Questions C1 (1562 questions)

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OCR C1 2006 June Q1
4 marks Easy -1.2
The points \(A(1, 3)\) and \(B(4, 21)\) lie on the curve \(y = x^2 + x + 1\).
  1. Find the gradient of the line \(AB\). [2]
  2. Find the gradient of the curve \(y = x^2 + x + 1\) at the point where \(x = 3\). [2]
OCR C1 2006 June Q2
6 marks Easy -1.2
  1. Evaluate \(27^{\frac{2}{3}}\). [2]
  2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
  3. Express \(\frac{1 - \sqrt{5}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]
OCR C1 2006 June Q3
7 marks Moderate -0.8
  1. Express \(2x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). [4]
  2. Solve \(2x^2 + 12x + 13 = 0\), giving your answers in simplified surd form. [3]
OCR C1 2006 June Q4
8 marks Easy -1.2
  1. By expanding the brackets, show that $$(x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12.$$ [3]
  2. Sketch the curve $$y = x^3 - 6x^2 + 5x + 12,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve $$y = -x^3 + 6x^2 - 5x - 12.$$ Label this curve \(C_2\). [2]
OCR C1 2006 June Q5
8 marks Moderate -0.8
Solve the inequalities
  1. \(1 < 4x - 9 < 5\), [3]
  2. \(y^2 \geq 4y + 5\). [5]
OCR C1 2006 June Q6
8 marks Moderate -0.3
  1. Solve the equation \(x^4 - 10x^2 + 25 = 0\). [4]
  2. Given that \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\), find \(\frac{dy}{dx}\). [2]
  3. Hence find the number of stationary points on the curve \(y = \frac{2}{5}x^5 - \frac{20}{3}x^3 + 50x + 3\). [2]
OCR C1 2006 June Q7
9 marks Moderate -0.3
  1. Solve the simultaneous equations $$y = x^2 - 5x + 4, \quad y = x - 1.$$ [4]
  2. State the number of points of intersection of the curve \(y = x^2 - 5x + 4\) and the line \(y = x - 1\). [1]
  3. Find the value of \(c\) for which the line \(y = x + c\) is a tangent to the curve \(y = x^2 - 5x + 4\). [4]
OCR C1 2006 June Q8
10 marks Moderate -0.3
A cuboid has a volume of \(8 \text{m}^3\). The base of the cuboid is square with sides of length \(x\) metres. The surface area of the cuboid is \(A \text{m}^2\).
  1. Show that \(A = 2x^2 + \frac{32}{x}\). [3]
  2. Find \(\frac{dA}{dx}\). [3]
  3. Find the value of \(x\) which gives the smallest surface area of the cuboid, justifying your answer. [4]
OCR C1 2006 June Q9
12 marks Easy -1.2
The points \(A\) and \(B\) have coordinates \((4, -2)\) and \((10, 6)\) respectively. \(C\) is the mid-point of \(AB\). Find
  1. the coordinates of \(C\), [2]
  2. the length of \(AC\), [2]
  3. the equation of the circle that has \(AB\) as a diameter, [3]
  4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(ax + by = c\). [5]
OCR C1 2013 June Q1
4 marks Easy -1.3
Express each of the following in the form \(a\sqrt{5}\), where \(a\) is an integer.
  1. \(4\sqrt{15} \times \sqrt{3}\) [2]
  2. \(\frac{20}{\sqrt{5}}\) [1]
  3. \(5^{\frac{3}{2}}\) [1]
OCR C1 2013 June Q2
5 marks Standard +0.3
Solve the equation \(8x^6 + 7x^3 - 1 = 0\). [5]
OCR C1 2013 June Q3
5 marks Moderate -0.8
It is given that \(f(x) = \frac{6}{x^2} + 2x\).
  1. Find \(f'(x)\). [3]
  2. Find \(f''(x)\). [2]
OCR C1 2013 June Q4
7 marks Moderate -0.8
  1. Express \(3x^2 + 9x + 10\) in the form \(3(x + p)^2 + q\). [3]
  2. State the coordinates of the minimum point of the curve \(y = 3x^2 + 9x + 10\). [2]
  3. Calculate the discriminant of \(3x^2 + 9x + 10\). [2]
OCR C1 2013 June Q5
6 marks Moderate -0.8
  1. Sketch the curve \(y = \frac{2}{x^2}\). [2]
  2. The curve \(y = \frac{2}{x^2}\) is translated by 5 units in the negative \(x\)-direction. Find the equation of the curve after it has been translated. [2]
  3. Describe a transformation that transforms the curve \(y = \frac{2}{x^2}\) to the curve \(y = \frac{1}{x^2}\). [2]
OCR C1 2013 June Q6
5 marks Moderate -0.8
A circle \(C\) has equation \(x^2 + y^2 + 8y - 24 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. The point \(A(2, 2)\) lies on the circumference of \(C\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
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]