Questions — OCR MEI C1 (499 questions)

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OCR MEI C1 2009 January Q11
14 marks Moderate -0.3
  1. Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
  2. Find the coordinates of the points of intersection of this circle with the \(y\)-axis. Give your answer in the form \(a \pm \sqrt { b }\).
  3. Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.
OCR MEI C1 Q10
12 marks Moderate -0.3
  1. Write down the equations of the circles A and B .
  2. Find the \(x\) coordinates of the points where the two curves intersect.
  3. Find the \(y\) coordinates of these points, giving your answers in surd form.
OCR MEI C1 2007 January Q11
12 marks Moderate -0.3
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 2009 January Q13
11 marks Moderate -0.3
13 Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x }\).
  1. On the insert, on the same axes, plot the graph of \(y = x ^ { 2 } - 5 x + 5\) for \(0 \leqslant x \leqslant 5\).
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac { 1 } { x }\) and \(y = x ^ { 2 } - 5 x + 5\) satisfy the equation \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\).
  3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x ^ { 3 } - 5 x ^ { 2 } + 5 x - 1 = 0\) is rational.
OCR MEI C1 Q4
12 marks Moderate -0.8
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\) (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 Q1
3 marks Easy -1.2
Solve the inequality \(2(x - 3) < 6x + 15\). [3]
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]