Questions C3 (1301 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 MEI C3 Q7
18 marks Standard +0.8
Fig. 8 shows part of the curve \(y = \text{f}(x)\), where \(\text{f}(x) = e^{-\frac{1}{5}x} \sin x\), for all \(x\). \includegraphics{figure_8}
  1. Sketch the graphs of (A) \(y = \text{f}(2x)\), (B) \(y = \text{f}(x + \pi)\). [4]
  2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P. [6]
  3. Show that \(\text{f}(x + \pi) = -e^{-\frac{1}{5}\pi}\text{f}(x)\). Hence, using the substitution \(u = x - \pi\), show that $$\int_{\pi}^{2\pi} \text{f}(x)\,dx = -e^{-\frac{1}{5}\pi} \int_{0}^{\pi} \text{f}(u)\,du.$$ Interpret this result graphically. [You should not attempt to integrate f(x).] [8]
OCR MEI C3 Q1
6 marks Moderate -0.3
  1. The function f(x) is defined by $$f(x) = \frac{1-x}{1+x}, x \neq -1.$$ Show that f(f(x)) = x. Hence write down \(f^{-1}(x)\). [3]
  2. The function g(x) is defined for all real x by $$g(x) = \frac{1-x^2}{1+x^2}.$$ Prove that g(x) is even. Interpret this result in terms of the graph of \(y = g(x)\). [3]
OCR MEI C3 Q2
18 marks Standard +0.3
Fig. 9 shows the curve \(y = f(x)\), where $$f(x) = (e^x - 2)^2 - 1, x \in \mathbb{R}.$$ The curve crosses the x-axis at O and P, and has a turning point at Q. \includegraphics{figure_9}
  1. Find the exact x-coordinate of P. [2]
  2. Show that the x-coordinate of Q is \(\ln 2\) and find its y-coordinate. [4]
  3. Find the exact area of the region enclosed by the curve and the x-axis. [5]
The domain of f(x) is now restricted to \(x \geqslant \ln 2\).
  1. Find the inverse function \(f^{-1}(x)\). Write down its domain and range, and sketch its graph on the copy of Fig. 9. [7]
OCR MEI C3 Q3
8 marks Standard +0.3
Fig. 7 shows the curve \(y = f(x)\), where \(f(x) = 1 + 2\arctan x, x \in \mathbb{R}\). The scales on the x- and y-axes are the same. \includegraphics{figure_7}
  1. Find the range of f, giving your answer in terms of \(\pi\). [3]
  2. Find \(f^{-1}(x)\), and add a sketch of the curve \(y = f^{-1}(x)\) to the copy of Fig. 7. [5]
OCR MEI C3 Q4
3 marks Moderate -0.8
Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function gf(x), expressing your answer as simply as possible. [3]
OCR MEI C3 Q5
4 marks Moderate -0.3
Write down the conditions for f(x) to be an odd function and for g(x) to be an even function. Hence prove that, if f(x) is odd and g(x) is even, then the composite function gf(x) is even. [4]
OCR MEI C3 Q6
6 marks Standard +0.3
The function f(x) is defined by $$f(x) = 1 + 2\sin 3x, \quad -\frac{\pi}{6} \leqslant x \leqslant \frac{\pi}{6}.$$ You are given that this function has an inverse, \(f^{-1}(x)\). Find \(f^{-1}(x)\) and its domain. [6]
OCR MEI C3 Q7
3 marks Moderate -0.5
Given that \(f(x) = \frac{1}{2}\ln(x - 1)\) and \(g(x) = 1 + e^{2x}\), show that g(x) is the inverse of f(x). [3]
OCR MEI C3 Q8
3 marks Moderate -0.8
Sketch the curve \(y = 2\arccos x\) for \(-1 \leqslant x \leqslant 1\). [3]
OCR MEI C3 Q1
5 marks Moderate -0.3
Find the exact value of \(\int_0^2 \sqrt{1+4x} \, dx\), showing your working. [5]
OCR MEI C3 Q2
18 marks Standard +0.3
Fig. 8 shows the line \(y = x\) and parts of the curves \(y = f(x)\) and \(y = g(x)\), where $$f(x) = e^{x-1}, \quad g(x) = 1 + \ln x.$$ The curves intersect the axes at the points A and B, as shown. The curves and the line \(y = x\) meet at the point C. \includegraphics{figure_8}
  1. Find the exact coordinates of A and B. Verify that the coordinates of C are \((1, 1)\). [5]
  2. Prove algebraically that \(g(x)\) is the inverse of \(f(x)\). [2]
  3. Evaluate \(\int_0^1 f(x) \, dx\), giving your answer in terms of \(e\). [3]
  4. Use integration by parts to find \(\int \ln x \, dx\). Hence show that \(\int_{e^{-1}}^1 g(x) \, dx = \frac{1}{e}\). [6]
  5. Find the area of the region enclosed by the lines OA and OB, and the arcs AC and BC. [2]
OCR MEI C3 Q3
19 marks Standard +0.3
A curve is defined by the equation \(y = 2x \ln(1 + x)\).
