Questions Pre-U 9794/1 (194 questions)

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Pre-U Pre-U 9794/1 Specimen Q15
12 marks Standard +0.8
15 In order to be accepted on a university course, a student needs to pass three exams.
The probability that the student passes the first exam is \(\frac { 3 } { 4 }\).
For each of the second and third exams, the probability of passing the exam is
  • the same as the probability of passing the preceding exam if the student passed the preceding exam,
  • half of the probability of passing the preceding exam if the student failed the preceding exam.
    1. Draw a tree diagram to represent the above information.
    2. Find the probability that the student passes all three exams.
    3. Find the probability that the student passes at least two of the exams.
    4. Find the probability that the student passes the third exam given that exactly two of the three exams are passed.
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 9794/1 2010 June Q1
3 marks Easy -1.2
Solve the equation \(2^x = 4^{2x+1}\). [3]
Pre-U Pre-U 9794/1 2010 June Q2
3 marks Standard +0.3
The equation \(x^3 - 5x + 3 = 0\) has a root between \(x = 0\) and \(x = 1\).
  1. The equation can be rearranged into the form \(x = g(x)\) where \(g(x) = px^3 + q\). State the values of \(p\) and \(q\). [1]
  2. By considering \(|g'(x)|\), show that the iterative form \(x_{n+1} = g(x_n)\) with a suitable starting value converges to the root between \(x = 0\) and \(x = 1\). [You are not required to find this root.] [2]
Pre-U Pre-U 9794/1 2010 June Q3
6 marks Moderate -0.3
Let \(f(x) = x^2(x - 2)\) and \(g(x) = 2x - 1\) for all real \(x\).
  1. Sketch the graph of \(y = f(x)\) and explain briefly why the function f has no inverse. [2]
  2. Write down \(g^{-1}(x)\). [1]
  3. On the same diagram, sketch the graphs of \(y = f(x - 1) - 3\) and \(y = g^{-1}(x)\) and state the number of real roots of the equation \(f(x - 1) - 3 = g^{-1}(x)\). [3]
Pre-U Pre-U 9794/1 2010 June Q4
5 marks Moderate -0.3
Using the substitution \(u = 1 + \sqrt{x}\), or otherwise, find \(\int \frac{1}{1 + \sqrt{x}} dx\) giving your answer in terms of \(x\). [5]
Pre-U Pre-U 9794/1 2010 June Q5
7 marks Standard +0.3
The parametric equations of a curve are \(x = \frac{1}{1 + t^2}\) and \(y = \frac{t}{1 + t^2}\), \(t \in \mathbb{R}\).
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [5]
  2. Hence find the coordinates of the stationary points of the curve. [2]
Pre-U Pre-U 9794/1 2010 June Q6
7 marks Standard +0.3
A geometric progression with common ratio \(r\) consists of positive terms. The sum of the first four terms is five times the sum of the first two terms.
  1. Find an equation in \(r\) and deduce that \(r = 2\). [3]
  2. Given that the fifth term is 192, find the value of the first term. [1]
  3. Find the smallest value of \(n\) such that the sum of the first \(n\) terms of the progression exceeds \(10^{64}\). [3]
Pre-U Pre-U 9794/1 2010 June Q7
9 marks Standard +0.3
Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)
  1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
  2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
  3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]
Pre-U Pre-U 9794/1 2010 June Q8
9 marks Standard +0.3
The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.
  1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
  2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]
Pre-U Pre-U 9794/1 2010 June Q9
9 marks Standard +0.3
A curve has equation \(x^2 - xy + y^2 = 1\).
  1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
  2. Find the coordinates of the points on the curve in the second and fourth quadrants where the tangent is parallel to \(y = x\). [5]
Pre-U Pre-U 9794/1 2010 June Q10
10 marks Standard +0.3
  1. Solve the equation \((2 + i)z = (4 + in)\). Give your answer in the form \(a + ib\), expressing \(a\) and \(b\) in terms of the real constant \(n\). [4]
  2. The roots of the equation \(z^2 + 8z + 25 = 0\) are denoted by \(z_1\) and \(z_2\).
    1. Find \(z_1\) and \(z_2\) and show these roots on an Argand diagram. [3]
    2. Find the modulus and argument in radians of each of \((z_1 + 1)\) and \((z_2 + 1)\). [3]
Pre-U Pre-U 9794/1 2010 June Q11
11 marks Challenging +1.2
  1. Write down an identity for \(\tan 2\theta\) in terms of \(\tan \theta\) and use this result to show that $$\tan 3\theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Given that \(0 < \theta < \frac{1}{2}\pi\) and \(\theta = \sin^{-1}\left(\frac{1}{\sqrt{10}}\right)\), show that \(\tan 3\theta = \frac{13}{3}\). [3]
  3. Show that the solutions of the equation $$\tan(3 \sin^{-1} x) = \frac{13}{3}$$ for \(0 < x < 2\pi\) are $$x = \frac{\sqrt{10}}{10} \quad \text{and} \quad x = \frac{\sqrt{10(1 + 3\sqrt{3})}}{20}.$$ [4]
Pre-U Pre-U 9794/1 2010 June Q12
7 marks Moderate -0.3
  1. Events \(A\) and \(B\) are such that \(\mathrm{P}(A' \cap B') = \frac{1}{6}\).
    1. Find \(\mathrm{P}(A \cup B)\). [2]
    2. Given that \(\mathrm{P}(A | B) = \frac{1}{4}\) and \(\mathrm{P}(B) = \frac{1}{3}\), find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A)\). [3]
  2. In playing the UK Lottery, a set of 6 different integers is chosen irrespective of order from the integers 1 to 49 inclusive. How many different sets of 6 integers can be chosen? [2]