Question 1

1 mark

It is a fact that

50=1^{2}+7^{2}=5^{2}+5^{2}=2\times 5\times 5.

Consider the following four statements. (1) Every positive integer that can be written as a sum of two positive squares in two different ways is odd. (2) If a prime

p

can be written as a sum of two positive squares, then

p

leaves remainder

1

on division by

4

or

p=2

. (3) No product of exactly three primes, not necessarily distinct, is a sum of two positive squares. (4) Every positive integer of the form

a^{2}+b^{2}

, where

a

and

b

are positive integers, has only odd prime factors. The fact above provides a counterexample to which of these statements?

A.none of them
B.(1) only
C.(1) and (3) only
D.(1) and (4) only
E.(1), (3) and (4) only
F.(1), (2) and (3) only
G.(2), (3) and (4) only
H.(1), (2), (3) and (4)

Answer: E

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Question 2

1 mark

The following is an attempted proof of the claim: if

x^{3}>x

then

x>1

. (I) Suppose

x^{3}>x

. Then

x^{3}-x>0

, so

x(x-1)(x+1)>0

. (II) So

x\ne 0

, and dividing by the positive number

x^{2}

gives

\dfrac{(x-1)(x+1)}{x}>0

. (III) Hence

x^{2}-1

and

x

have the same sign. (IV) Suppose both are negative. Then

x<0

and

x^{2}<1

. (V) But dividing

x^{3}>x

by

x

gives

x^{2}>1

, which contradicts

x^{2}<1

. So this case is impossible. (VI) Therefore both are positive, so

x>0

and

x^{2}>1

, which gives

x>1

. Which one of the following is the case?

A.The proof is correct.
B.The proof is incorrect, and the first error occurs in line (II).
C.The proof is incorrect, and the first error occurs in line (III).
D.The proof is incorrect, and the first error occurs in line (IV).
E.The proof is incorrect, and the first error occurs in line (V).
F.The proof is incorrect, and the first error occurs in line (VI).

Answer: E

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Question 3

1 mark

Let

k

be a real constant, and let

n

be the number of distinct real roots of

x^{3}-3x=k

. Consider the statements: I:

n=3

if

|k|<2

. II:

n=3

only if

|k|\le 2

. III:

n=1

if and only if

|k|>2

. IV:

n=2

only if

k=2

. Which of the statements are true?

A.I only
B.I and III only
C.I, II and III only
D.I, III and IV only
E.II and III only
F.I and II only
G.III only
H.I, II, III and IV

Answer: C

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Question 4

1 mark

Let

a

be a positive real number. Consider condition P:

\displaystyle\int_{0}^{a}\left(3x^{2}-2x\right)\mathrm{d}x=0

, and condition Q:

a=1

. Which of the following is true?

A.Q is necessary and sufficient for P.
B.Q is necessary but not sufficient for P.
C.Q is sufficient but not necessary for P.
D.Q is neither necessary nor sufficient for P.

Answer: A

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Question 5

1 mark

A function

f

is defined for all real

x

. Consider the two statements: (S) there is a real number

M

such that

f(x)\le M

for every real

x

; (T) there is a real number

a

such that

f(x)\le f(a)

for every real

x

. For which of the following functions is (S) true and (T) false?

A. $f(x)=x^{2}$
B. $f(x)=\sin x$
C. $f(x)=\dfrac{1}{1+x^{2}}$
D. $f(x)=-\dfrac{1}{1+x^{2}}$
E. $f(x)=-x^{2}$
F. $f(x)=2^{x}$

Answer: D

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Question 6

1 mark

The following three statements are all true. (i) If P then Q. (ii) R only if Q. (iii) Not P unless S. Which of the following must also be true? I: if not Q then neither P nor R. II: if Q then P or R. III: if P then both Q and S. IV: if not S then not Q.

A.I only
B.III only
C.I and III only
D.I and II only
E.I, III and IV only
F.II and IV only
G.I, II and III only
H.I, II, III and IV

Answer: C

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Question 7

1 mark

Consider the statement S: if not P then (Q or R). How many of the following five statements are logically equivalent to S? (1) If not Q and not R then P. (2) If Q or R then not P. (3) P or Q or R. (4) If not P and not Q then R. (5) If not Q or not R then P.

A. $1$
B. $2$
C. $3$
D. $4$
E. $5$

Answer: C

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Question 8

1 mark

A student claims that

2^{n}>n^{2}

for every positive integer

n

and offers this proof by induction. Base case:

2^{1}=2>1=1^{2}

. Inductive step: suppose

2^{k}>k^{2}

. Then

2^{k+1}=2\times 2^{k}>2k^{2}\ge (k+1)^{2}

, so the claim holds for

k+1

. Which of the following is correct?

A.The proof is correct and the claim is true.
B.The base case is wrong.
C.The inductive step is invalid for $k=1$ and $k=2$ only, and the claim is false exactly for $n=2,3,4$ .
D.The inductive step is invalid for $k=1$ only, and the claim is false exactly for $n=2$ .
E.The inductive step is invalid for every $k$ , and the claim is false for every $n\ge 2$ .
F.The inductive step is invalid for $k=1$ and $k=2$ only, but the claim is nevertheless true for all $n$ .

Answer: C

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Question 9

1 mark

Let

k

be a real constant. Consider the simultaneous equations

2^{x}+\log_{3}y=5,\qquad k\cdot 2^{x}-\log_{3}y=1,

in real unknowns

x

and

y

, subject to

y>1

. Which of the following is the necessary and sufficient condition on

k

for the system to have a solution?

