Questions — OCR (4907 questions)

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OCR C1 2012 January Q6
7 marks Moderate -0.8
6 Given that \(\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2\),
  1. find \(\mathrm { f } ^ { \prime } ( x )\),
  2. find \(\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)\).
OCR C1 2012 January Q7
12 marks Moderate -0.3
7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).
  1. Find the coordinates of the minimum point, justifying that it is a minimum.
  2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
  3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).
OCR C1 2012 January Q8
6 marks Standard +0.3
8 The line \(l\) has gradient - 2 and passes through the point \(A ( 3,5 ) . B\) is a point on the line \(l\) such that the distance \(A B\) is \(6 \sqrt { 5 }\). Find the coordinates of each of the possible points \(B\).
OCR C1 2012 January Q9
12 marks Moderate -0.3
9
  1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
  2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
  3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).
OCR C1 2012 January Q10
13 marks Moderate -0.3
10 A circle has centre \(C ( - 2,4 )\) and radius 5 .
  1. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  2. Show that the tangent to the circle at the point \(P ( - 5,8 )\) has equation \(3 x - 4 y + 47 = 0\).
  3. Verify that the point \(T ( 3,14 )\) lies on this tangent.
  4. Find the area of the triangle \(C P T\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}
OCR C1 2009 June Q1
5 marks Easy -1.2
1 Given that \(y = x ^ { 5 } + \frac { 1 } { x ^ { 2 } }\), find
  1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
OCR C1 2009 June Q2
4 marks Easy -1.2
2 Express \(\frac { 8 + \sqrt { 7 } } { 2 + \sqrt { 7 } }\) in the form \(a + b \sqrt { 7 }\), where \(a\) and \(b\) are integers.
OCR C1 2009 June Q3
4 marks Easy -1.8
3 Express each of the following in the form \(3 ^ { n }\) :
  1. \(\frac { 1 } { 9 }\),
  2. \(\sqrt [ 3 ] { 3 }\),
  3. \(3 ^ { 10 } \times 9 ^ { 15 }\).
OCR C1 2009 June Q4
6 marks Moderate -0.5
4 Solve the simultaneous equations $$4 x ^ { 2 } + y ^ { 2 } = 10 , \quad 2 x - y = 4$$
OCR C1 2009 June Q5
5 marks Moderate -0.8
5
  1. Expand and simplify \(( 2 x + 1 ) ( x - 3 ) ( x + 4 )\).
  2. Find the coefficient of \(x ^ { 4 }\) in the expansion of $$x \left( x ^ { 2 } + 2 x + 3 \right) \left( x ^ { 2 } + 7 x - 2 \right) .$$
OCR C1 2009 June Q6
6 marks Moderate -0.8
6
  1. Sketch the curve \(y = - \sqrt { x }\).
  2. Describe fully a transformation that transforms the curve \(y = - \sqrt { x }\) to the curve \(y = 5 - \sqrt { x }\).
  3. The curve \(y = - \sqrt { x }\) is stretched by a scale factor of 2 parallel to the \(x\)-axis. State the equation of the curve after it has been stretched.
OCR C1 2009 June Q7
6 marks Moderate -0.8
7
  1. Express \(x ^ { 2 } - 5 x + \frac { 1 } { 4 }\) in the form \(( x - a ) ^ { 2 } - b\).
  2. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 5 x + \frac { 1 } { 4 } = 0\).
OCR C1 2009 June Q8
6 marks Easy -1.2
8 Solve the inequalities
  1. \(- 35 < 6 x + 7 < 1\),
  2. \(3 x ^ { 2 } > 48\). \(9 \quad A\) is the point \(( 4 , - 3 )\) and \(B\) is the point \(( - 1,9 )\).
OCR C1 2009 June Q10
11 marks Moderate -0.8
10
  1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
  2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
  3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
  4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?
OCR C1 2009 June Q11
11 marks Standard +0.3
11 The point \(P\) on the curve \(y = k \sqrt { x }\) has \(x\)-coordinate 4 . The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\).
  1. Find the value of \(k\).
  2. This normal meets the \(x\)-axis at the point \(Q\). Calculate the area of the triangle \(O P Q\), where \(O\) is the point \(( 0,0 )\). RECOGNISING ACHIEVEMENT
OCR C1 2010 June Q2
5 marks Moderate -0.8
2
  1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
  2. Sketch the curve \(y = 3 - \frac { 1 } { x ^ { 2 } }\).
  3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 2 . State the equation of the transformed curve.
OCR C1 2010 June Q3
5 marks Easy -1.2
3
  1. Express \(\frac { 12 } { 3 + \sqrt { 5 } }\) in the form \(a - b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
  2. Express \(\sqrt { 18 } - \sqrt { 2 }\) in simplified surd form.
OCR C1 2010 June Q4
6 marks Moderate -0.8
4
  1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
  2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.
OCR C1 2010 June Q5
5 marks Moderate -0.8
5 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).
OCR C1 2010 June Q6
5 marks Easy -1.2
6 Find the gradient of the curve \(y = 2 x + \frac { 6 } { \sqrt { x } }\) at the point where \(x = 4\).
OCR C1 2010 June Q7
6 marks Moderate -0.3
7 Solve the simultaneous equations $$x + 2 y - 6 = 0 , \quad 2 x ^ { 2 } + y ^ { 2 } = 57 .$$
OCR C1 2010 June Q8
10 marks Moderate -0.8
8
  1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
  2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
  3. State the equation of the normal to the curve at its minimum point.
  4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).
OCR C1 2010 June Q9
13 marks Moderate -0.8
9
  1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\). \(A B\) is the diameter of a circle.
  2. Find the radius of the circle.
  3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
  4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).
OCR C1 2010 June Q10
14 marks Standard +0.3
10
  1. Find the coordinates of the stationary points of the curve \(y = 2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\).
  2. State the set of values for \(x\) for which \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\) is a decreasing function.
  3. Show that the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 }\) is \(10 x - 4 y - 7 = 0\).
  4. Hence, with the aid of a sketch, show that the equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x = \frac { 5 } { 2 } x - \frac { 7 } { 4 }\) has two distinct real roots.
OCR C1 2011 June Q1
4 marks Moderate -0.8
1 Express \(3 x ^ { 2 } - 18 x + 4\) in the form \(p ( x + q ) ^ { 2 } + r\).