  1. Find \(\frac{dy}{dx}\) and hence verify that the origin is a stationary point of the curve. [4]
  2. Find \(\frac{d^2y}{dx^2}\) and use this to verify that the origin is a minimum point. [5]
  3. Using the substitution \(u = 1 + x\), show that \(\int \frac{x^2}{1+x} \, dx = \int \left(u - 2 + \frac{1}{u}\right) du\). Hence evaluate \(\int_0^1 \frac{x^2}{1+x} \, dx\), giving your answer in an exact form. [6]
  4. Using integration by parts and your answer to part (iii), evaluate \(\int_0^1 2x \ln(1 + x) \, dx\). [4]
OCR MEI C3 Q4
4 marks Moderate -0.3
Find \(\int xe^{3x} \, dx\). [4]
OCR MEI C3 Q5
4 marks Moderate -0.3
Show that \(\int_1^4 \frac{x}{x^2 + 2} \, dx = \frac{1}{2} \ln 6\). [4]
OCR MEI C3 Q1
4 marks Standard +0.3
Either prove or disprove each of the following statements.
  1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.' [2]
  2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]
OCR MEI C3 Q2
4 marks Moderate -0.3
  1. Disprove the following statement: $$3^n + 2 \text{ is prime for all integers } n \geqslant 0.$$ [2]
  2. Prove that no number of the form \(3^n\) (where \(n\) is a positive integer) has 5 as its final digit. [2]
OCR MEI C3 Q3
4 marks Moderate -0.3
  1. Factorise fully \(n^3 - n\). [2]
  2. Hence prove that, if \(n\) is an integer, \(n^3 - n\) is divisible by 6. [2]
OCR MEI C3 Q5
3 marks Easy -1.2
Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}
OCR MEI C3 Q6
4 marks Standard +0.3
  1. Multiply out \((3^n + 1)(3^n - 1)\). [1]
  2. Hence prove that if \(n\) is a positive integer then \(3^{2n} - 1\) is divisible by 8. [3]
OCR MEI C3 Q7
3 marks Moderate -0.8
State whether the following statements are true or false; if false, provide a counter-example.
  1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
  2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
  3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational. [3]
OCR MEI C3 Q8
3 marks Moderate -0.8
  1. Disprove the following statement. $$\text{'If } p > q, \text{ then } \frac{1}{p} < \frac{1}{q}.$$ [2]
  2. State a condition on \(p\) and \(q\) so that the statement is true. [1]
OCR MEI C3 Q9
7 marks Standard +0.3
  1. Show that
    1. \((x - y)(x^2 + xy + y^2) = x^3 - y^3\),
    2. \((x + \frac{1}{2}y)^2 + \frac{3}{4}y^2 = x^2 + xy + y^2\). [4]
  2. Hence prove that, for all real numbers \(x\) and \(y\), if \(x > y\) then \(x^3 > y^3\). [3]
OCR MEI C3 Q10
4 marks Easy -1.2
  1. Verify the following statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{ less than 11'.} [2]
  2. Calculate \(23 \times 89\), and hence disprove this statement: $$\text{'} 2^p - 1 \text{ is a prime number for all prime numbers } p \text{'.} [2]
OCR MEI C3 Q11
3 marks Moderate -0.5
Use the method of exhaustion to prove the following result. No 1- or 2-digit perfect square ends in 2, 3, 7 or 8 State a generalisation of this result. [3]
OCR MEI C3 Q12
2 marks Moderate -0.8
Prove that the following statement is false. For all integers \(n\) greater than or equal to 1, \(n^2 + 3n + 1\) is a prime number. [2]