Questions Pre-U 9794/1 (194 questions)

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Pre-U Pre-U 9794/1 2014 June Q3
3 marks Easy -1.2
3 Solve the inequality \(| 2 x - 1 | < 3\).
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.8
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q5
4 marks Easy -1.2
5 A root of the equation \(z ^ { 2 } + p z + q = 0\) is \(3 + \mathrm { i }\), where \(p\) and \(q\) are real. Write down the other root of the equation and hence calculate the values of \(p\) and \(q\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Standard +0.3
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-3_648_679_342_733}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2014 June Q7
4 marks Standard +0.3
7 Taking \(x = 2\) as a first approximation, use the Newton-Raphson process to find a root of the equation \(\frac { 1 } { x ^ { 2 } } - 0.119 - 0.018 x = 0\). Give your answer correct to 3 significant figures.
Pre-U Pre-U 9794/1 2014 June Q8
4 marks Standard +0.3
8 The parametric equations of a curve are given by $$x = \mathrm { e } ^ { t } - 2 t , \quad y = \mathrm { e } ^ { t } - 5 t$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Show that \(t = - \ln 2\) at the point on the curve where the gradient is 3 .
Pre-U Pre-U 9794/1 2014 June Q9
9 marks Moderate -0.3
9 It is given that \(x , 6\) and \(x + 5\) are consecutive terms of a geometric progression.
  1. Show that \(x ^ { 2 } + 5 x - 36 = 0\) and find the possible values of \(x\).
  2. Hence find the possible values of the common ratio. Furthermore, \(x , 6\) and \(x + 5\) are the second, third and fourth terms of a geometric progression for which the sum to infinity exists.
  3. Find the first term and the sum to infinity.
Pre-U Pre-U 9794/1 2014 June Q10
4 marks Moderate -0.3
10
  1. Show that \(\int _ { 0 } ^ { 2 } \frac { x } { x ^ { 2 } + 5 } \mathrm {~d} x = \ln \left( \frac { 3 } { \sqrt { 5 } } \right)\).
  2. Find \(\int x \sqrt { x - 2 } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2014 June Q11
11 marks Standard +0.3
11 A differential equation is given by \(2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ( 1 - y )\).
  1. Express \(\frac { 2 } { y ( 1 - y ) }\) in partial fractions.
  2. Hence show by integration that \(\frac { y ^ { 2 } } { ( 1 - y ) ^ { 2 } } = A \mathrm { e } ^ { x }\).
  3. Given that \(x = 0\) when \(y = 2\), find the value of \(A\) and express \(y\) in terms of \(x\).
Pre-U Pre-U 9794/1 2014 June Q12
10 marks Standard +0.8
12
  1. Use the identity \(\tan 2 x \equiv \frac { 2 \tan x } { 1 - \tan ^ { 2 } x }\) to show that \(\tan 4 x \equiv \frac { 4 \left( 1 - \tan ^ { 2 } x \right) \tan x } { 1 - 6 \tan ^ { 2 } x + \tan ^ { 4 } x }\).
  2. Hence, given that \(x = \frac { 1 } { 16 } \pi\) is a root of the equation \(\tan ^ { 4 } x + p \tan ^ { 3 } x - 6 \tan ^ { 2 } x - p \tan x + 1 = 0\) where \(p\) is a positive constant, find the value of \(p\).
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.8
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Moderate -0.3
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2014 June Q4
2 marks Moderate -0.3
4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-2_949_1127_1041_507} Draw the graphs of
  1. \(\mathrm { f } ( x + 2 ) + 1\),
  2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2014 June Q6
7 marks Moderate -0.8
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-3_648_684_342_731}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2015 June Q1
3 marks Easy -1.2
1 Find the set of values of \(x\) for which \(x ^ { 2 } - x - 12 < 0\).
Pre-U Pre-U 9794/1 2015 June Q2
5 marks Moderate -0.8
2 Solve the following simultaneous equations. $$x ^ { 2 } + 2 y ^ { 2 } = 36 \quad x + 2 y = 10$$
Pre-U Pre-U 9794/1 2015 June Q3
3 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_305_825_630_660} The diagram shows a triangle \(A B C\) in which angle \(B = 39 ^ { \circ }\), angle \(C = 28 ^ { \circ } , A B = x \mathrm {~cm}\) and \(A C = ( 2 x - 1 ) \mathrm { cm }\). Find the value of \(x\).
Pre-U Pre-U 9794/1 2015 June Q4
6 marks Moderate -0.8
4 A population, \(P\), is modelled by the equation \(P = a \mathrm { e } ^ { b t }\) where \(t\) is time in years, and \(a\) and \(b\) are constants.
  1. By considering logarithms, show that a graph of \(\ln P\) against \(t\) is a straight line. State the intercept on the vertical axis and the gradient.
  2. Use the graph below to obtain values for \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_657_750_1530_740}
Pre-U Pre-U 9794/1 2015 June Q5
9 marks Moderate -0.8
5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).
  1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
  2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
  3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).
Pre-U Pre-U 9794/1 2015 June Q6
6 marks Moderate -0.3
6 The functions f and g are given by \(\mathrm { f } ( x ) = \frac { 3 } { x - 1 }\) for all \(x \neq 1\) and \(\mathrm { g } ( x ) = x + 2\) for all real \(x\).
  1. Find gf, stating its domain and range.
  2. Find \(( \mathrm { gf } ) ^ { - 1 }\), stating any values of \(x\) for which \(( \mathrm { gf } ) ^ { - 1 }\) is not defined.
Pre-U Pre-U 9794/1 2015 June Q7
9 marks Standard +0.3
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following vector equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$
  1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of their point of intersection.
  2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
Pre-U Pre-U 9794/1 2015 June Q8
11 marks Moderate -0.3
8 The complex numbers \(w\) and \(z\) are given by \(w = 3 - \mathrm { i }\) and \(z = 1 + \mathrm { i }\).
  1. Express \(\frac { z } { w }\) in the form \(p + \mathrm { i } q\) where \(p\) and \(q\) are real numbers.
  2. On the same Argand diagram, mark the points representing \(z , w\) and \(\frac { z } { w }\).
  3. Find the value in radians of \(\arg w\).
  4. Show that \(z + \frac { 2 } { z }\) is a real number.
Pre-U Pre-U 9794/1 2015 June Q9
7 marks Standard +0.3
9 A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x }\). Find the exact coordinates of the stationary points of the curve.
Pre-U Pre-U 9794/1 2015 June Q10
11 marks Standard +0.3
10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
  2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).
Pre-U Pre-U 9794/1 2015 June Q11
10 marks Standard +0.8
11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that $$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$ where \(A\) is an arbitrary constant and \(x > 0\).