Questions — OCR (4628 questions)

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OCR C1 2012 June Q1
3 marks Easy -1.2
1 Simplify \(( x - 5 ) \left( x ^ { 2 } + 3 \right) - ( x + 4 ) ( x - 1 )\).
OCR C1 2012 June Q2
5 marks
2 Express each of the following in the form \(7 ^ { k }\) :
  1. \(\sqrt [ 4 ] { 7 }\),
  2. \(\frac { 1 } { 7 \sqrt { 7 } }\),
  3. \(7 ^ { 4 } \times 49 ^ { 10 }\).
OCR C1 2012 June Q3
5 marks Easy -1.3
3
  1. Find the gradient of the line \(l\) which has equation \(3 x - 5 y - 20 = 0\).
  2. The line \(l\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
OCR C1 2012 June Q4
6 marks Moderate -0.8
4
  1. Express \(2 x ^ { 2 } - 20 x + 49\) in the form \(p ( x - q ) ^ { 2 } + r\).
  2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 20 x + 49\).
OCR C1 2012 June Q5
6 marks Moderate -0.8
5
  1. Sketch the curve \(y = \sqrt { x }\).
  2. Describe the transformation that transforms the curve \(y = \sqrt { x }\) to the curve \(y = \sqrt { x - 4 }\).
  3. The curve \(y = \sqrt { x }\) is stretched by a scale factor of 5 parallel to the \(x\)-axis. State the equation of the transformed curve.
OCR C1 2012 June Q6
7 marks Moderate -0.3
6 Find the equation of the normal to the curve \(y = \frac { 6 } { x ^ { 2 } } - 5\) at the point on the curve where \(x = 2\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
OCR C1 2012 June Q7
6 marks Standard +0.3
7 Solve the equation \(x - 6 x ^ { \frac { 1 } { 2 } } + 2 = 0\), giving your answers in the form \(p \pm q \sqrt { r }\), where \(p , q\) and \(r\) are integers.
OCR C1 2012 June Q8
8 marks Moderate -0.8
8
  1. Find the coordinates of the stationary point on the curve \(y = x ^ { 4 } + 32 x\).
  2. Determine whether this stationary point is a maximum or a minimum.
  3. For what values of \(x\) does \(x ^ { 4 } + 32 x\) increase as \(x\) increases?
OCR C1 2012 June Q9
11 marks Moderate -0.3
9
  1. A rectangular tile has length \(4 x \mathrm {~cm}\) and width \(( x + 3 ) \mathrm { cm }\). The area of the rectangle is less than \(112 \mathrm {~cm} ^ { 2 }\). By writing down and solving an inequality, determine the set of possible values of \(x\).
  2. A second rectangular tile of length \(4 y \mathrm {~cm}\) and width \(( y + 3 ) \mathrm { cm }\) has a rectangle of length \(2 y \mathrm {~cm}\) and width \(y \mathrm {~cm}\) removed from one corner as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662} Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of \(y\).
OCR C1 2012 June Q10
15 marks Moderate -0.8
10 A circle has equation \(( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
  1. Find the coordinates of the centre \(C\) and the length of the diameter.
  2. Find the equation of the line which passes through \(C\) and the point \(P ( 7,2 )\).
  3. Calculate the length of \(C P\) and hence determine whether \(P\) lies inside or outside the circle.
  4. Determine algebraically whether the line with equation \(y = 2 x\) meets the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR C1 2013 June Q1
4 marks Easy -1.8
1 Express each of the following in the form \(a \sqrt { 5 }\), where \(a\) is an integer.
  1. \(4 \sqrt { 15 } \times \sqrt { 3 }\)
  2. \(\frac { 20 } { \sqrt { 5 } }\)
  3. \(5 ^ { \frac { 3 } { 2 } }\)
