TMUA 2023 Paper 2

Given that \( \dfrac{1}{\sqrt{x} - 6} - \dfrac{1}{\sqrt{x} + 6} = \dfrac{3}{11} \) what is the value of \(x\)?

Evaluate \( \displaystyle\int_{9}^{16} \left(\dfrac{1}{\sqrt{x}} + \sqrt{x}\right)^2 d x - \displaystyle\int_{9}^{16} \left(\dfrac{1}{\sqrt{x}} - \sqrt{x}\right)^2 d x \)

Consider the claim: For all positive real numbers \(x\) and \(y\), \( \sqrt{x^y} = x^{\sqrt{y}} \) Which of the following is/are a counterexample to the claim?
I. \(x = 1, y = 16\)
II. \(x = 2, y = 8\)
III. \(x = 3, y = 4\)

A student attempts to answer the following question. What is the largest number of consecutive odd integers that are all prime? The student's attempt is as follows:
I. There are two consecutive odd integers that are prime (for example: 17, 19).
II. Any three consecutive odd integers can be written in the form \(n - 2\), \(n\), \(n + 2\) for some \(n\).
III. If \(n\) is one more than a multiple of 3, then \(n + 2\) is a multiple of 3.
IV. If \(n\) is two more than a multiple of 3, then \(n - 2\) is a multiple of 3.
V. The only other possibility is that \(n\) is a multiple of 3.
VI. In each case, one of the integers is a multiple of 3, so not prime.
VII. Therefore the largest number of consecutive odd integers that are all prime is two. Which of the following best describes this attempt?

Consider the two statements \(R\): \(k\) is an integer multiple of \(\pi\) \(S\): \( \displaystyle\int_{0}^{k} \sin 2x d x = 0 \) Which of the following statements is true?

Consider the following equation where \(a\) is a real number and \(a > 1\): \( (*) \quad a^x = x \) Which of the following equations must have the same number of real solutions as \((*)\)?
I. \(\log_a x = x\)
II. \(a^{2x} = x^2\)
III. \(a^{2x} = 2x\)

The graph of the line \(a x + b y = c\) is drawn, where \(a\), \(b\) and \(c\) are real non-zero constants. Which one of the following is a necessary but not sufficient condition for the line to have a positive gradient and a positive \(y\)-intercept?

A student draws a triangle that is acute-angled or obtuse-angled but not right-angled. The student counts the number of straight lines that divide the triangle into two triangles, at least one of which is right-angled. Which of the following statements is/are true?
I. The student can draw a triangle for which there is exactly 1 such straight line.
II. The student can draw a triangle for which there are exactly 2 such straight lines.
III. The student can draw a triangle for which there are exactly 3 such straight lines.

Consider the following statement about a pentagon \(P\): \((*)\) If at least one of the interior angles in \(P\) is \(108^{\circ}\), then all the interior angles in \(P\) form an arithmetic sequence. Which of the following is/are true?
I. The statement \((*)\)
II. The contrapositive of \((*)\)
III. The converse of \((*)\)

Here is an attempt to solve the inequality \(x^4 - 2x^2 - 3 < 0\) by completing the square: \( x^4 - 2x^2 - 3 < 0 \)
I. if and only if \(x^4 - 2x^2 + 1 < 4\)
II. if and only if \((x^2 - 1)^2 < 4\)
III. if and only if \(-2 < x^2 - 1 < 2\)
IV. if and only if \(x^2 - 1 < 2\)
V. if and only if \(x^2 < 3\)
VI. if and only if \(-\sqrt{3} < x < \sqrt{3}\) Which of the following statements is true?

In this question, \(k\) is a positive integer. Consider the following theorem: \( \text{If } 2^k + 1 \text{ is a prime, then } k \text{ is a power of 2.} \quad (*) \) Which of the following statements, taken individually, is/are equivalent to \((*)\)?
I. If \(k\) is a power of 2, then \(2^k + 1\) is prime.
II. \(2^k + 1\) is not prime only if \(k\) is not a power of 2.
III. A sufficient condition for \(k\) to be a power of 2 is that \(2^k + 1\) is prime.

In this question, \(p\) is a real constant. The equation \(\sin x \cos^2 x = p^2 \sin x\) has \(n\) distinct solutions in the range \(0 \leq x \leq 2 \pi\) Which of the following statements is/are true?
I. \(n = 3\) is sufficient for \(p > 1\)
II. \(n = 7\) only if \(-1 < p < 1\)

Let \(x\) be a real number. Which one of the following statements is a sufficient condition for exactly three of the other four statements?

Three lines are given by the equations: \( a x + b y + c = 0 \), \( b x + c y + a = 0 \), \( c x + a y + b = 0 \) where \(a\), \(b\) and \(c\) are non-zero real numbers. Which one of the following is correct?

The base 10 number \(0.03841\) has the value \( 0 \times 10^{-1} + 3 \times 10^{-2} + 8 \times 10^{-3} + 4 \times 10^{-4} + 1 \times 10^{-5} = 0.03841 \) Similarly, the base 2 number \(0.01101\) has the value \( 0 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3} + 0 \times 2^{-4} + 1 \times 2^{-5} = \dfrac{13}{32} \) What is the value of the recurring base 2 number \(0.001100110011...\) ?

A sequence is defined by: \( u_1 = a \), \( u_2 = b \), \( u_{n+2} = u_n + u_{n+1} \text{ for } n \geq 1 \) where \(a\) and \(b\) are positive integers. The highest common factor of \(a\) and \(b\) is 7. Which of the following statements must be true?
I. \(u_2023\) is a multiple of 7
II. If \(u_1\) is not a factor of \(u_2\), then \(u_1\) is not a factor of \(u_n\) for any \(n > 1\)
III. The highest common factor of \(u_1\) and \(u_5\) is 7

The ceiling of \(x\), written \(\lceil x \rceil\), is defined to be the value of \(x\) rounded up to the nearest integer. For example: \(\lceil \pi \rceil = 4\), \(\lceil 2.1 \rceil = 3\), \(\lceil 8 \rceil = 8\) What is the value of the following integral? \( \displaystyle\int_{0}^{99} 2^{\lceil x \rceil} d x \)

The equation \(x^4 + b x^2 + c = 0\) has four distinct real roots if and only if which of the following conditions is satisfied?

In this question, \(f(x)\) is a non-constant polynomial, and \(g(x) = x f'(x)\) \(f(x) = 0\) for exactly \(M\) real values of \(x\). \(g(x) = 0\) for exactly \(N\) real values of \(x\). Which of the following statements is/are true?
I. It is possible that \(M < N\)
II. It is possible that \(M = N\)
III. It is possible that \(M > N\)

Let \(f\) be a polynomial with real coefficients. The integral \(I_{p,q}\) where \(p < q\) is defined by \( I_{p,q} = \displaystyle\int_{p}^{q} (f(x))^2 - (f(|x|))^2 d x \) Which of the following statements must be true? 1. \(I_{p,q} = 0\) only if \(0 < p\) 2. \(f'(x) < 0\) for all \(x\) only if \(I_{p,q} < 0\) for all \(p < q < 0\) 3. \(I_{p,q} > 0\) only if \(p < 0\)