Questions C2 (1550 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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OCR C2 Q6
8 marks Standard +0.3
\includegraphics{figure_6} The diagram shows triangle \(ABC\) in which \(AC = 8\) cm and \(\angle BAC = \angle BCA = 30°\).
  1. Find the area of triangle \(ABC\) in the form \(k\sqrt{3}\). [4]
The point \(M\) is the mid-point of \(AC\) and the points \(N\) and \(O\) lie on \(AB\) and \(BC\) such that \(MN\) and \(MO\) are arcs of circles with centres \(A\) and \(C\) respectively.
  1. Show that the area of the shaded region \(BNMO\) is \(\frac{8}{3}(2\sqrt{3} - \pi)\) cm\(^2\). [4]
OCR C2 Q7
9 marks Moderate -0.8
  1. Expand \((2 + x)^4\) in ascending powers of \(x\), simplifying each coefficient. [4]
  2. Find the integers \(A\), \(B\) and \(C\) such that $$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
  3. Find the real values of \(x\) for which $$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]
OCR C2 Q8
12 marks Moderate -0.3
  1. The gradient of a curve is given by $$\frac{dy}{dx} = 3 - \frac{2}{x^2}, \quad x \neq 0.$$ Find an equation for the curve given that it passes through the point \((2, 6)\). [6]
  2. Show that $$\int_2^3 (6\sqrt{x} - \frac{4}{\sqrt{x}}) \, dx = k\sqrt{3},$$ where \(k\) is an integer to be found. [6]
OCR C2 Q9
12 marks Standard +0.3
The polynomial f(x) is given by $$\text{f}(x) = x^3 + kx^2 - 7x - 15,$$ where \(k\) is a constant. When f(x) is divided by \((x + 1)\) the remainder is \(r\). When f(x) is divided by \((x - 3)\) the remainder is \(3r\).
  1. Find the value of \(k\). [5]
  2. Find the value of \(r\). [1]
  3. Show that \((x - 5)\) is a factor of f(x). [2]
  4. Show that there is only one real solution to the equation f(x) = 0. [4]
OCR MEI C2 Q1
4 marks Easy -1.2
Differentiate \(x + \sqrt{x^3}\). [4]
OCR MEI C2 Q2
5 marks Moderate -0.8
The gradient of a curve is given by \(\frac{dy}{dx} = \frac{6}{x^3}\). The curve passes through \((1, 4)\). Find the equation of the curve. [5]
OCR MEI C2 Q3
3 marks Moderate -0.8
A and B are points on the curve \(y = 4\sqrt{x}\). Point A has coordinates \((9, 12)\) and point B has \(x\)-coordinate \(9.5\). Find the gradient of the chord AB. The gradient of AB is an approximation to the gradient of the curve at A. State the \(x\)-coordinate of a point C on the curve such that the gradient of AC is a closer approximation. [3]
OCR MEI C2 Q4
4 marks Moderate -0.3
Differentiate \(2x^3 + 9x^2 - 24x\). Hence find the set of values of \(x\) for which the function \(f(x) = 2x^3 + 9x^2 - 24x\) is increasing. [4]
OCR MEI C2 Q5
3 marks Moderate -0.8
Find the set of values of \(x\) for which \(x^2 - 7x\) is a decreasing function. [3]
OCR MEI C2 Q6
2 marks Easy -1.8
Differentiate \(10x^4 + 12\). [2]
OCR MEI C2 Q7
12 marks Moderate -0.3
\includegraphics{figure_7} Fig. 10 shows a solid cuboid with square base of side \(x\) cm and height \(h\) cm. Its volume is \(120\) cm\(^3\).
  1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A\) cm\(^2\), of the cuboid is given by $$A = 2x^2 + \frac{480}{x}.$$ [3]
  2. Find \(\frac{dA}{dx}\) and \(\frac{d^2A}{dx^2}\). [4]
  3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case. [5]
OCR MEI C2 Q8
2 marks Easy -1.2
Differentiate \(6x^{\frac{5}{2}} + 4\). [2]
OCR MEI C2 Q9
5 marks Moderate -0.8
A is the point \((2, 1)\) on the curve \(y = \frac{4}{x^2}\). B is the point on the same curve with \(x\)-coordinate \(2.1\).
  1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places. [2]
  2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A. [1]
  3. Use calculus to find the gradient of the curve at A. [2]
OCR MEI C2 Q10
3 marks Moderate -0.5
The gradient of a curve is given by \(\frac{dy}{dx} = x^2 - 6x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\). [3]
OCR MEI C2 Q11
4 marks Moderate -0.8
A curve has gradient given by \(\frac{dy}{dx} = 6x^2 + 8x\). The curve passes through the point \((1, 5)\). Find the equation of the curve. [4]
OCR MEI C2 Q1
13 marks Moderate -0.3
The gradient of a curve is given by \(\frac{dy}{dx} = 4x + 3\). The curve passes through the point \((2, 9)\).
  1. Find the equation of the tangent to the curve at the point \((2, 9)\). [3]
  2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve. [7]
  3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\). Write down the coordinates of the minimum point of the transformed curve. [3]
OCR MEI C2 Q2
5 marks Moderate -0.3
Find the equation of the normal to the curve \(y = 8x^4 + 4\) at the point where \(x = \frac{1}{2}\). [5]
OCR MEI C2 Q3
13 marks Moderate -0.3
  1. Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
  2. Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
    1. Expand \((2 + h)^4\). [3]
    2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
    3. Show how your result in part (iii) (B) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]
OCR MEI C2 Q4
12 marks Moderate -0.8
  1. Calculate the gradient of the chord joining the points on the curve \(y = x^2 - 7\) for which \(x = 3\) and \(x = 3.1\). [2]
  2. Given that \(f(x) = x^2 - 7\), find and simplify \(\frac{f(3 + h) - f(3)}{h}\). [3]
  3. Use your result in part (ii) to find the gradient of \(y = x^2 - 7\) at the point where \(x = 3\), showing your reasoning. [2]
  4. Find the equation of the tangent to the curve \(y = x^2 - 7\) at the point where \(x = 3\). [2]
  5. This tangent crosses the \(x\)-axis at the point P. The curve crosses the positive \(x\)-axis at the point Q. Find the distance PQ, giving your answer correct to 3 decimal places. [3]
OCR MEI C2 Q5
4 marks Moderate -0.8
In Fig. 5, A and B are the points on the curve \(y = 2^x\) with \(x\)-coordinates 3 and 3.1 respectively. \includegraphics{figure_5}
  1. Find the gradient of the chord AB. Give your answer correct to 2 decimal places. [2]
  2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2^x\) at A. [2]
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]