Questions — OCR C1 (333 questions)

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OCR C1 2013 June Q7
7 marks Moderate -0.8
Solve the inequalities
  1. \(3 - 8x > 4\), [2]
  2. \((2x - 4)(x - 3) \leq 12\). [5]
OCR C1 2013 June Q8
7 marks Moderate -0.3
\(A\) is the point \((-2, 6)\) and \(B\) is the point \((3, -8)\). The line \(l\) is perpendicular to the line \(x - 3y + 15 = 0\) and passes through the mid-point of \(AB\). Find the equation of \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]
OCR C1 2013 June Q9
12 marks Moderate -0.8
  1. Sketch the curve \(y = 2x^2 - x - 6\), giving the coordinates of all points of intersection with the axes. [5]
  2. Find the set of values of \(x\) for which \(2x^2 - x - 6\) is a decreasing function. [3]
  3. The line \(y = 4\) meets the curve \(y = 2x^2 - x - 6\) at the points \(P\) and \(Q\). Calculate the distance \(PQ\). [4]
OCR C1 2013 June Q10
14 marks Standard +0.3
The curve \(y = (1 - x)(x^2 + 4x + k)\) has a stationary point when \(x = -3\).
  1. Find the value of the constant \(k\). [7]
  2. Determine whether the stationary point is a maximum or minimum point. [2]
  3. Given that \(y = 9x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\). [5]
OCR C1 2014 June Q1
4 marks Moderate -0.8
Express \(5x^2 + 10x + 2\) in the form \(p(x + q)^2 + r\), where \(p\), \(q\) and \(r\) are integers. [4]
OCR C1 2014 June Q2
5 marks Easy -1.3
Express each of the following in the form \(k\sqrt{3}\), where \(k\) is an integer.
  1. \(\frac{6}{\sqrt{3}}\) [1]
  2. \(10\sqrt{3} - 6\sqrt{27}\) [2]
  3. \(3^{\frac{3}{2}}\) [2]
OCR C1 2014 June Q3
5 marks Standard +0.3
Find the real roots of the equation \(4x^4 + 3x^2 - 1 = 0\). [5]
OCR C1 2014 June Q4
4 marks Easy -1.3
The curve \(y = \text{f}(x)\) passes through the point \(P\) with coordinates \((2, 5)\).
  1. State the coordinates of the point corresponding to \(P\) on the curve \(y = \text{f}(x) + 2\). [1]
  2. State the coordinates of the point corresponding to \(P\) on the curve \(y = \text{f}(2x)\). [1]
  3. Describe the transformation that transforms the curve \(y = \text{f}(x)\) to the curve \(y = \text{f}(x + 4)\). [2]
OCR C1 2014 June Q5
8 marks Moderate -0.3
Solve the following inequalities.
  1. \(5 < 6x + 3 < 14\) [3]
  2. \(x(3x - 13) \geqslant 10\) [5]
OCR C1 2014 June Q6
6 marks Moderate -0.8
Given that \(y = 6x^3 + \frac{4}{\sqrt{x}} + 5x\), find
  1. \(\frac{\text{d}y}{\text{d}x}\), [4]
  2. \(\frac{\text{d}^2y}{\text{d}x^2}\). [2]
OCR C1 2014 June Q7
7 marks Moderate -0.8
\(A\) is the point \((5, 7)\) and \(B\) is the point \((-1, -5)\).
  1. Find the coordinates of the mid-point of the line segment \(AB\). [2]
  2. Find an equation of the line through \(A\) that is perpendicular to the line segment \(AB\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 2014 June Q8
9 marks Moderate -0.8
A curve has equation \(y = 3x^3 - 7x + \frac{2}{x}\).
  1. Verify that the curve has a stationary point when \(x = 1\). [5]
  2. Determine the nature of this stationary point. [2]
  3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]
OCR C1 2014 June Q9
12 marks Moderate -0.8
A circle with centre \(C\) has equation \((x - 2)^2 + (y + 5)^2 = 25\).
