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OCR MEI C1 Q2
3 marks Easy -1.2
Make \(r\) the subject of \(V = \frac{4}{3}\pi r^3\). [3]
OCR MEI C1 Q3
2 marks Easy -1.8
In each case, choose one of the statements $$P \Rightarrow Q \quad\quad P \Leftarrow Q \quad\quad P \Leftrightarrow Q$$ 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 [1]
  2. For triangle ABC: P: B is a right-angle Q: \(AB^2 + BC^2 = AC^2\) [1]
OCR MEI C1 Q4
4 marks Moderate -0.8
Find the coefficient of \(x^3\) in the expansion of \((2 + 3x)^5\). [4]
OCR MEI C1 Q5
4 marks Easy -1.8
Find the value of the following.
  1. \(\left(\frac{1}{3}\right)^{-2}\) [2]
  2. \(16^{\frac{1}{4}}\) [2]
OCR MEI C1 Q6
5 marks Moderate -0.8
The line \(L\) is parallel to \(y = -2x + 1\) and passes through the point \((5, 2)\). Find the coordinates of the points of intersection of \(L\) with the axes. [5]
OCR MEI C1 Q7
5 marks Easy -1.2
Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]
OCR MEI C1 Q8
5 marks Moderate -0.8
Find, in the form \(y = mx + c\), the equation of the line passing through A\((3, 7)\) and B\((5, -1)\). Show that the midpoint of AB lies on the line \(x + 2y = 10\). [5]
OCR MEI C1 Q9
5 marks Moderate -0.8
Simplify \((3 + \sqrt{2})(3 - \sqrt{2})\). Express \(\frac{1 + \sqrt{2}}{3 - \sqrt{2}}\) in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational. [5]
OCR MEI C1 Q10
12 marks Moderate -0.8
\includegraphics{figure_10} Fig. 10 shows a circle with centre C\((2, 1)\) and radius 5.
  1. Show that the equation of the circle may be written as $$x^2 + y^2 - 4x - 2y - 20 = 0.$$ [3]
  2. Find the coordinates of the points P and Q where the circle cuts the \(y\)-axis. Leave your answers in the form \(a \pm \sqrt{b}\). [3]
  3. Verify that the point A\((5, -3)\) lies on the circle. Show that the tangent to the circle at A has equation \(4y = 3x - 27\). [6]
OCR MEI C1 Q11
12 marks Moderate -0.3
A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).
  1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
  2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). 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. [5]
OCR MEI C1 Q12
12 marks Moderate -0.3
  1. Show that the graph of \(y = x^2 - 3x + 11\) is above the \(x\)-axis for all values of \(x\). [3]
  2. Find the set of values of \(x\) for which the graph of \(y = 2x^2 + x - 10\) is above the \(x\)-axis. [4]
  3. Find algebraically the coordinates of the points of intersection of the graphs of $$y = x^2 - 3x + 11 \quad\text{and}\quad y = 2x^2 + x - 10.$$ [5]
OCR MEI C1 2006 January Q1
2 marks Easy -1.2
\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]
OCR MEI C1 2006 January Q2
4 marks Moderate -0.8
\includegraphics{figure_2} Fig. 2 shows graphs \(A\) and \(B\).
  1. State the transformation which maps graph \(A\) onto graph \(B\). [2]
  2. The equation of graph \(A\) is \(y = f(x)\). Which one of the following is the equation of graph \(B\)? \(y = f(x) + 2\) \quad \(y = f(x) - 2\) \quad \(y = f(x + 2)\) \quad \(y = f(x - 2)\) \(y = 2f(x)\) \quad \(y = f(x + 3)\) \quad \(y = f(x - 3)\) \quad \(y = 3f(x)\) [2]
OCR MEI C1 2006 January Q3
4 marks Easy -1.8
Find the binomial expansion of \((2 + x)^4\), writing each term as simply as possible. [4]
OCR MEI C1 2006 January Q4
4 marks Easy -1.8
Solve the inequality \(\frac{3(2x + 1)}{4} > -6\). [4]
OCR MEI C1 2006 January Q5
4 marks Moderate -0.8
Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]
OCR MEI C1 2006 January Q6
3 marks Easy -1.2
When \(x^3 + 3x + k\) is divided by \(x - 1\), the remainder is 6. Find the value of \(k\). [3]
OCR MEI C1 2006 January Q7
5 marks Moderate -0.8
\includegraphics{figure_7} The line AB has equation \(y = 4x - 5\) and passes through the point B(2, 3), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C. Find the equation of the line BC and the \(x\)-coordinate of C. [5]
OCR MEI C1 2006 January Q8
5 marks Easy -1.3
  1. Simplify \(5\sqrt{8} + 4\sqrt{50}\). Express your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
  2. Express \(\frac{\sqrt{3}}{6 - \sqrt{3}}\) in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are rational. [3]
OCR MEI C1 2006 January Q9
5 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
  2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]
OCR MEI C1 2006 January Q10
10 marks Moderate -0.8
A circle has equation \(x^2 + y^2 = 45\).
  1. State the centre and radius of this circle. [2]
  2. The circle intersects the line with equation \(x + y = 3\) at two points, A and B. Find algebraically the coordinates of A and B. Show that the distance AB is \(\sqrt{162}\). [8]
OCR MEI C1 2006 January Q11
13 marks Moderate -0.8
  1. Write \(x^2 - 7x + 6\) in the form \((x - a)^2 + b\). [3]
  2. State the coordinates of the minimum point on the graph of \(y = x^2 - 7x + 6\). [2]
  3. Find the coordinates of the points where the graph of \(y = x^2 - 7x + 6\) crosses the axes and sketch the graph. [5]
  4. Show that the graphs of \(y = x^2 - 7x + 6\) and \(y = x^2 - 3x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. [3]
OCR MEI C1 2006 January Q12
13 marks Standard +0.3
  1. Sketch the graph of \(y = x(x - 3)^2\). [3]
  2. Show that the equation \(x(x - 3)^2 = 2\) can be expressed as \(x^3 - 6x^2 + 9x - 2 = 0\). [2]
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i). [8]
OCR MEI C1 2006 June Q1
3 marks Easy -1.2
The volume of a cone is given by the formula \(V = \frac{1}{3}\pi r^2 h\). Make \(r\) the subject of this formula. [3]
OCR MEI C1 2006 June Q2
2 marks Moderate -0.8
One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]