Questions — OCR C1 (333 questions)

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OCR C1 2013 January Q1
5 marks Moderate -0.8
  1. Solve the equation \(x^2 - 6x - 2 = 0\), giving your answers in simplified surd form. [3]
  2. Find the gradient of the curve \(y = x^2 - 6x - 2\) at the point where \(x = -5\). [2]
OCR C1 2013 January Q2
6 marks Easy -1.3
Solve the equations
  1. \(3^n = 1\), [1]
  2. \(t^{-3} = 64\), [2]
  3. \((8p^6)^{\frac{1}{3}} = 8\). [3]
OCR C1 2013 January Q3
5 marks Moderate -0.3
  1. Sketch the curve \(y = (1 + x)(2 - x)(3 + x)\), giving the coordinates of all points of intersection with the axes. [3]
  2. Describe the transformation that transforms the curve \(y = (1 + x)(2 - x)(3 + x)\) to the curve \(y = (1 - x)(2 + x)(3 - x)\). [2]
OCR C1 2013 January Q4
6 marks Moderate -0.3
  1. Solve the simultaneous equations $$y = 2x^2 - 3x - 5, \quad 10x + 2y + 11 = 0.$$ [5]
  2. What can you deduce from the answer to part (i) about the curve \(y = 2x^2 - 3x - 5\) and the line \(10x + 2y + 11 = 0\)? [1]
OCR C1 2013 January Q5
6 marks Easy -1.3
  1. Simplify \((x + 4)(5x - 3) - 3(x - 2)^2\). [3]
  2. The coefficient of \(x^2\) in the expansion of $$(x + 3)(x + k)(2x - 5)$$ is \(-3\). Find the value of the constant \(k\). [3]
OCR C1 2013 January Q6
10 marks Easy -1.3
  1. The line joining the points \((-2, 7)\) and \((-4, p)\) has gradient 4. Find the value of \(p\). [3]
  2. The line segment joining the points \((-2, 7)\) and \((6, q)\) has mid-point \((m, 5)\). Find \(m\) and \(q\). [3]
  3. The line segment joining the points \((-2, 7)\) and \((d, 3)\) has length \(2\sqrt{13}\). Find the two possible values of \(d\). [4]
OCR C1 2013 January Q7
8 marks Easy -1.3
Find \(\frac{dy}{dx}\) in each of the following cases:
  1. \(y = \frac{(3x)^2 \times x^4}{x}\), [3]
  2. \(y = ^3\sqrt{x}\), [3]
  3. \(y = \frac{1}{2x^3}\). [2]
OCR C1 2013 January Q8
7 marks Standard +0.3
The quadratic equation \(kx^2 + (3k - 1)x - 4 = 0\) has no real roots. Find the set of possible values of \(k\). [7]
OCR C1 2013 January Q9
9 marks Moderate -0.3
A circle with centre \(C\) has equation \(x^2 + y^2 - 2x + 10y - 19 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle. [3]
  2. Verify that the point \((7, -2)\) lies on the circumference of the circle. [1]
  3. Find the equation of the tangent to the circle at the point \((7, -2)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
OCR C1 2013 January Q10
10 marks Standard +0.3
Find the coordinates of the points on the curve \(y = \frac{1}{3}x^3 + \frac{9}{x}\) at which the tangent is parallel to the line \(y = 8x + 3\). [10]
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]