).
Pre-U Pre-U 9795 Specimen Q1
4 marks Standard +0.3
The region \(R\) of an Argand diagram is defined by the inequalities $$0 \leqslant \arg(z + 4\mathrm{i}) \leqslant \frac{1}{4}\pi \quad \text{and} \quad |z| \leqslant 4.$$ Draw a clearly labelled diagram to illustrate \(R\). [4]
Pre-U Pre-U 9795 Specimen Q2
6 marks Challenging +1.2
It is given that $$\mathrm{f}(n) = 7^n (6n + 1) - 1.$$ By considering \(\mathrm{f}(n + 1) - \mathrm{f}(n)\), prove by induction that \(\mathrm{f}(n)\) is divisible by 12 for all positive integers \(n\). [6]
Pre-U Pre-U 9795 Specimen Q3
6 marks Standard +0.3
Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]
Pre-U Pre-U 9795 Specimen Q4
6 marks Standard +0.3
Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]
Pre-U Pre-U 9795 Specimen Q5
7 marks Standard +0.8
Find the general solution of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 6\frac{\mathrm{d}y}{\mathrm{d}x} + 9y = 72\mathrm{e}^{3x}.$$ [7]
Pre-U Pre-U 9795 Specimen Q6
8 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows a sketch of the curve \(C\) with polar equation \(r = a \cos^2 \theta\), where \(a\) is a positive constant and \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\). [2]
  2. The equation of \(C\) may be expressed in the form \(r = \frac{1}{2}a(1 + \cos 2\theta)\). Using this form, show that the area of the region enclosed by \(C\) is given by $$\frac{1}{16}a^2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (3 + 4 \cos 2\theta + \cos 4\theta) \, \mathrm{d}\theta,$$ and find this area. [6]
Pre-U Pre-U 9795 Specimen Q7
8 marks Challenging +1.2
The equation $$8x^3 + 12x^2 + 4x - 1 = 0$$ has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(2\alpha + 1, 2\beta + 1, 2\gamma + 1\) is $$y^3 - y - 1 = 0.$$ [3] The sum \((2\alpha + 1)^n + (2\beta + 1)^n + (2\gamma + 1)^n\) is denoted by \(S_n\). Find the values of \(S_3\) and \(S_{-2}\). [5]
Pre-U Pre-U 9795 Specimen Q8
9 marks Standard +0.3
The curve \(C\) has equation $$y = \frac{x^2 - 2x - 3}{x + 2}.$$
  1. Find the equations of the asymptotes of \(C\). [4]
  2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]
Pre-U Pre-U 9795 Specimen Q9
9 marks Challenging +1.3
Given that \(w_n = 3^{-n} \cos 2n\theta\) for \(n = 1, 2, 3, \ldots\), use de Moivre's theorem to show that $$1 + w_1 + w_2 + w_3 + \ldots + w_{N-1} = \frac{9 - 3\cos 2\theta + 3^{-N+1} \cos 2(N-1)\theta - 3^{-N+2} \cos 2N\theta}{10 - 6\cos 2\theta}.$$ [7] Hence show that the infinite series $$1 + w_1 + w_2 + w_3 + \ldots$$ is convergent for all values of \(\theta\), and find the sum to infinity. [2]
Pre-U Pre-U 9795 Specimen Q10
10 marks Standard +0.3
  1. Find the inverse of the matrix \(\begin{pmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & 2 \end{pmatrix}\), and hence solve the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 10,
    3x + 8y + 2z &= 8. \end{align} [5]
  2. Find the value of \(k\) for which the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 15,
    3x + 8y + 3z &= k, \end{align} is consistent. Find the solution in this case and interpret it geometrically. [5]
Pre-U Pre-U 9795 Specimen Q11
10 marks Challenging +1.8
A group \(G\) has distinct elements \(e, a, b, c, \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation. Prove that if $$a \circ a = b, \quad b \circ b = a$$ then the set of elements \(\{e, a, b\}\) forms a subgroup of \(G\). [5] Prove that if $$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$ then the set of elements \(\{e, a, b, c\}\) does not form a subgroup of \(G\). [5]
Pre-U Pre-U 9795 Specimen Q12
12 marks Standard +0.3
With respect to an origin \(O\), the points \(A, B, C, D\) have position vectors $$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$ respectively. Find
  1. a vector perpendicular to the plane \(OAB\), [2]
  2. the acute angle between the planes \(OAB\) and \(OCD\), correct to the nearest \(0.1°\), [3]
  3. the shortest distance between the line which passes through \(A\) and \(B\) and the line which passes through \(C\) and \(D\), [4]
  4. the perpendicular distance from the point \(A\) to the line which passes through \(C\) and \(D\). [3]