A. $k>-1$
B. $k\ne -1$
C. $k>0$
D. $k>\tfrac{1}{5}$
E. $k\ge\tfrac{1}{5}$
F. $-1<k<\tfrac{1}{5}$
G. $k>5$

Answer: D

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Question 10

1 mark

Let

k

be a real constant. The equation

\cos x\,(\cos x-k)=0

has exactly

n

distinct solutions in the interval

0\le x\le 2\pi

. Consider the statements: I:

n=4

only if

0<|k|\le 1

. II:

n=3

if

k=-1

. III:

n=2

only if

|k|>1

.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: E

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Question 11

1 mark

Let

p

be a prime number greater than

3

. Which of the following statements must be true? I: at least one of

p+2

and

p+4

is prime. II:

p^{2}-1

is divisible by

24

. III:

p^{2}+2

is not prime.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: G

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Question 12

1 mark

A function

f

defined for all real numbers is called restless if for every real

x

there exists a real

y

with

|y-x|<1

and

f(y)>f(x)

. Which of the following is equivalent to

f

being not restless?

A.There exists a real $x$ such that for every real $y$ with $|y-x|<1$ we have $f(y)\le f(x)$ .
B.There exists a real $x$ such that for every real $y$ with $|y-x|<1$ we have $f(y)<f(x)$ .
C.For every real $x$ there exists a real $y$ with $|y-x|<1$ and $f(y)\le f(x)$ .
D.There exists a real $x$ such that for every real $y$ we have $|y-x|\ge 1$ and $f(y)\le f(x)$ .
E.There exists a real $x$ and there exists a real $y$ with $|y-x|<1$ and $f(y)\le f(x)$ .
F.For every real $x$ and for every real $y$ with $|y-x|<1$ we have $f(y)\le f(x)$ .

Answer: A

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Question 13

1 mark

The function

f

is differentiable for all real

x

. Which of the following statements must be true? I: if

f'(x)>0

for all

x

then

f(x)>0

for some

x

. II: if

f(x)>0

for all

x

then

\displaystyle\int_{0}^{1}f(x)\,\mathrm{d}x>0

. III: if

\displaystyle\int_{0}^{1}f(x)\,\mathrm{d}x>0

then

f(x)>0

for some

x

with

0\le x\le 1

.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: G

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Question 14

1 mark

The function

f

is defined by

f(x)=\dfrac{1}{x-2}

for all real

x\ne 2

. Which of the following statements are true? I: if

a<b

are both in the domain of

f

then

f(a)>f(b)

. II:

f

takes every real value except

0

. III:

f(f(x))

is defined for every real

x\ne 2

.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: C

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Question 15

1 mark

Let

x

be a real number and consider the five statements: (1)

x>2

. (2)

x^{2}>4

. (3)

x^{3}>8

. (4)

|x|>2

. (5)

x>1

. As

x

varies, which of the following is the complete list of possible values for the number of these statements that are true?

A. $0,1,2,3,4,5$
B. $0,1,2,5$
C. $0,1,3,5$
D. $0,2,3,5$
E. $1,2,5$
F. $0,1,2,4,5$

Answer: B

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Question 16

1 mark

Let

b

and

c

be real numbers. Which of the following conditions is sufficient but not necessary for the equation

x^{2}+bx+c=0

to have two distinct positive real roots?

A. $b<0$ and $c>0$
B. $b^{2}>4c$
C. $b<-2$ and $0<c<1$
D. $b<0$ , $c>0$ and $b^{2}>4c$
E. $c>0$ and $b^{2}>4c$
F. $b<-2$ and $c>1$

Answer: C

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Question 17

1 mark

Let

a

and

b

be integers. Five of the following six statements are equivalent to one another. Which one is not equivalent to the others?

A. $ab$ is odd.
B. $a$ and $b$ are both odd.
C. $a^{2}b^{2}$ is odd.
D. $a+b$ is even and $ab$ is odd.
E. $a+ab$ is even and $b+ab$ is even.
F.Neither $a$ nor $b$ is even.

Answer: E

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Question 18

1 mark

Let

a

,

b

and

c

be non-zero real numbers. Which of the following statements must be true? I: if

a,b,c

is both an arithmetic and a geometric progression then

a=b=c

. II: if

a,b,c

is a geometric progression then so is

a^{2},b^{2},c^{2}

. III: if

a,b,c

is an arithmetic progression then so is

a^{2},b^{2},c^{2}

.

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: E

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Question 19

1 mark

Consider the claim: if

n

is prime then

2^{n}-1

is prime. What is the smallest value of

n

that is a counterexample to the claim?

A. $7$
B. $9$
C. $11$
D. $13$
E. $23$
F.there is no counterexample

Answer: C

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Question 20

1 mark

Nine people attend a party, and some pairs of them shake hands, each pair at most once. Afterwards each person reports how many hands they shook. Which of the following lists of nine reports is impossible?

A. $2,2,2,2,2,2,2,2,2$
B. $0,1,2,3,4,5,6,7,8$
C. $1,1,1,1,1,1,1,1,8$
D. $4,4,4,4,4,4,4,4,4$
E. $1,1,1,1,1,1,1,1,2$

Answer: B

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TMUA Challenge Paper 2 Tmua-p2-challenge-3

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20