  1. Show that no part of the circle lies above the \(x\)-axis. [3]
  2. The point \(P\) has coordinates \((6, k)\) and lies inside the circle. Find the set of possible values of \(k\). [5]
  3. Prove that the line \(2y = x\) does not meet the circle. [4]
OCR C1 2014 June Q10
12 marks Moderate -0.3
A curve has equation \(y = (x + 2)^2(2x - 3)\).
  1. Sketch the curve, giving the coordinates of all points of intersection with the axes. [3]
  2. Find an equation of the tangent to the curve at the point where \(x = -1\). Give your answer in the form \(ax + by + c = 0\). [9]
OCR C1 Q1
4 marks Easy -1.8
  1. Express \(\frac{21}{\sqrt{7}}\) in the form \(k\sqrt{7}\). [2]
  2. Express \(8^{-1}\) as an exact fraction in its simplest form. [2]
OCR C1 Q2
4 marks Easy -1.3
Find \(\frac{dy}{dx}\) when
  1. \(y = x - 2x^2\), [2]
  2. \(y = \frac{3}{x^2}\). [2]
OCR C1 Q3
6 marks Moderate -0.8
  1. Express \(x^2 - 10x + 27\) in the form \((x + p)^2 + q\). [3]
  2. Sketch the curve with equation \(y = x^2 - 10x + 27\), showing on your sketch
    1. the coordinates of the vertex of the curve,
    2. the coordinates of any points where the curve meets the coordinate axes. [3]
OCR C1 Q4
8 marks Moderate -0.8
The straight line \(l_1\) has gradient 2 and passes through the point with coordinates \((4, -5)\).
  1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]
The straight line \(l_2\) is perpendicular to the line with equation \(3x - y = 4\) and passes through the point with coordinates \((3, 0)\).
  1. Find an equation for \(l_2\). [3]
  2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]
OCR C1 Q5
8 marks Moderate -0.3
Given that the equation $$4x^2 - kx + k - 3 = 0,$$ where \(k\) is a constant, has real roots,
  1. show that $$k^2 - 16k + 48 \geq 0, \quad [2]$$
  2. find the set of possible values of \(k\), [3]
  3. state the smallest value of \(k\) for which the roots are equal and solve the equation when \(k\) takes this value. [3]
OCR C1 Q6
9 marks Moderate -0.8
The points \(P\) and \(Q\) have coordinates \((-2, 6)\) and \((4, -1)\) respectively. Given that \(PQ\) is a diameter of circle \(C\),
  1. find the coordinates of the centre of \(C\), [2]
  2. show that \(C\) has the equation $$x^2 + y^2 - 2x - 5y - 14 = 0. \quad [5]$$
The point \(R\) has coordinates \((2, 7)\).
  1. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle PRQ\) in degrees. [2]
OCR C1 Q7
10 marks Standard +0.3
  1. Describe fully the single transformation that maps the graph of \(y = f(x)\) onto the graph of \(y = f(x - 1)\). [2]
  2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
  3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]
OCR C1 Q8
11 marks Moderate -0.3
\includegraphics{figure_8} The diagram shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
The line \(l\) is the tangent to \(C\) at \(O\).
  1. Find an equation for \(l\). [4]
  2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]
OCR C1 Q9
12 marks Moderate -0.3
The curve with equation \(y = 2x^3 - 8x^{\frac{1}{3}}\) has a minimum at the point \(A\).
  1. Find \(\frac{dy}{dx}\). [3]
  2. Find the \(x\)-coordinate of \(A\). [3]
The point \(B\) on the curve has \(x\)-coordinate 2.
  1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]
OCR C1 Q1
3 marks Easy -1.2
  1. Calculate the discriminant of \(2x^2 + 8x + 8\). [2]
  2. State the number of real roots of the equation \(2x^2 + 8x + 8 = 0\). [1]
OCR C1 Q2
4 marks Moderate -0.3
Find the set of values of \(x\) for which $$(x - 1)(x - 2) < 20.$$ [4]