Questions — OCR (4628 questions)

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OCR C1 2014 June Q8
9 marks Moderate -0.3
8 A curve has equation \(y = 3 x ^ { 3 } - 7 x + \frac { 2 } { x }\).
  1. Verify that the curve has a stationary point when \(x = 1\).
  2. Determine the nature of this stationary point.
  3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\).
OCR C1 2014 June Q9
12 marks Moderate -0.3
9 A circle with centre \(C\) has equation \(( x - 2 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 25\).
  1. Show that no part of the circle lies above the \(x\)-axis.
  2. The point \(P\) has coordinates \(( 6 , k )\) and lies inside the circle. Find the set of possible values of \(k\).
  3. Prove that the line \(2 y = x\) does not meet the circle.
OCR C1 2014 June Q10
12 marks Moderate -0.3
10 A curve has equation \(y = ( x + 2 ) ^ { 2 } ( 2 x - 3 )\).
  1. Sketch the curve, giving the coordinates of all points of intersection with the axes.
  2. Find an equation of the tangent to the curve at the point where \(x = - 1\). Give your answer in the form \(a x + b y + c = 0\). \section*{OCR}
OCR C1 2015 June Q1
3 marks Easy -1.2
1 Express \(\frac { 8 } { \sqrt { 3 } - 1 }\) in the form \(a \sqrt { 3 } + b\), where \(a\) and \(b\) are integers.
OCR C1 2015 June Q2
6 marks Moderate -0.8
2
  1. Sketch the curve \(y = - \frac { 1 } { x }\).
  2. The curve \(y = - \frac { 1 } { x }\) is translated by 2 units parallel to the \(x\)-axis in the positive direction. State the equation of the transformed curve.
  3. Describe a transformation that transforms the curve \(y = - \frac { 1 } { x }\) to the curve \(y = - \frac { 1 } { 3 x }\).
OCR C1 2015 June Q3
5 marks Easy -1.3
3 Express each of the following in the form \(5 ^ { k }\).
  1. \(25 ^ { 4 }\)
  2. \(\frac { 1 } { \sqrt [ 4 ] { 5 } }\)
  3. \(( 5 \sqrt { 5 } ) ^ { 3 }\)
OCR C1 2015 June Q4
5 marks Moderate -0.3
4 Solve the equation \(x ^ { \frac { 2 } { 3 } } - x ^ { \frac { 1 } { 3 } } - 6 = 0\).
OCR C1 2015 June Q5
9 marks Moderate -0.3
5 The points \(A\) and \(B\) have coordinates \(( 2,1 )\) and \(( 5 , - 3 )\) respectively.
  1. Find the length of \(A B\).
  2. Find an equation of the line through the mid-point of \(A B\) which is perpendicular to \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 2015 June Q6
5 marks Moderate -0.3
6 Solve the simultaneous equations $$2 x + y - 5 = 0 , \quad x ^ { 2 } - y ^ { 2 } = 3$$
OCR C1 2015 June Q7
8 marks Moderate -0.8
7
  1. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 \right) ( 5 - x )\), find \(\mathrm { f } ^ { \prime } ( x )\).
  2. Find the gradient of the curve \(y = x ^ { - \frac { 1 } { 3 } }\) at the point where \(x = - 8\).
OCR C1 2015 June Q8
9 marks Moderate -0.3
8
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
  2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
  3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).
OCR C1 2015 June Q9
10 marks Moderate -0.3
9 The curve \(y = 2 x ^ { 3 } - a x ^ { 2 } + 8 x + 2\) passes through the point \(B\) where \(x = 4\).
  1. Given that \(B\) is a stationary point of the curve, find the value of the constant \(a\).
  2. Determine whether the stationary point \(B\) is a maximum point or a minimum point.
  3. Find the \(x\)-coordinate of the other stationary point of the curve.
OCR C1 2015 June Q10
12 marks Standard +0.3
10 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 4 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle.
  2. Show that the tangent to the circle at the point \(P ( 8,2 )\) has equation \(3 x + 4 y = 32\).
  3. The circle meets the \(y\)-axis at \(Q\) and the tangent meets the \(y\)-axis at \(R\). Find the area of triangle \(P Q R\).
OCR C1 2016 June Q1
4 marks Easy -1.2
1
  1. Simplify \(( 2 x - 3 ) ^ { 2 } - 2 ( 3 - x ) ^ { 2 }\).
  2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 x ^ { 2 } - 3 x + 4 \right) \left( 5 - 2 x - x ^ { 3 } \right)\).
OCR C1 2016 June Q2
4 marks Easy -1.2
2 Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).
OCR C1 2016 June Q3
5 marks Moderate -0.5
3 Solve the simultaneous equations $$x ^ { 2 } + y ^ { 2 } = 34 , \quad 3 x - y + 4 = 0$$
OCR C1 2016 June Q4
5 marks Standard +0.3
4 Solve the equation \(2 y ^ { \frac { 1 } { 2 } } - 7 y ^ { \frac { 1 } { 4 } } + 3 = 0\).
OCR C1 2016 June Q5
5 marks Easy -1.3
5 Express the following in the form \(2 ^ { p }\).
  1. \(\left( 2 ^ { 5 } \div 2 ^ { 7 } \right) ^ { 3 }\)
  2. \(5 \times 4 ^ { \frac { 2 } { 3 } } + 3 \times 16 ^ { \frac { 1 } { 3 } }\)
OCR C1 2016 June Q6
6 marks Moderate -0.8
6
  1. Express \(4 + 12 x - 2 x ^ { 2 }\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the coordinates of the maximum point of the curve \(y = 4 + 12 x - 2 x ^ { 2 }\).
OCR C1 2016 June Q7
7 marks Moderate -0.3
7
  1. Sketch the curve \(y = x ^ { 2 } ( 3 - x )\) stating the coordinates of points of intersection with the axes.
  2. The curve \(y = x ^ { 2 } ( 3 - x )\) is translated by 2 units in the positive direction parallel to the \(x\)-axis. State the equation of the curve after it has been translated.
  3. Describe fully a transformation that transforms the curve \(y = x ^ { 2 } ( 3 - x )\) to \(y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )\).
OCR C1 2016 June Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 2 }\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates 5 and \(5 + h\) respectively, where \(h > 0\).
  1. Show that the gradient of the line \(A B\) is \(20 + 2 h\).
  2. Explain how the answer to part (i) relates to the gradient of the curve at \(A\).
  3. The normal to the curve at \(A\) meets the \(y\)-axis at the point \(C\). Find the \(y\)-coordinate of \(C\).
OCR C1 2016 June Q9
7 marks Standard +0.3
9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.
OCR C1 2016 June Q10
14 marks Moderate -0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).
  1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
  2. Find the equation of the tangent to the circle at \(A\).
  3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
  4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).
OCR C1 2016 June Q11
8 marks Standard +0.3
11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .
OCR FP3 2009 June Q1
4 marks Standard +0.3
1 Find the cube roots of \(\frac { 1 } { 2 } \sqrt { 3 } + \frac { 1 } { 2 } \mathrm { i }\), giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(0 \leqslant \theta < 2 \pi\).