Questions — OCR MEI C2 (480 questions)

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OCR MEI C2 Q1
13 marks Moderate -0.3
\includegraphics{figure_1} Fig. 9 shows a sketch of the graph of \(y = x^3 - 10x^2 + 12x + 72\).
  1. Write down \(\frac{dy}{dx}\). [2]
  2. Find the equation of the tangent to the curve at the point on the curve where \(x = 2\). [4]
  3. Show that the curve crosses the \(x\)-axis at \(x = -2\). Show also that the curve touches the \(x\)-axis at \(x = 6\). [3]
  4. Find the area of the finite region bounded by the curve and the \(x\)-axis, shown shaded in Fig. 9. [4]
OCR MEI C2 Q2
11 marks Standard +0.3
Fig. 10 shows a sketch of the curve \(y = x^2 - 4x + 3\). The point A on the curve has \(x\)-coordinate 4. At point B the curve crosses the \(x\)-axis. \includegraphics{figure_2}
  1. Use calculus to find the equation of the normal to the curve at A and show that this normal intersects the \(x\)-axis at C (16, 0). [6]
  2. Find the area of the region ABC bounded by the curve, the normal at A and the \(x\)-axis. [5]
OCR MEI C2 Q3
12 marks Moderate -0.3
The point A has \(x\)-coordinate 5 and lies on the curve \(y = x^2 - 4x + 3\).
  1. Sketch the curve. [2]
  2. Use calculus to find the equation of the tangent to the curve at A. [4]
  3. Show that the equation of the normal to the curve at A is \(x + 6y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again. [6]
OCR MEI C2 Q4
12 marks Moderate -0.3
\includegraphics{figure_3} A is the point with coordinates (1, 4) on the curve \(y = 4x^2\). B is the point with coordinates (0, 1), as shown in Fig. 10.
  1. The line through A and B intersects the curve again at the point C. Show that the coordinates of C are \(\left(-\frac{1}{4}, \frac{1}{4}\right)\). [4]
  2. Use calculus to find the equation of the tangent to the curve at A and verify that the equation of the tangent at C is \(y = -2x - \frac{1}{4}\). [6]
  3. The two tangents intersect at the point D. Find the \(y\)-coordinate of D. [2]
OCR MEI C2 Q5
5 marks Moderate -0.5
Find the equation of the tangent to the curve \(y = 6\sqrt{x}\) at the point where \(x = 16\). [5]
OCR MEI C2 Q1
3 marks Easy -1.8
Find \(\int 7x^2 dx\). [3]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{18}{x} + 2\). The curve passes through the point \((3, 6)\). Find the equation of the curve. [5]
OCR MEI C2 Q3
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 6x^{\frac{1}{2}} - 5\). Given also that the curve passes through the point \((4, 20)\), find the equation of the curve. [5]
OCR MEI C2 Q4
3 marks Moderate -0.8
Find \(\int_2^5 (2x^3 + 3) dx\). [3]
OCR MEI C2 Q5
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = 6\sqrt{x} - 2\). Given also that the curve passes through the point \((9, 4)\), find the equation of the curve. [5]
OCR MEI C2 Q6
4 marks Moderate -0.3
Find \(\int_2^5 \left(1 - \frac{6}{x^3}\right) dx\). [4]
OCR MEI C2 Q7
4 marks Easy -1.2
Find \(\int_1^2 (12x^5 + 5) dx\). [4]
OCR MEI C2 Q8
5 marks Moderate -0.8
The gradient of a curve is \(3\sqrt{x} - 5\). The curve passes through the point \((4, 6)\). Find the equation of the curve. [5]
OCR MEI C2 Q9
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6\sqrt{x}\). Find the equation of the curve, given that it passes through the point \((9, 105)\). [4]
OCR MEI C2 Q10
5 marks Moderate -0.8
Find \(\int_1^2 \left(x^4 - \frac{3}{x^2} + 1\right) dx\), showing your working. [5]
OCR MEI C2 Q11
3 marks Easy -1.8
Find \(\int 30x^2 dx\). [3]
OCR MEI C2 Q12
4 marks Easy -1.2
Find \(\int (x^5 + 10x^3) dx\). [4]
OCR MEI C2 Q1
4 marks Easy -1.2
Find \(\int (3x^5 + 2x^{-\frac{1}{2}}) dx\). [4]
OCR MEI C2 Q2
11 marks Moderate -0.3
Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}
  1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]
OCR MEI C2 Q3
3 marks Easy -1.2
Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]
OCR MEI C2 Q4
4 marks Easy -1.2
Find \(\int (20x^4 + 6x^{-\frac{2}{3}}) dx\). [4]
OCR MEI C2 Q5
10 marks Easy -1.2
Find \(\int (12x^5 + \sqrt[5]{x} + 7) dx\). [5]
OCR MEI C2 Q6
5 marks Moderate -0.8
Find \(\int \left(x^{\frac{1}{2}} + \frac{6}{x^3}\right) dx\). [5]
OCR MEI C2 Q7
4 marks Easy -1.2
Find \(\int \left(x^4 + \frac{1}{x^3}\right) dx\). [4]
OCR MEI C2 Q8
5 marks Easy -1.3
  1. Differentiate \(12\sqrt{x}\). [2]
  2. Integrate \(\frac{6}{x^5}\). [3]