Questions — Pre-U (885 questions)

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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 9794/1 Specimen Q1
3 marks Moderate -0.8
1 It is given that \(8 ^ { 4 x } = 4 ^ { 3 x - 6 }\).
  1. By expressing each side as a power of 2 , find the value of \(x\).
  2. Write down the value of \(\log _ { 4 } | x |\).
Pre-U Pre-U 9794/1 Specimen Q2
4 marks Moderate -0.8
2 Find the value of the positive constant \(k\) for which \(\int _ { 1 } ^ { k } ( 2 x - 1 ) \mathrm { d } x = 6\).
Pre-U Pre-U 9794/1 Specimen Q3
5 marks Moderate -0.3
3
  1. Find the value of \(a\) for which ( \(x - 2\) ) is a factor of \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8\).
  2. Show that, for this value of \(a\), the cubic equation \(5 x ^ { 3 } + a x ^ { 2 } + 6 a x - 8 = 0\) has only one real root.
Pre-U Pre-U 9794/1 Specimen Q4
4 marks Standard +0.8
4
  1. Sketch the graph of \(y = \sqrt { 2 } \sin x\) for \(0 \leqslant x \leqslant 2 \pi\). The points \(P\) and \(Q\) on the graph have \(x\)-coordinates \(\frac { 1 } { 4 } \pi\) and \(\frac { 3 } { 4 } \pi\), respectively.
  2. Determine the equation of the tangent to the curve at \(P\). The normals to the curve at \(P\) and \(Q\) intersect at the point \(R\).
  3. Determine the exact coordinates of \(R\).
Pre-U Pre-U 9794/1 Specimen Q5
10 marks Moderate -0.3
5 The complex number \(z\) satisfies the equation \(2 z - \mathrm { i } = \mathrm { i } z + 2\).
  1. Express \(z\) in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are rational numbers.
  2. Find the exact value of \(| z |\) and of \(\arg ( z )\).
  3. Express \(z ^ { 2 }\) in the form \(c + \mathrm { i } d\) where \(c\) and \(d\) are rational numbers.
  4. Verify that \(\tan ( 2 \arg ( z ) ) = \tan \left( \arg \left( z ^ { 2 } \right) \right)\) using an appropriate trigonometrical identity.
Pre-U Pre-U 9794/1 Specimen Q6
9 marks Standard +0.3
6
  1. (a) Using the substitution \(u = \frac { 1 } { 2 } \pi - x\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 2 } x \mathrm {~d} x = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { 2 } u \mathrm {~d} u$$ (b) Hence find the common value of these definite integrals.
  2. Find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 2 } x \mathrm {~d} x$$
Pre-U Pre-U 9794/1 Specimen Q7
11 marks Standard +0.3
7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, where \(\mathbf { a } = 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }\) and \(\mathbf { b } = - \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the vector equations $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } , \quad \mathbf { r } = 2 \mathbf { b } + \mu \mathbf { a }$$ respectively.
  1. Determine whether or not \(L _ { 1 }\) and \(L _ { 2 }\) intersect.
  2. Find the acute angle between the directions of \(L _ { 1 }\) and \(L _ { 2 }\). The point \(C\) has position vector \(\mathbf { c } = p \mathbf { i } + \mathbf { j } + r \mathbf { k }\).
  3. Given that \(O C\) is perpendicular to the triangle \(O A B\), determine \(p\) and \(r\).
  4. Determine the volume of the tetrahedron \(O A B C\).
Pre-U Pre-U 9794/1 Specimen Q8
14 marks Standard +0.3
8
  1. The sum of the first \(n\) terms of the arithmetic series \(1 + 3 + 5 + \ldots\) exceeds the sum of the first \(n\) terms of the arithmetic series \(100 + 97 + 94 + \ldots\). Find the least possible value of \(n\).
  2. \(3 \sqrt { 2 }\) and \(2 - \sqrt { 2 }\) are the first two terms of a geometric progression.
    1. Show that the third term is \(\sqrt { 2 } - \frac { 4 } { 3 }\).
    2. Find the index \(n\) of the first term that is less than 0.01.
    3. Find the exact value of the sum to infinity of this progression.
    4. Which of the terms 'alternating', 'periodic', 'convergent' apply to the sequences generated by the following \(n\)th terms, where \(n\) is a positive integer?
      (a) \(1 - \left( \frac { 3 } { 4 } \right) ^ { n }\) (b) \(\frac { 1 } { n } \cos n \pi\) (c) \(\sec n \pi\)
Pre-U Pre-U 9794/1 Specimen Q9
16 marks Challenging +1.8
9 The cubic polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are real, is denoted by \(\mathrm { p } ( x )\).
  1. Give a reason why the equation \(\mathrm { p } ( x ) = 0\) has at least one real root.
  2. Suppose that the curve with equation \(y = \mathrm { p } ( x )\) has a local minimum point and a local maximum point with \(y\)-coordinates \(y _ { \text {min } }\) and \(y _ { \text {max } }\) respectively.
