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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Edexcel C2 Q7
11 marks Moderate -0.3
$$f(x) = 2x^3 - 5x^2 + x + 2.$$
  1. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
  2. Fully factorise \(f(x)\). [4]
  3. Solve the equation \(f(x) = 0\). [1]
  4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2\pi\) for which $$2\sin^3 \theta - 5\sin^2 \theta + \sin \theta + 2 = 0,$$ giving your answers in terms of \(\pi\). [4]
Edexcel C2 Q8
13 marks Standard +0.3
The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
  2. Find the exact coordinates of the stationary point of \(C\). [5]
  3. Determine the nature of the stationary point. [2]
  4. Sketch the curve \(C\). [2]
Edexcel C2 Q9
14 marks Standard +0.3
\includegraphics{figure_2} Figure 2 shows the curve \(C\) with equation \(y = 3x - 4\sqrt{x} + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,
  1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2x - 2\). [6]
The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the positive coordinate axes.
  1. Find the area of the shaded region. [8]
OCR C2 Q1
4 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the sector \(OAB\) of a circle of radius 9.2 cm and centre \(O\). Given that the area of the sector is 37.4 cm\(^2\), find to 3 significant figures
  1. the size of \(\angle AOB\) in radians, [2]
  2. the perimeter of the sector. [2]
OCR C2 Q2
6 marks Moderate -0.3
$$f(x) = x^3 + kx - 20.$$ Given that f(x) is exactly divisible by \((x + 1)\),
  1. find the value of the constant \(k\), [2]
  2. solve the equation \(f(x) = 0\). [4]
OCR C2 Q3
7 marks Moderate -0.3
Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{4}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]
OCR C2 Q4
8 marks Moderate -0.3
A geometric progression has third term 36 and fourth term 27. Find
  1. the common ratio, [2]
  2. the fifth term, [2]
  3. the sum to infinity. [4]
OCR C2 Q5
8 marks Standard +0.3
  1. Solve the equation $$\log_2 (6 - x) = 3 - \log_2 x.$$ [4]
  2. Find the smallest integer \(n\) such that $$3^{n-2} > 8^{250}.$$ [4]
OCR C2 Q6
8 marks Moderate -0.8
$$f(x) = \cos 2x, \quad 0 \leq x \leq \pi.$$
  1. Sketch the curve \(y = f(x)\). [2]
  2. Write down the coordinates of any points where the curve \(y = f(x)\) meets the coordinate axes. [3]
  3. Solve the equation \(f(x) = 0.5\), giving your answers in terms of \(\pi\). [3]
OCR C2 Q7
9 marks Moderate -0.8
  1. Find $$\int \left( x + 5 + \frac{3}{\sqrt{x}} \right) dx.$$ [4]
  2. Evaluate $$\int_{-2}^{0} (3x - 1)^2 dx.$$ [5]
OCR C2 Q8
11 marks Moderate -0.3
  1. An arithmetic series has a common difference of 7. Given that the sum of the first 20 terms of the series is 530, find
    1. the first term of the series, [3]
    2. the smallest positive term of the series. [2]
  2. The terms of a sequence are given by $$u_n = (n + k)^2, \quad n \geq 1,$$ where \(k\) is a positive constant. Given that \(u_2 = 2u_1\),
    1. find the value of \(k\), [4]
    2. show that \(u_3 = 11 + 6\sqrt{2}\). [2]
OCR C2 Q9
11 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows the curve \(y = 2x^2 + 6x + 7\) and the straight line \(y = 2x + 13\).
  1. Find the coordinates of the points where the curve and line intersect. [4]
  2. Show that the area of the shaded region bounded by the curve and line is given by $$\int_{-3}^{1} (6 - 4x - 2x^2) dx.$$ [2]
  3. Hence find the area of the shaded region. [5]
OCR C2 Q2
4 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows the curve with equation \(y = 2^x\). Use the trapezium rule with four intervals, each of width 1, to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = -2\) and \(x = 2\). [4]
OCR C2 Q3
6 marks Moderate -0.8
  1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
  2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]
OCR C2 Q4
7 marks Moderate -0.3
  1. Given that \(y = \log_2 x\), find expressions in terms of \(y\) for
    1. \(\log_2 \left(\frac{x}{2}\right)\), [2]
    2. \(\log_2 (\sqrt{x})\). [2]
  2. Hence, or otherwise, solve the equation $$2 \log_2 \left(\frac{x}{2}\right) + \log_2 (\sqrt{x}) = 8.$$ [3]
OCR C2 Q5
8 marks Moderate -0.3
\includegraphics{figure_5} The diagram shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,
  1. find the length \(OA\), [2]
  2. find the perimeter and the area of the shaded segment. [6]
OCR C2 Q6
8 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}} - x\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \((a, 0)\).
  1. Show that \(a = 8\). [3]
  2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis. [5]
OCR C2 Q7
10 marks Moderate -0.3
  1. Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]
    1. Write down the formula for the sum of the first \(n\) positive integers. [1]
    2. Using this formula, find the sum of the integers from 100 to 200 inclusive. [3]
    3. Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3. [2]
OCR C2 Q8
12 marks Standard +0.3
The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.
  1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
  2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]
Using \(x = 6\),
  1. find the common ratio of the series, [1]
  2. find the sum of the first eight terms of the series. [2]
OCR C2 Q9
13 marks Moderate -0.3
  1. Evaluate $$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$ giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [6]
  2. The gradient of a curve is given by $$\frac{dy}{dx} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that the curve passes through the points \((0, -2)\) and \((2, 18)\), show that \(k = 2\) and find an equation for the curve. [7]
OCR C2 Q1
4 marks Easy -1.2
A geometric progression has first term 75 and second term \(-15\).
  1. Find the common ratio. [2]
  2. Find the sum to infinity. [2]
OCR C2 Q2
6 marks Moderate -0.3
Find the area of the finite region enclosed by the curve \(y = 5x - x^2\) and the \(x\)-axis. [6]
OCR C2 Q3
6 marks Moderate -0.8
During one day, a biological culture is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times (1.06)^t.$$ Using this model,
  1. find the number of bacteria present at 11 a.m., [2]
  2. find, to the nearest minute, the time when the initial number of bacteria will have doubled. [4]
OCR C2 Q4
7 marks Moderate -0.3
\includegraphics{figure_4} The diagram shows the curve with equation \(y = (x - \log_{10} x)^2\), \(x > 0\).
  1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.
    \(x\)23456
    \(y\)2.896.36
    [2]
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).
  1. Use the trapezium rule with all the values in your table to estimate the area of the shaded region. [3]
  2. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area. [2]
OCR C2 Q5
8 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos 3x = \frac{\sqrt{3}}{2}.$$ [5]