Questions — OCR MEI C1 (499 questions)

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OCR MEI C1 2013 June Q9
4 marks Moderate -0.8
\(n - 1\), \(n\) and \(n + 1\) are any three consecutive integers.
  1. Show that the sum of these integers is always divisible by 3. [1]
  2. Find the sum of the squares of these three consecutive integers and explain how this shows that the sum of the squares of any three consecutive integers is never divisible by 3. [3]
OCR MEI C1 2013 June Q10
12 marks Moderate -0.8
The circle \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.
  1. Write down the radius of the circle and the coordinates of C. [2]
  2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
  3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]
OCR MEI C1 2013 June Q11
12 marks Moderate -0.8
You are given that \(\text{f}(x) = (2x - 3)(x + 2)(x + 4)\).
  1. Sketch the graph of \(y = \text{f}(x)\). [3]
  2. State the roots of \(\text{f}(x - 2) = 0\). [2]
  3. You are also given that \(\text{g}(x) = \text{f}(x) + 15\).
    1. Show that \(\text{g}(x) = 2x^3 + 9x^2 - 2x - 9\). [2]
    2. Show that \(\text{g}(1) = 0\) and hence factorise \(\text{g}(x)\) completely. [5]
OCR MEI C1 2013 June Q12
12 marks Standard +0.3
\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{1}{x - 2}\).
  1. Draw accurately the graph of \(y = 2x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\). [3]
  2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = 2x + 3\) satisfy the equation \(2x^2 - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection. [5]
  3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac{1}{x - 2}\) and \(y = -x + k\). Hence find the exact values of \(k\) for which \(y = -x + k\) is a tangent to \(y = \frac{1}{x - 2}\). [4]
OCR MEI C1 Q1
2 marks Easy -1.8
Make \(r\) the subject of the formula \(A = \pi r^2(x+y)\), where \(r > 0\). [2]
OCR MEI C1 Q2
5 marks Moderate -0.8
Fig. 8 shows a right-angled triangle with base \(2x + 1\), height \(h\) and hypotenuse \(3x\). \includegraphics{figure_1}
  1. Show that \(h^2 = 5x^2 - 4x - 1\). [2]
  2. Given that \(h = \sqrt{7}\), find the value of \(x\), giving your answer in surd form. [3]
OCR MEI C1 Q3
12 marks Moderate -0.3
  1. Find the set of values of \(k\) for which the line \(y = 2x + k\) intersects the curve \(y = 3x^2 + 12x + 13\) at two distinct points. [5]
  2. Express \(3x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). Hence show that the curve \(y = 3x^2 + 12x + 13\) lies completely above the \(x\)-axis. [5]
  3. Find the value of \(k\) for which the line \(y = 2x + k\) passes through the minimum point of the curve \(y = 3x^2 + 12x + 13\). [2]
OCR MEI C1 Q4
4 marks Moderate -0.5
Make \(a\) the subject of \(3(a + 4) = ac + 5f\). [4]
OCR MEI C1 Q5
4 marks Moderate -0.8
Find the coordinates of the point of intersection of the lines \(y = 3x - 2\) and \(x + 3y = 1\). [4]
OCR MEI C1 Q6
5 marks Moderate -0.8
Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]
OCR MEI C1 Q7
3 marks Easy -1.2
Simplify \(\frac{(4x^5 y)^3}{(2xy^2) \times (8x^{10}y^4)}\). [3]
OCR MEI C1 Q8
4 marks Moderate -0.8
You are given that \(f(x) = x^2 + kx + c\). Given also that \(f(2) = 0\) and \(f(-3) = 35\), find the values of the constants \(k\) and \(c\). [4]
OCR MEI C1 Q9
4 marks Moderate -0.8
Rearrange the equation \(5c + 9t = a(2c + t)\) to make \(c\) the subject. [4]
OCR MEI C1 Q10
3 marks Easy -1.2
Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]
OCR MEI C1 Q11
3 marks Easy -1.8
Rearrange the following equation to make \(h\) the subject. $$4h + 5 = 9a - ha^2$$ [3]
OCR MEI C1 Q1
3 marks Moderate -0.8
Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]
OCR MEI C1 Q2
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 Q3
4 marks Moderate -0.5
Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]
OCR MEI C1 Q4
3 marks Standard +0.3
Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]
OCR MEI C1 Q5
4 marks Moderate -0.5
Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]
OCR MEI C1 Q6
3 marks Moderate -0.8
Rearrange the formula \(c = \sqrt{\frac{a + b}{2}}\) to make \(a\) the subject. [3]
OCR MEI C1 Q7
3 marks Easy -1.2
Make \(a\) the subject of the formula \(s = ut + \frac{1}{2}at^2\). [3]
OCR MEI C1 Q8
3 marks Moderate -0.5
Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]
OCR MEI C1 Q11
3 marks Easy -1.2
Solve the equation \(\frac{3x + 1}{2x} = 4\). [3]
OCR MEI C1 Q12
4 marks Standard +0.3
Find the range of values of \(k\) for which the equation \(2x^2 + kx + 18 = 0\) does not have real roots. [4]