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OCR MEI C1 2016 June Q11
11
  1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
  2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
  3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}
OCR MEI C1 Q1
1 Solve the inequality \(2 ( x - 3 ) < 6 x + 15\).
OCR MEI C1 Q2
2 Make \(r\) the subject of \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).
OCR MEI C1 Q3
3 In each case, choose one of the statements $$\mathbf { P } \Rightarrow \mathbf { Q } \quad \mathbf { P } \Leftarrow \mathbf { Q } \quad \mathbf { P } \Leftrightarrow \mathbf { Q }$$ to describe the complete relationship between P and Q .
  1. For \(n\) an integer: P: \(n\) is an even number
    Q: \(n\) is a multiple of 4
  2. For triangle ABC : P: \(\quad \mathrm { B }\) is a right-angle
    Q: \(\quad \mathrm { AB } ^ { 2 } + \mathrm { BC } ^ { 2 } = \mathrm { AC } ^ { 2 }\)
OCR MEI C1 Q4
4 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 2 + 3 x ) ^ { 5 }\).
OCR MEI C1 Q5
5 Find the value of the following.
  1. \(\left( \frac { 1 } { 3 } \right) ^ { - 2 }\)
  2. \(16 ^ { \frac { 3 } { 4 } }\)
OCR MEI C1 Q6
6 The line \(L\) is parallel to \(y = - 2 x + 1\) and passes through the point \(( 5,2 )\).
Find the coordinates of the points of intersection of \(L\) with the axes.
OCR MEI C1 Q7
7 Express \(x ^ { 2 } - 6 x\) in the form \(( x - a ) ^ { 2 } - b\).
Sketch the graph of \(y = x ^ { 2 } - 6 x\), giving the coordinates of its minimum point and the intersections with the axes.
OCR MEI C1 Q8
8 Find, in the form \(y = m x + c\), the equation of the line passing through \(\mathrm { A } ( 3,7 )\) and \(\mathrm { B } ( 5 , - 1 )\).
Show that the midpoint of AB lies on the line \(x + 2 y = 10\).
OCR MEI C1 Q11
11 A cubic polynomial is given by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8\).
  1. Show that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\). Factorise \(\mathrm { f } ( x )\) fully.
    Sketch the graph of \(y = f ( x )\).
  2. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { - 3 } { 0 }\). Write down an equation for the resulting graph. You need not simplify your answer.
    Find also the intercept on the \(y\)-axis of the resulting graph.
AQA C1 Q7
7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$
  1. Find:
    1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
    2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
  2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
    1. Verify that \(V\) has a stationary value when \(t = 1\).
    2. Determine whether this is a maximum or minimum value.
AQA C1 Q8
25 marks
8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\).
\includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).
  1. Find the area of the rectangle \(A B C D\).
    1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
    2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
  3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
    1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
    2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
    3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
  4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).
AQA C1 2005 January Q1
1 The point \(A\) has coordinates \(( 11,2 )\) and the point \(B\) has coordinates \(( - 1 , - 1 )\).
    1. Find the gradient of \(A B\).
    2. Hence, or otherwise, show that the line \(A B\) has equation $$x - 4 y = 3$$
  2. The line with equation \(3 x + 5 y = 26\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
AQA C1 2005 January Q2
2 A curve has equation \(y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. The point \(P\) on the curve has coordinates \(( 2,3 )\).
    1. Show that the gradient of the curve at \(P\) is 5 .
    2. Hence find an equation of the normal to the curve at \(P\), expressing your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
  3. Determine whether \(y\) is increasing or decreasing when \(x = 1\).
AQA C1 2005 January Q3
3 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0\).
  1. By completing the square, express the equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. Write down:
    1. the coordinates of the centre of the circle;
    2. the radius of the circle.
  3. The line with equation \(y = x + 4\) intersects the circle at the points \(P\) and \(Q\).
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } - 5 x + 6 = 0$$
    2. Find the coordinates of \(P\) and \(Q\).
AQA C1 2005 January Q4
4
  1. The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
    2. Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
    3. Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
  2. The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below.
    \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
    1. The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
    2. The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
    1. Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
    2. Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.
AQA C1 2005 January Q5
5
  1. Simplify \(( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )\).
  2. Express \(\sqrt { 12 }\) in the form \(m \sqrt { 3 }\), where \(m\) is an integer.
  3. Express \(\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
AQA C1 2005 January Q6
6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm .
\includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side \(x \mathrm {~cm}\) is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height \(x \mathrm {~cm}\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
    1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
    2. Show that any stationary values of \(V\) must occur when \(x ^ { 2 } - 11 x + 18 = 0\).
    3. Solve the equation \(x ^ { 2 } - 11 x + 18 = 0\).
    4. Explain why there is only one value of \(x\) for which \(V\) is stationary.
    1. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
    2. Hence determine whether the stationary value is a maximum or minimum.
AQA C1 2005 January Q7
7
  1. Simplify \(( k + 5 ) ^ { 2 } - 12 k ( k + 2 )\).
  2. The quadratic equation \(3 ( k + 2 ) x ^ { 2 } + ( k + 5 ) x + k = 0\) has real roots.
    1. Show that \(( k - 1 ) ( 11 k + 25 ) \leqslant 0\).
    2. Hence find the possible values of \(k\).
AQA C1 2006 January Q1
1
  1. Simplify \(( \sqrt { 5 } + 2 ) ( \sqrt { 5 } - 2 )\).
  2. Express \(\sqrt { 8 } + \sqrt { 18 }\) in the form \(n \sqrt { 2 }\), where \(n\) is an integer.
AQA C1 2006 January Q2
2 The point \(A\) has coordinates \(( 1,1 )\) and the point \(B\) has coordinates \(( 5 , k )\). The line \(A B\) has equation \(3 x + 4 y = 7\).
    1. Show that \(k = - 2\).
    2. Hence find the coordinates of the mid-point of \(A B\).
  2. Find the gradient of \(A B\).
  3. The line \(A C\) is perpendicular to the line \(A B\).
    1. Find the gradient of \(A C\).
    2. Hence find an equation of the line \(A C\).
    3. Given that the point \(C\) lies on the \(x\)-axis, find its \(x\)-coordinate.
AQA C1 2006 January Q3
3
    1. Express \(x ^ { 2 } - 4 x + 9\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
    2. Hence, or otherwise, state the coordinates of the minimum point of the curve with equation \(y = x ^ { 2 } - 4 x + 9\).
  2. The line \(L\) has equation \(y + 2 x = 12\) and the curve \(C\) has equation \(y = x ^ { 2 } - 4 x + 9\).
    1. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation $$x ^ { 2 } - 2 x - 3 = 0$$
    2. Hence find the coordinates of the points of intersection of \(L\) and \(C\).
AQA C1 2006 January Q4
4 The quadratic equation \(x ^ { 2 } + ( m + 4 ) x + ( 4 m + 1 ) = 0\), where \(m\) is a constant, has equal roots.
  1. Show that \(m ^ { 2 } - 8 m + 12 = 0\).
  2. Hence find the possible values of \(m\).
AQA C1 2006 January Q5
5 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x + 6 y = 11\).
  1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
  3. The point \(O\) has coordinates \(( 0,0 )\).
    1. Find the length of CO .
    2. Hence determine whether the point \(O\) lies inside or outside the circle, giving a reason for your answer.
AQA C1 2006 January Q6
6 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$
    1. Using the factor theorem, show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
    2. Hence express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Sketch the curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 10 x + 8\), showing the coordinates of the points where the curve cuts the axes.
    (You are not required to calculate the coordinates of the stationary points.)