TMUA 2020 Paper 2

Find the complete set of values of \(k\) for which the line \(y = x - 2\) crosses or touches the curve \(y = x^2 + k x + 2\)

Given that \(\tan \theta = 2\) and \(180^{\circ} < \theta < 360^{\circ}\), find the value of \(\cos \theta\)

A student makes the following claim: For all integers \(n\), the expression \(4 \left(\dfrac{9 n + 1}{2} - \dfrac{3 n - 1}{2}\right)\) is divisible by 3. Here is the student's argument: (I) \(4 \left(\dfrac{9 n + 1}{2} - \dfrac{3 n - 1}{2}\right) = 2 (2 \left(\dfrac{9 n + 1}{2} - \dfrac{3 n - 1}{2}\right))\) (II) \(= 2 (9 n + 1 - 3 n - 1)\) (III) \(= 2 (6 n)\) (IV) \(= 12 n\) (V) \(= 3 (4 n)\) (VI) which is always a multiple of 3. So the expression \(4 \left(\dfrac{9 n + 1}{2} - \dfrac{3 n - 1}{2}\right)\) is always divisible by 3. Which one of the following is true?

Consider the following statement: Every positive integer \(N\) that is greater than 6 can be written as the sum of two non-prime integers that are greater than 1. Which of the following is/are counterexample(s) to this statement?
I. \(N = 5\)
II. \(N = 7\)
III. \(N = 9\)

Which one of the following shows the graph of \(y = \dfrac{2^x}{1 + 2^x}\) (Dotted lines indicate asymptotes.)

The function \(f(x)\) is defined for all real values of \(x\). Which of the following conditions on \(f(x)\) is/are necessary to ensure that \(\displaystyle\int_{-5}^0 f(x) d x = \displaystyle\int_{0}^{5} f(x) d x\) Condition I: \(f(x) = f(-x)\) for \(-5 \leq x \leq 5\) Condition II: \(f(x) = c\) for \(-5 \leq x \leq 5\), where \(c\) is a constant Condition III: \(f(x) = -f(-x)\) for \(-5 \leq x \leq 5\)

Consider the following conditions on a parallelogram \(P Q R S\), labelled anticlockwise:
I. length of \(P Q\) = length of \(Q R\)
II. The diagonal \(P R\) intersects the diagonal \(Q S\) at right angles
III. \(\angle P Q R = \angle Q R S\) Which of these conditions is/are individually sufficient for the parallelogram \(P Q R S\) to be a square?

A student is asked to prove whether the following statement \((*)\) is true or false: \((*)\) For all real numbers \(a\) and \(b\), \(|a + b| < |a| + |b|\) The student's proof is as follows: Statement \((*)\) is false. A counterexample is \(a = 3\), \(b = 4\), as \(|3 + 4| = 7\) and \(|3| + |4| = 7\), but \(7 < 7\) is false. Which of the following best describes the student's proof?

A student wishes to evaluate the function \(f(x) = x \sin x\), where \(x\) is in radians, but has a calculator that only works in degrees. What could the student type into their calculator to correctly evaluate \(f(4)\) ?

The real numbers \(a\), \(b\), \(c\) and \(d\) satisfy both \(0 < a + b < c + d\) and \(0 < a + c < b + d\) Which of the following inequalities must be true?
I. \(a < d\)
II. \(b < c\)
III. \(a + b + c + d > 0\)

A spiral line is drawn as shown. This spiral pattern continues indefinitely. Which one of the following points is not on the spiral line?

Which one of A–F correctly completes the following statement? Given that \(a < b\), and \(f(x) > 0\) for all \(x\) with \(a < x < b\), the trapezium rule produces an overestimate for \(\displaystyle\int_{a}^{b} f(x) d x\) ...

\(f(x)\) is a function for which \(\displaystyle\int_{0}^{3} (f(x))^2 d x + \displaystyle\int_{0}^{3} f(x) d x = \displaystyle\int_{0}^{1} f(x) d x\) Which of the following claims about \(f(x)\) is/are necessarily true?
I. \(f(x) \leq 0\) for some \(x\) with \(1 \leq x \leq 3\)
II. \(\displaystyle\int_{0}^{3} f(x) d x \leq \displaystyle\int_{0}^{1} f(x) d x\)

An arithmetic sequence \(T\) has first term \(a\) and common difference \(d\), where \(a\) and \(d\) are non-zero integers. Property \(P\) is: For some positive integer \(m\), the sum of the first \(m\) terms of the sequence is equal to the sum of the first \(2 m\) terms of the sequence. For example, when \(a = 11\) and \(d = -2\), the sequence \(T\) has property \(P\), because \(11 + 9 + 7 + 5 = 11 + 9 + 7 + 5 + 3 + 1 + (-1) + (-3)\) i.e. the sum of the first 4 terms equals the sum of the first 8 terms. Which of the following statements is/are true?
I. For \(T\) to have property \(P\), it is sufficient that \(a d < 0\).
II. For \(T\) to have property \(P\), it is necessary that \(d\) is even.

Which one of the following is a necessary and sufficient condition for \(\displaystyle\sum_{k=1}^n \sin\left(\dfrac{k \pi}{3}\right) = \dfrac{\sqrt{3}}{2}\) to be true?

The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial \(f\): \(\dfrac{d}{d x} (\displaystyle\int_{0}^{x} f(t) d t) = f(x)\) A student calculates \(\dfrac{d}{d x} \displaystyle\int_{x}^{2 x} t^2 d t\) as follows: (I) \(\displaystyle\int_{x}^{2 x} t^2 d t = \displaystyle\int_{0}^{2 x} t^2 d t - \displaystyle\int_{0}^{x} t^2 d t\) (II) By FTC, \(\dfrac{d}{d x} (\displaystyle\int_{0}^{x} t^2 d t) = x^2\) (III) By FTC, \(\dfrac{d}{d x} (\displaystyle\int_{0}^{2 x} t^2 d t) = (2 x)^2 = 4 x^2\) (IV) So \(\dfrac{d}{d x} (\displaystyle\int_{x}^{2 x} t^2 d t) = 4 x^2 - x^2\) (V) giving \(\dfrac{d}{d x} (\displaystyle\int_{x}^{2 x} t^2 d t) = 3 x^2\) Which of the following best describes the student's calculation?

There are two sets of three integers. The first set of three integers has a mean of 10 and a median of 8. The second set of three integers has a mean of 12 and a median of 9. What is the smallest possible range of the set of all six integers?

In this question, \(f(x) = a x^3 + b x^2 + c x + d\) and \(g(x) = p x^3 + q x^2 + r x + s\) are cubic polynomials. If \(f(x) - g(x) > 0\) for every real \(x\), which of the following is/are necessarily true?
I. \(a > p\)
II. if \(b = q\) then \(c = r\)
III. \(d > s\)

Nine people are sitting in the squares of a 3 by 3 grid, one in each square, as shown. Two people are called neighbours if they are sitting in squares that share a side. (People in diagonally adjacent squares, which only have a point in common, are not called neighbours.) Each of the nine people in the grid is either a truth-teller who always tells the truth, or a liar who always lies. Every person in the grid says: 'My neighbours are all liars'. Given only this information, what are the smallest number and the largest number of people who could be telling the truth?

\(x\) is a real number and \(f\) is a function. Given that exactly one of the following statements is true, which one is it?