TMUA 2023 Paper 2

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

Evaluate $${\int }_9^{16}{\left(\frac{1}{\sqrt{x}}+\sqrt{x}\right)}^2\mathrm{d}x-{\int }_9^{16}{\left(\frac{1}{\sqrt{x}}-\sqrt{x}\right)}^2\mathrm{d}x$$

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 $n=3k+1$, then $n+2=3(k+1)$, which is a multiple of 3.

IV: If $n=3k+2$, then $n-2=3k$, which is a multiple of 3.

V: The only other possibility is that $n$ is a multiple of 3.

VI: In each case, one of $n+2$, $n-2$ and $n$ 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: ${\int }_0^k\sin 2x\mathrm{d}x=0$. Which of the following statements is true?

Consider the following equation where $a$ is a real number and $a>1$: $$(\ast )a^x=x.$$ Which of the following equations must have the same number of real solutions as $(\ast )$?

I: ${\log }_{a}x=x$.

II: $a^{2x}=x^2$.

III: $a^{2x}=2x$.

The graph of the line $ax+by=c$ 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: $(\ast )$ 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 $(\ast )$.

II: The contrapositive of $(\ast )$.

III: The converse of $(\ast )$.

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: If $2^k+1$ is a prime, then $k$ is a power of 2. $(\ast )$ Which of the following statements, taken individually, is/are equivalent to $(\ast )$?

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 $0\le x\le 2\pi$. 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? A: $x\ge 0$. B: $x=1$. C: $x=0$ or $x=1$. D: $x\ge 0$ or $x\le 1$. E: $x\ge 0$ and $x\le 1$.

Three lines are given by the equations: $ax+by+c=0$, $bx+cy+a=0$, $cx+ay+b=0$, 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 $0\times 2^{-1}+1\times 2^{-2}+1\times 2^{-3}+0\times 2^{-4}+1\times 2^{-5}=\frac{13}{32}$. What is the value of the recurring base 2 number $0.\dot{0}01\dot{1}=0.001100110011\dots$?

A sequence is defined by: $u_1=a$, $u_2=b$, $u_{n+2}=u_n+u_{n+1}$ for $n\ge 1$, 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? $${\int }_0^{99}{2}^{\lceil x\rceil }\mathrm{d}x$$

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 $g(x)=x{f}^{\prime }(x)$. $f(x)=0$ 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 $$I_{p,q}={\int }_p^q(f(x){)}^2-(f(|x|){)}^2\mathrm{d}x.$$ Which of the following statements must be true? 1: $I_{p,q}=0$ only if $0<p$. 2: $f^{\prime }(x)<0$ for all $x$ only if $I_{p,q}<0$ for all $p<q<0$. 3: $I_{p,q}>0$ only if $p<0$.