TMUA Challenge Paper 2 Tmua-p2-challenge-2

• A.$n=1$
• B.$n=10$
• C.$n=20$
• D.$n=30$
• E.$n=40$
• F.$n=39$
• G.$n=50$
• H.there is none

Answer: E

• A.If $a+b$ is rational then $a$ and $b$ are both rational.
• B.If $a$ and $b$ are not both rational then $a+b$ is irrational.
• C.If $a+b$ is irrational then at least one of $a$ and $b$ is irrational.
• D.If $a+b$ is irrational then neither $a$ nor $b$ is rational.
• E.If $a$ or $b$ is irrational then $a+b$ is irrational.
• F.If $a$ and $b$ are both irrational then $a+b$ is irrational.
• G.If $a+b$ is rational then at least one of $a$ and $b$ is rational.
• H.If at least one of $a$ and $b$ is irrational then $a+b$ is not rational.

Answer: C

• A.necessary but not sufficient for two distinct real roots.
• B.necessary and sufficient for two distinct real roots.
• C.neither necessary nor sufficient for two distinct real roots.
• D.sufficient for two distinct real roots only when $b=0$.
• E.necessary but not sufficient for two distinct real roots of opposite sign.
• F.sufficient but not necessary for two distinct real roots.
• G.equivalent to the discriminant being positive.
• H.necessary and sufficient for two distinct real roots of the same sign.

Answer: F

Consider these three statements about real numbers.

Which are true?

• A.none of them
• B.$P$ only
• C.$Q$ only
• D.$R$ only
• E.$P$ and $Q$
• F.$P$ and $R$
• G.$Q$ and $R$
• H.$P$, $Q$ and $R$

Answer: G

• A.It is $0$.
• B.It is $1$.
• C.It is odd.
• D.It is even.
• E.It is a multiple of $1013$.
• F.It is at least $1013$.
• G.It is prime.
• H.None of these must be true.

Answer: C

The following argument is offered, where $a$ and $b$ are non-zero real numbers.

At which step does the argument first fail?

• A.step (2)
• B.step (3)
• C.step (4)
• D.step (5)
• E.step (6)
• F.step (1)
• G.more than one step fails
• H.it does not fail

Answer: D

• A.$n^{3}$ and $n$ always have the same parity.
• B.$6$ divides $n^{3}$ for every integer $n$.
• C.$n^{3}-n=n(n^{2}-1)$, and $n^{2}-1$ is always even.
• D.The result holds for $n=1,2,3$ and therefore for all $n$.
• E.$n^{3}\equiv n \pmod 6$ because $6$ has no repeated prime factor.
• F.$n^{3}-n=(n-1)n(n+1)$, and a product of three integers is always divisible by $6$.
• G.$n^{3}-n$ is even, and every even number is divisible by $6$.
• H.$n^{3}-n=(n-1)n(n+1)$ is a product of three consecutive integers, so one of them is a multiple of $3$ and at least one is even.

Answer: H

• A.True: a square is congruent to $0$ or $1$ modulo $3$, so a sum of two squares is divisible by $3$ only when both squares are.
• B.True, but only when $a$ and $b$ are both positive.
• C.False, as $a=1$, $b=2$ shows.
• D.False, as $a=3$, $b=6$ shows.
• E.True, and it follows at once from the fact that $3$ is prime, with no further argument needed.
• F.False, as $a=1$, $b=5$ shows.
• G.True, because $a^{2}+b^{2}=(a+b)^{2}-2ab$ and $3$ must therefore divide $2ab$.
• H.False, as $a=2$, $b=4$ shows.

Answer: A

A sequence $(a_n)$ and a number $L$ are given. Consider the statement

Which of the following is its negation?

• A.For every $\varepsilon>0$ there exists $N$ such that $|a_n-L|\ge\varepsilon$ for all $n>N$.
• B.There exists $\varepsilon>0$ such that for every $N$ there exists $n>N$ with $|a_n-L|\ge\varepsilon$.
• C.There exist $\varepsilon>0$ and $N$ such that $|a_n-L|\ge\varepsilon$ for all $n>N$.
• D.For every $\varepsilon>0$ and every $N$ there exists $n>N$ with $|a_n-L|\ge\varepsilon$.
• E.There exists $\varepsilon>0$ such that $|a_n-L|\ge\varepsilon$ for every $n$.
• F.For every $N$ there exist $\varepsilon>0$ and $n>N$ with $|a_n-L|\ge\varepsilon$.
• G.There exists $N$ such that for every $\varepsilon>0$ there exists $n>N$ with $|a_n-L|\ge\varepsilon$.
• H.For every $\varepsilon>0$ there exists $N$ such that $|a_n-L|>\varepsilon$ for some $n>N$.

Answer: B

• A.$100$ is not divisible by $4$, so no tiling exists.
• B.Colour the board like a chessboard. Each piece covers two squares of each colour, and the two colours occur unequally often.
• C.No piece can be placed in a corner of the board.
• D.The board has an even number of squares, so any tiling would use an odd number of pieces.
• E.Nothing establishes it: a tiling does exist.
• F.Colour the square in row $i$ and column $j$ with $i+j$ modulo $4$. Each piece covers all four colours once, so a tiling would need $25$ squares of each colour; in fact they occur $25$, $26$, $25$ and $24$ times.
• G.Colour the columns in the repeating pattern $1,2,3,4,1,2,\dots$ Each piece covers all four colours, and the colours occur $30$, $30$, $20$ and $20$ times.
• H.A tiling would use $25$ pieces, and $25$ is odd.

