Questions Pre-U 9794/1 (194 questions)

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Pre-U Pre-U 9794/1 2012 June Q1
5 marks Easy -1.2
1 The first term of a geometric progression is 16 and the common ratio is 0.8 .
  1. Calculate the sum of the first 12 terms.
  2. Find the sum to infinity.
Pre-U Pre-U 9794/1 2012 June Q2
6 marks Moderate -0.8
2 Let \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\).
  1. Show that \(\mathrm { f } ( 1 ) = 0\) and hence factorise \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\) completely.
  2. Hence solve the equation \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0\).
Pre-U Pre-U 9794/1 2012 June Q3
6 marks Moderate -0.8
3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the stationary points on the curve.
Pre-U Pre-U 9794/1 2012 June Q4
5 marks Moderate -0.8
4
  1. Show that the equation \(x ^ { 3 } - 6 x + 2 = 0\) has a root between \(x = 0\) and \(x = 1\).
  2. Use the iterative formula \(x _ { n + 1 } = \frac { 2 + x _ { n } ^ { 3 } } { 6 }\) with \(x _ { 0 } = 0.5\) to find this root correct to 4 decimal places, showing the result of each iteration.
Pre-U Pre-U 9794/1 2012 June Q5
5 marks Easy -1.3
5 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).
  1. Give a reason why f has no inverse function.
  2. Write down an expression for \(\mathrm { gf } ( x )\).
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
Pre-U Pre-U 9794/1 2012 June Q6
5 marks Moderate -0.8
6 The roots of the equation \(z ^ { 2 } - 6 z + 10 = 0\) are \(z _ { 1 }\) and \(z _ { 2 }\), where \(z _ { 1 } = 3 + \mathrm { i }\).
  1. Write down the value of \(z _ { 2 }\).
  2. Show \(z _ { 1 }\) and \(z _ { 2 }\) on an Argand diagram.
  3. Show that \(z _ { 1 } ^ { 2 } = 8 + 6 \mathrm { i }\).
Pre-U Pre-U 9794/1 2012 June Q7
9 marks Moderate -0.3
7
  1. Show that the first three terms in the expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(1 - x - \frac { 1 } { 2 } x ^ { 2 }\) and find the next term.
  2. State the range of values of \(x\) for which this expansion is valid.
  3. Hence show that the first four terms in the expansion of \(( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(2 - x + a x ^ { 2 } + b x ^ { 3 }\) and state the values of \(a\) and \(b\).
Pre-U Pre-U 9794/1 2012 June Q8
9 marks Moderate -0.3
8
  1. Given that \(\frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \equiv \frac { A } { 2 x + 1 } + \frac { B } { x + 3 }\), find the values of the constants \(A\) and \(B\).
  2. Hence show that \(\int _ { 0 } ^ { 2 } \frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \mathrm { d } x = \ln 15\).
Pre-U Pre-U 9794/1 2012 June Q9
10 marks Standard +0.3
9 Three points \(A , B\) and \(C\) have coordinates \(( 1,0,7 ) , ( 13,9,1 )\) and \(( 2 , - 1 , - 7 )\) respectively.
  1. Use a scalar product to find angle \(A C B\).
  2. Hence find the area of triangle \(A C B\).
  3. Show that a vector equation of the line \(A B\) is given by \(\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\), where \(\lambda\) is a scalar parameter.
Pre-U Pre-U 9794/1 2012 June Q10
9 marks Standard +0.3
10
  1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
  2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
Pre-U Pre-U 9794/1 2012 June Q11
11 marks Standard +0.3
11
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
  2. Find
    1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
    2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2012 Specimen Q1
2 marks Easy -1.8
1 Write down the coordinates of the centre and the radius of the circle with equation $$( x + 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 36 .$$
Pre-U Pre-U 9794/1 2012 Specimen Q2
5 marks Moderate -0.8
2
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2012 Specimen Q3
6 marks Easy -1.2
3
  1. In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
  2. A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.
Pre-U Pre-U 9794/1 2012 Specimen Q4
7 marks Moderate -0.3
4 A sector \(A O B\) of a circle has radius \(r \mathrm {~cm}\) and the angle \(A O B\) is \(\theta\) radians. The perimeter of the sector is 40 cm and its area is \(100 \mathrm {~cm} ^ { 2 }\).
  1. Write down equations for the perimeter and area of the sector in terms of \(r\) and \(\theta\).
  2. Use your equations to show that \(r ^ { 2 } - 20 r + 100 = 0\) and hence find the value of \(r\) and of \(\theta\).
Pre-U Pre-U 9794/1 2012 Specimen Q5
8 marks Moderate -0.3
5
  1. Find \(\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\) giving your answer in its simplest form.
  2. Use integration by parts to find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2012 Specimen Q6
7 marks Moderate -0.3
6
  1. Find and simplify the first four terms in the expansion of \(( 1 - 2 x ) ^ { 9 }\) in ascending powers of \(x\).
  2. In the expansion of $$( 2 + a x ) ( 1 - 2 x ) ^ { 9 }$$ the coefficient of \(x ^ { 2 }\) is 66 . Find the value of \(a\).
Pre-U Pre-U 9794/1 2012 Specimen Q7
4 marks Standard +0.3
7 Given that the equation \(x = 2 - \frac { 1 } { ( x + 1 ) ^ { 2 } }\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { 0 } = 2\) to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2012 Specimen Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(x\)-coordinates of the stationary points of the curve.
  3. By using the second derivative, determine whether each of the stationary points is a maximum or a minimum.
Pre-U Pre-U 9794/1 2012 Specimen Q9
10 marks Moderate -0.3
9
  1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
  2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
Pre-U Pre-U 9794/1 2012 Specimen Q10
6 marks Moderate -0.3
10 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).
  1. Find the length of \(A B\).
  2. Use a scalar product to find angle \(O A B\).
Pre-U Pre-U 9794/1 2012 Specimen Q11
6 marks Standard +0.3
11 Solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y\) given that \(y = \frac { \pi } { 6 }\) when \(x = 4\) giving your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2012 Specimen Q12
11 marks Challenging +1.2
12 Calculate the maximum and minimum values of \(\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }\) and the smallest positive values of \(\theta\) for which they occur.
Pre-U Pre-U 9794/1 2013 June Q2
4 marks Easy -1.2
2 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).
Pre-U Pre-U 9794/1 2013 June Q3
5 marks Moderate -0.8
3 A sector, \(P O Q\), of a circle centre \(O\) has radius 7 cm and angle 1.7 radians (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{41d9ff74-82de-4ac5-928f-f6ab008319d2-2_469_723_662_712}
  1. Find the length of the line \(P Q\).
  2. Hence find the perimeter of the shaded area.