    1. Prove that if \(y _ { \text {min } } y _ { \text {max } } < 0\), then the equation \(\mathrm { p } ( x ) = 0\) has three real roots.
    2. Comment on the number of distinct real roots of the equation \(\mathrm { p } ( x ) = 0\) in the case \(y _ { \text {min } } y _ { \text {max } } = 0\).
    3. Suppose instead that the equation \(\mathrm { p } ( x ) = 0\) has only one real root for all values of \(c\). Prove that \(a ^ { 2 } \leqslant 3 b\).
    4. The iterative scheme $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 1 } { 3 x _ { n } ^ { 2 } + 1 } , \quad x _ { 0 } = 0$$ converges to a root of the cubic equation \(\mathrm { p } ( x ) = 0\).
      (a) Find \(\mathrm { p } ( x )\).
      (b) Find the limit of the iteration, correct to 4 decimal places.
    5. Determine the rate of convergence of the iterative scheme.
Pre-U Pre-U 9794/1 Specimen Q10
9 marks Standard +0.3
10 Cheeky Cola is sold in bottles of two sizes, small and large. For each size, the content of a randomly chosen bottle is normally distributed with mean and standard deviation, in litres, as given in the table.
MeanStandard deviation
Small bottle0.50.01
Large bottle1.5\(x\)
  1. Find the probability that a randomly chosen small bottle contains more than 0.51 litres.
  2. Find \(x\) if the probability that a randomly chosen large bottle contains less than 1.45 litres is 0.1 . The manufacturer introduces a new size of bottle of Cheeky Cola, called the mega bottle. It is found that the probabilities that a randomly chosen mega bottle contains less than 2.97 litres or more than 3.05 litres are both 0.05 .
  3. Assuming that the contents of the mega bottle are normally distributed, find the mean and variance of the distribution.
Pre-U Pre-U 9794/1 Specimen Q11
11 marks Standard +0.8
11 During the 30-day month of April, the probability that it will rain on any given day is 0.25 .
  1. Find the probability that the first rainy day in April is the 7th of April, explaining any modelling assumptions you have made.
  2. Given that it does not rain on the first 6 days of April, find the probability that it first rains on the 10th of April.
  3. The probability that it first rains on the \(n\)th of April and next on the ( \(n + 3\) )th of April is 0.02 , correct to 1 significant figure. Determine \(n\).
  4. Determine the expected number of dry days in April, given that it first rains on the 8th of April.
Pre-U Pre-U 9794/1 Specimen Q12
10 marks Standard +0.3
12 A faulty random number generator generates odd digits according to the probability distribution for the random variable \(X\) given in the following table.
\(x\)13579
\(\mathrm { P } ( X ) = x\)0.3\(p\)0.2\(2 p\)0.2
  1. Find \(p\).
  2. Find \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\).
  3. Deduce the value of \(\operatorname { Var } ( X )\). A second random number generator generates odd digits each with equal probability. Both random generators are operated once.
  4. Find the probability that both generate a prime number.
  5. Given that the first generates 1, 3 or 5, find the probability that both generate a power of 3 . 1315 pupils, including two sisters, are placed in random order in a line.
  6. What is the probability that the sisters are not next to each other?
  7. How many arrangements are there with 9 pupils between the sisters? A team of 5 is chosen from the 15 pupils.
  8. How many ways are there of choosing the team if no more than one of the sisters can be in the team? Having chosen the first team, a second team of 5 pupils is chosen from the remaining 10 pupils.
  9. How many ways are there of choosing the teams if each sister is in one or other of the teams?
Pre-U Pre-U 9795/1 Specimen Q2
7 marks Standard +0.3
2
  1. Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find
  2. the sum to \(N\) terms,
  3. the sum to infinity.
Pre-U Pre-U 9795/1 Specimen Q3
5 marks Challenging +1.2
3 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\).
  1. Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { 1 + x ^ { 4 } } \mathrm {~d} x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\).
  2. Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { 2 } - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]
Pre-U Pre-U 9795/1 Specimen Q4
14 marks Standard +0.3
4
  1. Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  2. On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\).
  3. Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.
Pre-U Pre-U 9795/1 Specimen Q5
6 marks Challenging +1.2
5 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\).
  1. Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
  2. Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$
Pre-U Pre-U 9795/1 Specimen Q6
9 marks Standard +0.8
6
  1. Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73 .$$
  2. Show that, whatever the initial conditions, \(\frac { y } { x ^ { 2 } } \rightarrow 1\) as \(x \rightarrow \infty\).