TMUA 2017 Paper 2

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

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

The first term of a geometric progression is $2\sqrt{3}$ and the fourth term is $\frac{9}{4}$. 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 $\sin 2x=\frac{\sqrt{3}}{2}$. Which one of the following statements is correct?

Consider the following three statements:
1. $10p^2+1$ and $10p^2-1$ 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 $u_0,u_1,u_2,\dots$ is defined as follows: $$u_0=1,u_n={\int }_0^{1}4x{u}_{n-1}dx\text{for }n⩾1.$$ What is the value of $u_{1000}$?

The graphs of two functions are shown: $y=a^x$ is shown with a solid line, where $a$ is a positive real number, and $y=f(x)$ is shown with a dashed line. The dashed curve lies above the solid curve for $x>0$ and below it for $x<0$, with both passing through $(0,1)$. 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. $f(x)=a^{kx}$ for some $k>1$
4. $f(x)=a^{kx}$ for some $k<1$

Which one of the following numbers is smallest in value? $$\text{A: }{\log }_{2}7\text{B: }(2^{-3}+2^{-2}{)}^{-1}\text{C: }{2}^{\pi /3}\text{D: }\frac{1}{4(\sqrt{2}-1{)}^3}\text{E: }4{\sin }^2\left(\frac{\pi }{4}\right)$$

Consider the following attempt to prove this true 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 $a^3=(c-b)(c^2+cb+b^2)$.

III. It follows that $a=c-b$ and $a^2=c^2+cb+b^2$.

IV. Eliminating $a$, we have $(c-b{)}^2=c^2+cb+b^2$.

V. Multiplying out, we have $c^2-2cb+b^2=c^2+cb+b^2$.

VI. Hence $3cb=0$ so one of $b$ and $c$ is zero.

But this is a contradiction since 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 ${\int }_1^{3}f(x)dx=0$?

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 $g(x)=f(x)+2f(b)$. 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 $-\frac{\pi }{2}<x<\frac{\pi }{2}$. 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 $y=x^2-2bx+c$. The vertex of this graph is at the point $P$. Which one of the following could be the graph of $y=x^2-2Bx+c$, where $B>b$?

(A) Graph

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

Consider the following statement: $(\ast )$ If $f(x)$ is an integer for every integer $x$, then $f^{\prime }(x)$ is an integer for every integer $x$. Which one of the following is a counterexample to $(\ast )$?

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 ${\left(\frac{1}{4}\right)}^n<{\left(\frac{1}{32}\right)}^{10}$, where $n$ is a positive integer. A student produces the following argument: $${\left(\frac{1}{4}\right)}^n<{\left(\frac{1}{32}\right)}^{10}(\text{I})$$ $${\log }_{\frac{1}{2}}{\left(\frac{1}{4}\right)}^n<{\log }_{\frac{1}{2}}{\left(\frac{1}{32}\right)}^{10}(\text{II})$$ $$n{\log }_{\frac{1}{2}}\left(\frac{1}{4}\right)<10{\log }_{\frac{1}{2}}\left(\frac{1}{32}\right)(\text{III})$$ $$n<\frac{10{\log }_{\frac{1}{2}}\left(\frac{1}{32}\right)}{{\log }_{\frac{1}{2}}\left(\frac{1}{4}\right)}(\text{IV})$$ $$n<\frac{10\times 5}{2}=25(\text{V})$$ $$1⩽n⩽24$$ Which step (if any) in the argument is invalid?

Which one of the following is a sufficient condition for the equation $x^3-3x^2+a=0$, 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?