TMUA 2018 Paper 2

The function \(f\) is given, for \(x > 0\), by \( f(x) = \dfrac{x^3 - 4 x}{2 \sqrt{x}} \) Find the value of \(f'(4)\).

Find the value of the constant term in the expansion of \( \left(x^6 - \dfrac{1}{x^2}\right)^{12} \)

Consider the following statement: A car journey consists of two parts. In the first part, the average speed is \(u\) km/h. In the second part, the average speed is \(v\) km/h. Hence the average speed for the whole journey is \(\dfrac{1}{2}(u + v)\) km/h. Which of the following examples of car journeys provide(s) a *counterexample* to the statement?
I. In the first part of the journey, the car travels at a constant speed of \(50\) km/h for \(100\) km. In the second part of the journey, the car travels at a constant speed of \(40\) km/h for \(100\) km.
II. In the first part of the journey, the car travels at a constant speed of \(50\) km/h for one hour. In the second part of the journey, the car travels at a constant speed of \(40\) km/h for one hour.
III. In the first part of the journey, the car travels at a constant speed of \(50\) km/h for \(80\) km. In the second part of the journey, the car travels at a constant speed of \(40\) km/h for \(100\) km.

The non-zero real number \(c\) is such that the equation \(\cos x = c\) has two solutions for \(0 < x < \dfrac{3}{2} \pi\). How many solutions of the equation \(\cos^2 2x = c^2\) are there in the range \(0 < x < \dfrac{3}{2} \pi\) ?

The two diagonals of the quadrilateral \(Q\) are perpendicular. Consider the following statements:
I. One of the diagonals of \(Q\) is a line of symmetry of \(Q\).
II. The midpoints of the sides of \(Q\) are the vertices of a square. Which of these statements is/are *necessarily* true for the quadrilateral \(Q\)?

Which one of the following functions provides a *counterexample* to the statement: *if* \(f'(x) > 0\) for all real \(x\), *then* \(f(x) > 0\) for all real \(x\).

Sequence 1 is an arithmetic progression with first term \(11\) and common difference \(3\). Sequence 2 is an arithmetic progression with first term \(2\) and common difference \(5\). Some numbers that appear in Sequence 1 also appear in Sequence 2. Let \(N\) be the 20th such number. What is the remainder when \(N\) is divided by \(7\) ?

The diagram shows an example of a mountain profile. This consists of upstrokes which go upwards from left to right, and downstrokes which go downwards from left to right. The example shown has six upstrokes and six downstrokes. The horizontal line at the bottom is known as sea level. A mountain profile of order \(n\) consists of \(n\) upstrokes and \(n\) downstrokes, with the condition that the profile begins and ends at sea level and *never* goes *below* sea level (although it might reach sea level at any point). So the example shown is a mountain profile of order \(6\). Mountain profiles can be coded by using U to indicate an upstroke and D to indicate a downstroke. The example shown has the code UDUUUDUDDUDD. A sequence of U's and D's obtained from a mountain profile in this way is known as a valid code. Which of the following statements is/are true?
I. If a valid code is written in reverse order, the result is always a valid code.
II. If each U in a valid code is replaced by D and each D by U, the result is always a valid code.
III. If U is added at the beginning of a valid code and D is added at the end of the code, the result is always a valid code.

Consider the following attempt to solve the equation \(4 x \sqrt{2 x - 1} = 10 x - 5\) : \(4 x \sqrt{2 x - 1} = 10 x - 5\) \(\quad\) (I) \(4 x \sqrt{2 x - 1} = 5(2 x - 1)\) \(\quad\) (II) \(16 x^2 (2 x - 1) = 25(2 x - 1)^2\) \(\quad\) (III) \(16 x^2 = 25(2 x - 1)\) \(\quad\) (IV) \(16 x^2 - 50 x + 25 = 0\) \(\quad\) (V) The solutions of the original equation are \(x = \dfrac{5}{8}\) and \(x = \dfrac{5}{2}\). Which one of the following is true?

