TMUA 2017 Paper 2

Given that \(y = \dfrac{(1 - 3x)^2}{2 x^{\dfrac{3}{2}}}\), which one of the following is a correct expression for \(\dfrac{d y}{d x}\)?

\(P Q R S\) is a rectangle. The coordinates of \(P\) and \(Q\) are \((0, 6)\) and \((1, 8)\) respectively. The perpendicular to \(P Q\) at \(Q\) meets the \(x\)-axis at \(R\). What is the area of \(P Q R S\)?

The first term of a geometric progression is \(2 - \sqrt{3}\) and the fourth term is \(\dfrac{9}{4}\). What is the sum to infinity of this geometric progression?

The following question appeared in an examination: Given that \(\tan x = \sqrt{3}\), find the possible values of \(\sin 2x\). A student gave the following answer: \(\tan x = \sqrt{3}\) so \(x = 60^{\circ}\) and \(2x = 120^{\circ}\), therefore \(\sin 2x = \dfrac{\sqrt{3}}{2}\). Which one of the following statements is correct?

Consider the following three statements: 1. \(10 p^2 + 1\) and \(10 p^2 - 1\) are both prime when \(p\) is an odd prime. 2. Every prime greater than 5 is of the form \(6n + 1\) for some integer \(n\). 3. No multiple of 7 greater than 7 is prime. The result \(91 = 7 \times 13\) can be used to provide a counterexample to which of the above statements?

A sequence \(u_0, u_1, u_2, ...\) is defined as follows: \(u_0 = 1\) \(u_n = \displaystyle\int_{0}^{1} 4 x u_{n-1} d x\) for \(n \geq 1\) What is the value of \(u_1000\)?

The graphs of two functions are shown here: - \(y = a^x\) is shown with a solid line, where \(a\) is a positive real number - \(y = f(x)\) is shown with a dashed line Which of the following statements (1, 2, 3,
4) could be true? 1. \(f(x) = b^x\) for some \(b > a\) 2. \(f(x) = b^x\) for some \(b < a\) 3. \(f(x) = a^{k x}\) for some \(k > 1\) 4. \(f(x) = a^{k x}\) for some \(k < 1\)

Which one of the following numbers is smallest in value?

Consider the following attempt to prove this true theorem: Theorem: \(a^3 + b^3 = c^3\) has no solutions with \(a\), \(b\) and \(c\) positive integers. Attempted proof: Suppose that there are positive integers \(a\), \(b\) and \(c\) such that \(a^3 + b^3 = c^3\).
I. We have \(a^3 = c^3 - b^3\).
II. Hence \(a^3 = (c - b)(c^2 + c b + b^2)\).
III. It follows that \(a = c - b\) and \(a^2 = c^2 + c b + b^2\), since \(a | a^2\) and \(c - b | c^2 + c b + b^2\).
IV. Eliminating \(a\), we have \((c - b)^2 = c^2 + c b + b^2\).
V. Multiplying out, we have \(c^2 - 2 c b + b^2 = c^2 + c b + b^2\).
VI. Hence \(3 c b = 0\) so one of \(b\) and \(c\) is zero. But this is a contradiction to the original assumption that all of \(a\), \(b\) and \(c\) are positive. It follows that the equation has no solutions. Comment on this proof by choosing one of the following options:

\(f(x)\) is a function defined for all real values of \(x\). Which one of the following is a *sufficient* condition for \(\displaystyle\int_{1}^{3} f(x) d x = 0\)?

The function \(f(x)\) is increasing and \(f(0) = 0\). The positive constants \(a\) and \(b\) are such that \(a < b\). The area of the region enclosed by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = a\) and \(x = b\) is denoted by \(R\). The function \(g(x)\) is defined by \(g(x) = f(x) + 2 f(b)\). Which of the following is an expression for the area enclosed by the curve \(y = g(x)\), the \(x\)-axis and the lines \(x = a\) and \(x = b\)?

The diagram shows the graphs of \(y = \sin 2x\) and \(y = \cos 2x\) for \(-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}\). Which one of the following is *not* true?

The positive real numbers \(a \times 10^{-3}\), \(b \times 10^{-2}\) and \(c \times 10^{-1}\) are each in standard form, and \((a \times 10^{-3}) + (b \times 10^{-2}) = (c \times 10^{-1})\). Which of the following statements (I, II, III,
IV) must be true?
I. \(a > 9\)
II. \(b > 9\)
III. \(a < c\)
IV. \(b < c\)

The diagram below shows the graph of \(y = x^2 - 2 b x + c\). The vertex of this graph is at the point \(P\). Which one of the following could be the graph of \(y = x^2 - 2 B x + c\), where \(B > b\)?

The function \(f\) is defined on the positive integers as follows: \(f(1) = 5\), and for \(n \geq 1\): \(f(n+1) = 3 f(n) + 1\) if \(f(n)\) is odd, \(f(n+1) = \dfrac{1}{2} f(n)\) if \(f(n)\) is even The function \(g\) is defined on the positive integers as follows: \(g(1) = 3\), and for \(n \geq 1\): \(g(n+1) = g(n) + 5\) if \(g(n)\) is odd, \(g(n+1) = \dfrac{1}{2} g(n)\) if \(g(n)\) is even What is the value of \(f(1000) - g(1000)\)?

Consider the following statement: (\(*\)) *If* \(f(x)\) is an integer for every integer \(x\), *then* \(f'(x)\) is an integer for every integer \(x\). Which one of the following is a *counterexample* to (\(*\))?

A set \(S\) of whole numbers is called stapled *if and only if* for every whole number \(a\) which is in \(S\) there exists a prime factor of \(a\) which divides at least one other number in \(S\). Let \(T\) be a set of whole numbers. Which of the following is true *if and only if* \(T\) is *not* stapled?

Consider the following problem: Solve the inequality \(\left(\dfrac{1}{4}\right)^n < \left(\dfrac{1}{32}\right)^{10}\), where \(n\) is a positive integer. A student produces the following argument:
I. \(\left(\dfrac{1}{4}\right)^n < \left(\dfrac{1}{32}\right)^{10}\)
II. \(\log_{\dfrac{1}{2}} \left(\dfrac{1}{4}\right)^n < \log_{\dfrac{1}{2}} \left(\dfrac{1}{32}\right)^{10}\)
III. \(n \log_{\dfrac{1}{2}} \left(\dfrac{1}{4}\right) < 10 \log_{\dfrac{1}{2}} \left(\dfrac{1}{32}\right)\)
IV. \(n < \dfrac{10 \log_{\dfrac{1}{2}} \left(\dfrac{1}{32}\right)}{\log_{\dfrac{1}{2}} \left(\dfrac{1}{4}\right)}\)
V. \(n < \dfrac{10 \times 5}{2} = 25\) Which step (if any) in the argument is invalid?

Which one of the following is a *sufficient* condition for the equation \(x^3 - 3 x^2 + a = 0\), where \(a\) is a constant, to have exactly one real root?

I have forgotten my 5-character computer password, but I know that it consists of the letters \(a, b, c, d, e\) in some order. When I enter a potential password into the computer, it tells me exactly how many of the letters are in the correct position. When I enter abcde, it tells me that none of the letters are in the correct position. The same happens when I enter cdbea and eadbc. Using the best strategy, how many *further* attempts must I make in order to *guarantee* that I can *deduce* the correct password?