Questions — OCR C1 (324 questions)

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OCR C1 2012 January Q3
3 Given that $$5 x ^ { 2 } + p x - 8 = q ( x - 1 ) ^ { 2 } + r$$ for all values of \(x\), find the values of the constants \(p , q\) and \(r\).
OCR C1 2012 January Q4
4 Evaluate
  1. \(3 ^ { - 2 }\),
  2. \(16 ^ { \frac { 3 } { 4 } }\),
  3. \(\frac { \sqrt { 200 } } { \sqrt { 8 } }\).
OCR C1 2012 January Q5
5 Find the real roots of the equation \(\frac { 3 } { y ^ { 4 } } - \frac { 10 } { y ^ { 2 } } - 8 = 0\).
OCR C1 2012 January Q6
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
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
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
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
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 2013 January Q1
1
  1. Solve the equation \(x ^ { 2 } - 6 x - 2 = 0\), giving your answers in simplified surd form.
  2. Find the gradient of the curve \(y = x ^ { 2 } - 6 x - 2\) at the point where \(x = - 5\).
OCR C1 2013 January Q2
2 Solve the equations
  1. \(3 ^ { n } = 1\),
  2. \(t ^ { - 3 } = 64\),
  3. \(\left( 8 p ^ { 6 } \right) ^ { \frac { 1 } { 3 } } = 8\).
OCR C1 2013 January Q3
3
  1. Sketch the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\), giving the coordinates of all points of intersection with the axes.
  2. Describe the transformation that transforms the curve \(y = ( 1 + x ) ( 2 - x ) ( 3 + x )\) to the curve \(y = ( 1 - x ) ( 2 + x ) ( 3 - x )\).
OCR C1 2013 January Q4
4
  1. Solve the simultaneous equations $$y = 2 x ^ { 2 } - 3 x - 5 , \quad 10 x + 2 y + 11 = 0$$
  2. What can you deduce from the answer to part (i) about the curve \(y = 2 x ^ { 2 } - 3 x - 5\) and the line \(10 x + 2 y + 11 = 0\) ?
  3. Simplify \(( x + 4 ) ( 5 x - 3 ) - 3 ( x - 2 ) ^ { 2 }\).
  4. The coefficient of \(x ^ { 2 }\) in the expansion of $$( x + 3 ) ( x + k ) ( 2 x - 5 )$$ is - 3 . Find the value of the constant \(k\).
OCR C1 2013 January Q6
6
  1. The line joining the points ( \(- 2,7\) ) and ( \(- 4 , p\) ) has gradient 4 . Find the value of \(p\).
  2. The line segment joining the points \(( - 2,7 )\) and \(( 6 , q )\) has mid-point \(( m , 5 )\). Find \(m\) and \(q\).
  3. The line segment joining the points \(( - 2,7 )\) and \(( d , 3 )\) has length \(2 \sqrt { 13 }\). Find the two possible values of \(d\).
OCR C1 2013 January Q7
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \frac { ( 3 x ) ^ { 2 } \times x ^ { 4 } } { x }\),
  2. \(y = \sqrt [ 3 ] { x }\),
  3. \(y = \frac { 1 } { 2 x ^ { 3 } }\).
OCR C1 2013 January Q8
8 The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).
OCR C1 2013 January Q9
9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 2 x + 10 y - 19 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle.
  2. Verify that the point \(( 7 , - 2 )\) lies on the circumference of the circle.
  3. Find the equation of the tangent to the circle at the point \(( 7 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C1 2013 January Q10
10 Find the coordinates of the points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } + \frac { 9 } { x }\) at which the tangent is parallel to the line \(y = 8 x + 3\).
OCR C1 2009 June Q1
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
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
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
4 Solve the simultaneous equations $$4 x ^ { 2 } + y ^ { 2 } = 10 , \quad 2 x - y = 4$$
OCR C1 2009 June Q5
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
  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
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
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 \(( - 1,9 )\).