Answer: F

The diagram shows a sketch of $y=f'(x)$, the derivative of a function $f$ defined for all real $x$. The curve meets the $x$-axis at $x=-2$ and $x=1$, and touches it at $x=3$.

Which statement about $f$ is true?

• A.$f$ has a local minimum at $x=-2$ and a local maximum at $x=1$.
• B.$f$ has exactly two stationary points.
• C.$f$ has a local maximum at $x=1$ and a local minimum at $x=-2$.
• D.$f$ has a local maximum at $x=-2$, a local minimum at $x=1$, and a stationary point of inflection at $x=3$.
• E.$f$ is decreasing for $x>3$.
• F.$f$ has a local maximum at $x=3$.
• G.$f$ has a local minimum at $x=3$.
• H.$f$ is increasing for $-2<x<1$.

Answer: D

• A.Divide the square into four squares of side $1$; two of the points lie in the same one, and its diagonal is $\sqrt2$.
• B.The five points cannot all lie at corners of the square.
• C.The average of the ten distances between the points is less than $\sqrt2$.
• D.Divide the square into five regions; each point lies in one of them.
• E.No argument works: the four corners together with the centre is a counterexample.
• F.The square has diagonal $2\sqrt2$, and half of $2\sqrt2$ is $\sqrt2$.
• G.Divide the square into two rectangles; three of the points lie in the same one.
• H.Any five points in the plane include two at distance at most $\sqrt2$.

Answer: A

• A.All $45$ red.
• B.All $45$ green.
• C.All $45$ blue.
• D.$15$ of each colour.
• E.$22$ red, $22$ green and $1$ blue.
• F.Any one of "all red", "all green" and "all blue".
• G."All green", but neither of the other two single-colour states.
• H.None of the states listed above.

Answer: H

• A.$b^{2}>4c$
• B.$b^{2}<4c$
• C.$c>0$
• D.$b^{2}\le 4c$
• E.$b^{2}<4c$ and $b>0$
• F.$c>0$ and $b>0$
• G.$b^{2}<4c$ or $c>0$
• H.$b^{2}\ge 4c$

Answer: B

Two squares at opposite ends of a diagonal are removed from an $8\times 8$ board, as shown. Can the remaining $62$ squares be covered exactly by $31$ dominoes, each covering two squares that share an edge?

• A.Yes, and an explicit covering can be written down.
• B.Yes, because $62$ is even.
• C.No, because $62$ is not divisible by $4$.
• D.No, because no domino can be placed in a corner.
• E.Yes, because $62=2\times31$.
• F.No, because an odd number of rows is left uncovered.
• G.No: the two removed squares are the same colour, so $32$ squares of one colour and $30$ of the other remain, while every domino covers one of each.
• H.It depends on which pair of opposite corners is removed.

Answer: G

• A.$S$ contains two elements whose sum is $100$.
• B.$S$ contains three elements in arithmetic progression.
• C.$S$ contains two elements whose difference is $10$.
• D.The elements of $S$ have sum at least $100$.
• E.$S$ has two disjoint non-empty subsets with the same sum.
• F.$S$ contains two elements, one of which divides the other.
• G.$S$ contains a multiple of $10$.
• H.The largest element of $S$ is at least $55$.

Answer: E

• A.The statement and its converse are both true.
• B.The statement is false, as $x=\tfrac12$ shows.
• C.The converse is true and the contrapositive is false.
• D.The statement and its contrapositive are true, but the converse is false, as $x=-1$ shows.
• E.The statement, its converse and its contrapositive are all false.
• F.The converse and the contrapositive are both true.
• G.The statement is true and its contrapositive is false.
• H.The statement is true, and its converse is false, as $x=2$ shows.

Answer: D

• A.$4$
• B.$5$
• C.$6$
• D.$7$
• E.$8$
• F.$9$
• G.$10$
• H.$3$

Answer: C

• A.$(a-b)^{2}\ge 0$
• B.$(a+b)^{2}\ge 0$
• C.$a^{2}+b^{2}\ge 0$
• D.$ab>0$
• E.$\tfrac{a}{b}>0$ and $\tfrac{b}{a}>0$
• F.$a>0$ and $b>0$
• G.$a^{2}+b^{2}>ab$
• H.$(a-b)^{2}>0$

Answer: A

• A.It is false, as $n=3$ shows.
• B.It is true, vacuously: $n^{2}+n=n(n+1)$ is always even, so there is no such $n$.
• C.It is false, as $n=1$ shows.
• D.It is true, and every such $n$ happens to be a multiple of $7$.
• E.It is neither true nor false, because no such $n$ exists.
• F.It is false, as $n=7$ shows.
• G.It is true, but only because $7$ is prime.
• H.It cannot be decided without knowing which integers $n$ are allowed.

Answer: B