The function \(f(x)\) is defined for all real numbers. Consider the following three conditions, where \(a\) is a real constant:
I. \(f(a - x) = f(a + x)\) for all real \(x\).
II. \(f(2 a - x) = f(x)\) for all real \(x\).
III. \(f(a - x) = f(x)\) for all real \(x\). Which of these conditions is/are *necessary and sufficient* for the graph of \(y = f(x)\) to have reflection symmetry in the line \(x = a\)?

Consider the equation \(2^x = m x + c\), where \(m\) and \(c\) are real constants. Which of the following statements is/are true?
I. The equation has a negative real solution *only if* \(c > 1\).
II. The equation has two distinct real solutions *if* \(c > 1\).
III. The equation has two distinct positive real solutions *if and only if* \(c \leq 1\).

Consider the following statement: For any positive integer \(N\) there is a positive integer \(K\) such that \(N(K m + 1) - 1\) is not prime for any positive integer \(m\). Which one of the following is the negation of this statement?

The following is an attempted proof of the conjecture: *if* \(\tan \theta > 0\), *then* \(\sin \theta + \cos \theta > 1\). Suppose \(\tan \theta > 0\), so in particular \(\cos \theta \neq 0\). Since \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}\), then \(\sin \theta \cos \theta = \tan \theta \cos^2 \theta > 0\). \(\quad\) (I) It follows that \(1 + 2 \sin \theta \cos \theta > 1\). \(\quad\) (II) Therefore \(\sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta > 1\), \(\quad\) (III) which factorises to give \((\sin \theta + \cos \theta)^2 > 1\). \(\quad\) (IV) Therefore \(\sin \theta + \cos \theta > 1\). \(\quad\) (V) Which one of the following is the case?

In the triangle \(P Q R\), \(P R = 2\), \(Q R = p\) and \(\angle R P Q = 30^{\circ}\). What is the set of *all* the values of \(p\) for which this information uniquely determines the length of \(P Q\)?

It is given that \(f(x) = x^3 + 3 q x^2 + 2\), where \(q\) is a real constant. The equation \(f(x) = 0\) has 3 distinct real roots. Which of the following statements is/are *necessarily* true?
I. The equation \(f(x) + 1 = 0\) has 3 distinct real roots.
II. The equation \(f(x + 1) = 0\) has 3 distinct real roots.
III. The equation \(f(-x) - 1 = 0\) has 3 distinct real roots.

In this question, \(x_1, x_2, x_3, ...\) is an *arithmetic progression*, all of whose terms are integers. Let \(n\) be a positive integer. If the median of the first \(n\) terms of the sequence is an integer, which of the following three statements *must* be true?
I. The median of the first \(n + 2\) terms is an integer.
II. The median of the first \(2 n\) terms is an integer.
III. The median of \(x_2, x_4, x_6, ..., x_{2 n}\) is an integer.

A positive integer is called a squaresum *if and only if* it can be written as the sum of the squares of two integers. For example, \(61\) and \(9\) are both squaresums since \(61 = 5^2 + 6^2\) and \(9 = 3^2 + 0^2\). A prime number is called awkward *if and only if* it has a remainder of \(3\) when divided by \(4\). For example, \(23\) is awkward since \(23 = 5 \times 4 + 3\). A (true) theorem due to Fermat states that: A positive integer is a squaresum *if and only if* each of its awkward prime factors occurs to an even power in its prime factorisation. It follows that \(5 \times 23^2\) is a squaresum, since \(23\) occurs to the power \(2\), but \(5 \times 23^3\) is not, since \(23\) occurs to the power \(3\). Which one of the following statements is *not* true?

\(f(x)\) is a polynomial function defined for all real \(x\). Which of the following is a *necessary* condition for the inequality \( \dfrac{f(a) + f(b)}{2} \geq f\left(\dfrac{a + b}{2}\right) \) to be true for all real numbers \(a\) and \(b\) with \(a < b\) ?

Three *real* numbers \(x\), \(y\) and \(z\) satisfy \(x > y > z > 1\). Which one of the following statements *must* be true?

It is given that the equation \(\sqrt{x + p} + \sqrt{x} = p\) has at least one real solution for \(x\), where \(p\) is a real constant. What is the complete set of possible values for \(p\)?