Questions — OCR MEI C1 (499 questions)

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OCR MEI C1 2009 June Q9
5 marks Easy -1.2
  1. Express \(x^2 + 6x + 5\) in the form \((x + a)^2 + b\). [3]
  2. Write down the coordinates of the minimum point on the graph of \(y = x^2 + 6x + 5\). [2]
OCR MEI C1 2009 June Q10
4 marks Moderate -0.3
Find the real roots of the equation \(x^4 - 5x^2 - 36 = 0\) by considering it as a quadratic equation in \(x^2\). [4]
OCR MEI C1 2009 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows the line joining the points A \((0, 3)\) and B \((6, 1)\).
  1. Find the equation of the line perpendicular to AB that passes through the origin, O. [2]
  2. Find the coordinates of the point where this perpendicular meets AB. [4]
  3. Show that the perpendicular distance of AB from the origin is \(\frac{9\sqrt{10}}{10}\). [2]
  4. Find the length of AB, expressing your answer in the form \(a\sqrt{10}\). [2]
  5. Find the area of triangle OAB. [2]
OCR MEI C1 2009 June Q12
13 marks Moderate -0.8
  1. You are given that \(\text{f}(x) = (x + 1)(x - 2)(x - 4)\).
    1. Show that \(\text{f}(x) = x^3 - 5x^2 + 2x + 8\). [2]
    2. Sketch the graph of \(y = \text{f}(x)\). [3]
    3. The graph of \(y = \text{f}(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). State an equation for the resulting graph. You need not simplify your answer. Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis. [3]
  2. Show that 3 is a root of \(x^3 - 5x^2 + 2x + 8 = -4\). Hence solve this equation completely, giving the other roots in surd form. [5]
OCR MEI C1 2009 June Q13
11 marks Moderate -0.8
A circle has equation \((x - 5)^2 + (y - 2)^2 = 20\).
  1. State the coordinates of the centre and the radius of this circle. [2]
  2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
  3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
  4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]
OCR MEI C1 2010 June Q1
3 marks Easy -1.2
Find the equation of the line which is parallel to \(y = 3x + 1\) and which passes through the point with coordinates \((4, 5)\). [3]
OCR MEI C1 2010 June Q2
5 marks Easy -1.8
  1. Simplify \((5a^2b)^3 \times 2b^4\). [2]
  2. Evaluate \(\left(\frac{1}{16}\right)^{-1}\). [1]
  3. Evaluate \((16)^{\frac{1}{2}}\). [2]
OCR MEI C1 2010 June Q3
3 marks Easy -1.2
Make \(y\) the subject of the formula \(a = \frac{\sqrt{y} - 5}{c}\). [3]
OCR MEI C1 2010 June Q4
5 marks Easy -1.2
Solve the following inequalities.
  1. \(2(1 - x) > 6x + 5\) [3]
  2. \((2x - 1)(x + 4) < 0\) [2]
OCR MEI C1 2010 June Q5
5 marks Easy -1.3
  1. Express \(\sqrt{48} + \sqrt{27}\) in the form \(a\sqrt{3}\). [2]
  2. Simplify \(\frac{5\sqrt{7}}{3 - \sqrt{2}}\). Give your answer in the form \(\frac{b + c\sqrt{7}}{d}\). [3]
OCR MEI C1 2010 June Q6
5 marks Moderate -0.3
You are given that • the coefficient of \(x^3\) in the expansion of \((5 + 2x^2)(x^3 + kx + m)\) is 29, • when \(x^3 + kx + m\) is divided by \((x - 3)\), the remainder is 59. Find the values of \(k\) and \(m\). [5]
OCR MEI C1 2010 June Q7
4 marks Easy -1.2
Expand \((1 + \frac{1}{2}x)^4\), simplifying the coefficients. [4]
OCR MEI C1 2010 June Q8
4 marks Moderate -0.8
Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]
OCR MEI C1 2010 June Q9
2 marks Easy -1.8
Show that the following statement is false. $$x - 5 = 0 \Leftrightarrow x^2 = 25$$ [2]
OCR MEI C1 2010 June Q10
12 marks Moderate -0.3
  1. Solve, by factorising, the equation \(2x^2 - x - 3 = 0\). [3]
  2. Sketch the graph of \(y = 2x^2 - x - 3\). [3]
  3. Show that the equation \(x^2 - 5x + 10 = 0\) has no real roots. [2]
  4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2x^2 - x - 3\) and \(y = x^2 - 5x + 10\). Give your answer in the form \(a \pm \sqrt{b}\). [4]
OCR MEI C1 2010 June Q11
12 marks Moderate -0.3
\includegraphics{figure_11} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).
  1. Find the equation of the line through A and B. [3]
  2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac{32}{3}\) square units. [2]
  3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
  4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle. [4]
OCR MEI C1 2010 June Q12
12 marks Moderate -0.3
You are given that \(f(x) = x^3 + 6x^2 - x - 30\).
  1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
  2. Sketch the graph of \(y = f(x)\). [3]
  3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]
OCR MEI C1 2011 June Q1
3 marks Easy -1.8
Solve the inequality \(6(x + 3) > 2x + 5\). [3]
OCR MEI C1 2011 June Q2
2 marks Easy -1.3
A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]
OCR MEI C1 2011 June Q3
5 marks Easy -1.3
  1. Evaluate \(\left(\frac{9}{16}\right)^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{(2ac^2)^3 \times 9a^2c}{36a^4c^{12}}\). [3]
OCR MEI C1 2011 June Q4
4 marks Easy -1.3
The point P \((5, 4)\) is on the curve \(y = f(x)\). State the coordinates of the image of P when the graph of \(y = f(x)\) is transformed to the graph of
  1. \(y = f(x - 5)\), [2]
  2. \(y = f(x) + 7\). [2]
OCR MEI C1 2011 June Q5
4 marks Moderate -0.8
Find the coefficient of \(x^4\) in the binomial expansion of \((5 + 2x)^6\). [4]
OCR MEI C1 2011 June Q6
3 marks Easy -1.2
Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]
OCR MEI C1 2011 June Q7
3 marks Easy -1.2
Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]
OCR MEI C1 2011 June Q8
4 marks Moderate -0.8
Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]