OCR C1 2013 June Q2
5 marks Standard +0.3
2 Solve the equation \(8 x ^ { 6 } + 7 x ^ { 3 } - 1 = 0\).
OCR C1 2013 June Q3
5 marks Moderate -0.8
3 It is given that \(\mathrm { f } ( x ) = \frac { 6 } { x ^ { 2 } } + 2 x\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
OCR C1 2013 June Q4
7 marks Moderate -0.8
4
  1. Express \(3 x ^ { 2 } + 9 x + 10\) in the form \(3 ( x + p ) ^ { 2 } + q\).
  2. State the coordinates of the minimum point of the curve \(y = 3 x ^ { 2 } + 9 x + 10\).
  3. Calculate the discriminant of \(3 x ^ { 2 } + 9 x + 10\).
OCR C1 2013 June Q5
6 marks Moderate -0.8
5
  1. Sketch the curve \(y = \frac { 2 } { x ^ { 2 } }\).
  2. The curve \(y = \frac { 2 } { x ^ { 2 } }\) is translated by 5 units in the negative \(x\)-direction. Find the equation of the curve after it has been translated.
  3. Describe a transformation that transforms the curve \(y = \frac { 2 } { x ^ { 2 } }\) to the curve \(y = \frac { 1 } { x ^ { 2 } }\).
OCR C1 2013 June Q6
5 marks Moderate -0.8
6 A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 y - 24 = 0\).
  1. Find the centre and radius of the circle.
  2. The point \(A ( 2,2 )\) lies on the circumference of \(C\). Given that \(A B\) is a diameter of the circle, find the coordinates of \(B\).
OCR C1 2013 June Q7
7 marks Moderate -0.8
7 Solve the inequalities
  1. \(3 - 8 x > 4\),
  2. \(( 2 x - 4 ) ( x - 3 ) \leqslant 12\). \(8 \quad A\) is the point \(( - 2,6 )\) and \(B\) is the point \(( 3 , - 8 )\). The line \(l\) is perpendicular to the line \(x - 3 y + 15 = 0\) and passes through the mid-point of \(A B\). Find the equation of \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2013 June Q9
12 marks Moderate -0.3
9
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 6\), giving the coordinates of all points of intersection with the axes.
  2. Find the set of values of \(x\) for which \(2 x ^ { 2 } - x - 6\) is a decreasing function.
  3. The line \(y = 4\) meets the curve \(y = 2 x ^ { 2 } - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(P Q\).
OCR C1 2013 June Q10
14 marks Standard +0.3
10 The curve \(y = ( 1 - x ) \left( x ^ { 2 } + 4 x + k \right)\) has a stationary point when \(x = - 3\).
  1. Find the value of the constant \(k\).
  2. Determine whether the stationary point is a maximum or minimum point.
  3. Given that \(y = 9 x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\).
OCR C1 2014 June Q1
4 marks Easy -1.2
1 Express \(5 x ^ { 2 } + 10 x + 2\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
OCR C1 2014 June Q2
5 marks Easy -1.3
2 Express each of the following in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  1. \(\frac { 6 } { \sqrt { 3 } }\)
  2. \(10 \sqrt { 3 } - 6 \sqrt { 27 }\)
  3. \(3 ^ { \frac { 5 } { 2 } }\)
OCR C1 2014 June Q3
5 marks Moderate -0.3
3 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).
OCR C1 2014 June Q4
4 marks Easy -1.3
4 The curve \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates \(( 2,5 )\).
  1. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( x ) + 2\).
  2. State the coordinates of the point corresponding to \(P\) on the curve \(y = \mathrm { f } ( 2 x )\).
  3. Describe the transformation that transforms the curve \(y = \mathrm { f } ( x )\) to the curve \(y = \mathrm { f } ( x + 4 )\).
OCR C1 2014 June Q5
8 marks Moderate -0.8
5 Solve the following inequalities.
  1. \(5 < 6 x + 3 < 14\)
  2. \(x ( 3 x - 13 ) \geqslant 10\)
OCR C1 2014 June Q6
6 marks Easy -1.3
6 Given that \(y = 6 x ^ { 3 } + \frac { 4 } { \sqrt { x } } + 5 x\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). \(7 \quad A\) is the point \(( 5,7 )\) and \(B\) is the point \(( - 1 , - 5 )\).
  3. Find the coordinates of the mid-point of the line segment \(A B\).
  4. Find an equation of the line through \(A\) that is perpendicular to the line segment \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.