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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | GCSE Higher |
If a, b and c are positive integers use the following statements to find the values of a, b and c.
$$ (ab^c)^3 = 27b^{21} $$ $$ b= 9a $$2. | IB Analysis and Approaches |
It is given that \( \log_{10} x = \frac{1}{4} \), where \( x > 0 \).
Find the value of
(a) \( \log_{10} \left(\frac{1}{x}\right) \);
(b) \( \log_{1000} x \).
3. | IB Standard |
Find the value of the following:
(a) \(log_464\);
(b) \(log_7\frac17\);
(c) \(log_{25}5\);
(d) Use the solutions to the previous parts of this question to help solve:
$$log_464+log_7\frac17-log_{25}5=log_4x$$4. | IB Standard |
Evaluate the following, giving your answers as integers.
(a) \(\log _5 25\)
(b) \(\log _6 3 + \log _6 12\)
(c) \(\log _2 12 - \log _2 6\)
5. | IB Standard |
Find the value of
(a) \(\log _4 2 + \log _4 8\)
(b) \(\log_2 60-\log_2 15\)
(c) \(27^{\log_3 4}\)
6. | IB Applications and Interpretation |
In an old science fiction book the author described the intensity of reverse polarity, \(P\) measured in treckons, is a function of the nebula thrust, \(N\) measures in whovians. The intensity level is given by the following formula.
$$P = 7 \log_{10}(N \times 10^{8}), N \ge 0$$(a) An space shuttle has a nebula thrust of \(9.1 × 10^{-3}\) whovians. Calculate the intensity level, \(P\) of the shuttle.
(b) A different space shuttle has an intensity level of 112 trekons. Find its nebula thrust, \(N\).
7. | IB Standard |
(a) Solve \(4x^2 - 8x - 5 = 0\)
(b) Hence solve \(4 \times 25^x - 8 \times 5^x = 5\)
8. | IB Analysis and Approaches |
(a) Show that \( \log_4 (\sin 2x +2) = \log_2 \sqrt{\sin 2x + 2 }\)
(b) Hence or otherwise solve \( \log_2 (2 \cos x) = \log_4 (\sin 2x + 2) \) to show that \(x = \frac12 \tan^{-1} 2 \).
9. | IB Standard |
An arithmetic sequence has \(u_1 = \log_h(j)\) and \(u_2 = \log_h(jk)\), where \(h > 1\) and \(j, k \gt 0\).
(a) Show that the common difference, \(d = \log_h(k)\).
(b) Let \(j = h^5\) and \(k = h^7\). Find the value of \( \sum_{n=1}^{16} u_n \).
10. | IB Standard |
Consider the function \(f (x) = \log_p(24x - 18x^2)\) , for \(0 \lt x \lt 1\), where \(p \gt 0\).
The equation \(f (x) = 3\) has exactly one solution. Find the value of \(p\).
11. | IB Standard |
Two functions are defined as follows: \(f(x) = 2\ln x\) and \(g(x) = \ln \frac{x^2}{3}\).
(a) Express \(g(x)\) in the form \(f(x) - \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .
(b) The graph of \(g(x)\) is a transformation of the graph of \(f(x)\) . Give a full geometric description of this transformation.
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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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