TMUA 2022 D513/02: Questions & Worked Solutions

Question 1

1 mark

Determine the number of stationary points on the curve with equation

y = 3x^4 + 4x^3 + 6x^2 - 5

A.0
B.1
C.2
D.3
E.4

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 2

1 mark

Find the coefficient of the

x^5

term in the expansion of

(1+x)^5 \times \sum_{i=0}^5 x^i

A.1
B.5
C.16
D.25
E.32

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 3

1 mark

Consider the following statement about the positive integer

n

if

n

is prime, then

n^2 + 2

is not prime
Which of the following is a counterexample to this statement?
I

n = 2

II

n = 3

III

n = 4

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

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 4

1 mark

The point

P

has coordinates

(p, q)

, and the equation of a circle is

x^2 + 2fx + y^2 + 2gy + h = 0

where

f, g, h, p

and

q

are all real constants.
Let

L

be the distance between the centre of the circle and the point

P

.
Which one of the following is sufficient on its own to be able to calculate

L

?

A.the values of $f, g$ and $h$
B.the values of $f, g, p$ and $q$
C.the values of $f, h, p$ and $q$
D.the values of $g, h, p$ and $q$
E.none of the options A-D is sufficient on its own

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 5

1 mark

A straight line

L

passes through

(1,2)

.
Let

P

be the statement
if the

y

-intercept of

L

is negative, then the

x

-intercept of

L

is positive.
Which of the following statements must be true?
I

P

II the converse of

P

III the contrapositive of

P

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: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 6

1 mark

A list consists of

n

integers.
Consider the following statements:

P

:

n

is odd.

Q

: The median of the list is one of the numbers in the list.
Which one of the following is true?

A.P is necessary and sufficient for Q.
B.P is necessary but not sufficient for Q.
C.P is sufficient but not necessary for Q.
D.P is not necessary and not sufficient for Q.

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 7

1 mark

Consider the following claim:
The difference between two consecutive positive cube numbers is always prime.
Here is an attempted proof of this claim:
I

(x + 1)^3 = x^3 + 3x^2 + 3x + 1

II Taking

x

to be a positive integer, the difference between two consecutive cube numbers can be expressed as

(x + 1)^3 - x^3 = 3x^2 + 3x + 1

III It is impossible to factorise

3x^2 + 3x + 1

into two linear factors with integer coefficients because its discriminant is negative.
IV Therefore for every positive integer value of

x

the integer

3x^2 + 3x + 1

cannot be factorised.
V Hence, the difference between two consecutive cube numbers will always be prime.
Which of the following best describes this proof?

A.The proof is completely correct, and the claim is true.
B.The proof is completely correct, but there are counterexamples to the claim.
C.The proof is wrong, and the first error occurs on line I.
D.The proof is wrong, and the first error occurs on line II.
E.The proof is wrong, and the first error occurs on line III.
F.The proof is wrong, and the first error occurs on line IV.
G.The proof is wrong, and the first error occurs on line V.

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 8

1 mark

A selection,

S

, of

n

terms is taken from the arithmetic sequence

1, 4, 7, 10, ..., 70

.
Consider the following statement:
(*) There are two distinct terms in

S

whose sum is

74

.
What is the smallest value of

n

for which (*) is necessarily true?

A.12
B.13
C.14
D.21
E.22
F.23

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 9

1 mark

Consider the following statement:
(*) For all real numbers

x

, if

x < k

then

x^2 < k

What is the complete set of values of

k

for which (*) is true?

A.no real numbers
B. $k>0$
C. $k<1$
D. $k \le 1$
E. $0<k<1$
F. $0<k \le 1$
G.all real numbers

Answer: A

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 10

1 mark

Which of the following statements is/are true?
I For all real numbers

x

and for all positive integers

n

,

x < n

II For all real numbers

x

, there exists a positive integer

n

such that

x < n

III There exists a real number

x

such that for all positive integers

n

,

x < n

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

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 11

1 mark

The diagram shows a kite

PQRS

whose diagonals meet at

O

.

OP = x

OQ = y

OR = x

OS = z

Which of the following is necessary and sufficient for angle

SPQ

to be a right angle?

A. $x = y = z$
B. $2x = y + z$
C. $x^2 = yz$
D. $y = z$
E. $y^2 = x^2 + z^2$

Answer: C

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 12

1 mark

Place the following integrals in order of size, starting with the smallest.

P = \int_0^1 2^{\sqrt{x}} \, dx

Q = \int_0^1 2^x \, dx

R = \int_0^1 (\sqrt{2})^x \, dx

A. $P < Q < R$
B. $P < R < Q$
C. $Q < P < R$
D. $Q < R < P$
E. $R < P < Q$
F. $R < Q < P$

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 13

1 mark

Consider the statement (*) about a real number

x

:
(*) There exists a real number

y

such that

x - xy + y

is negative.
For how many real values of

x

is (*) true?

A.no values of $x$
B.exactly one value of $x$
C.exactly two values of $x$
D.all except exactly two values of $x$
E.all except exactly one value of $x$
F.all values of $x$

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 14

1 mark

Consider the two inequalities:

|x+5|<|x+11|

|x+11|<|x+1|

Which one of the following is correct?

A.There is no real number for which both inequalities are true.
B.There is exactly one real number for which both inequalities are true.
C.The real numbers for which both inequalities are true form an interval of length 1.
D.The real numbers for which both inequalities are true form an interval of length 2.
E.The real numbers for which both inequalities are true form an interval of length 3.
F.The real numbers for which both inequalities are true form an interval of length 4.
G.The real numbers for which both inequalities are true form an interval of length 5.

Answer: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 15

1 mark

The real numbers

x, y

and

z

are all greater than

1

, and satisfy the equations

\log_x y = z

and

\log_y z = x

Which one of the following equations for

\log_z x

must be true?

A. $\log_z x = y$
B. $\log_z x = \frac{1}{y}$
C. $\log_z x = xy$
D. $\log_z x = \frac{1}{xy}$
E. $\log_z x = xz$
F. $\log_z x = \frac{1}{xz}$
G. $\log_z x = yz$
H. $\log_z x = \frac{1}{yz}$

Answer: F

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 16

1 mark

In this question,

a_1, ..., a_{100}

and

b_1, ..., b_{100}

and

c_1, ..., c_{100}

are three sequences of integers such that

a_n \le b_n + c_n

for each

n

.
Which of the following statements must be true?
I

(\text{minimum of } a_1, ..., a_{100}) \le (\text{minimum of } b_1, ..., b_{100}) + (\text{minimum of } c_1, ..., c_{100})

II

(\text{minimum of } a_1, ..., a_{100}) \ge (\text{minimum of } b_1, ..., b_{100}) + (\text{minimum of } c_1, ..., c_{100})

III

(\text{maximum of } a_1, ..., a_{100}) \le (\text{maximum of } b_1, ..., b_{100}) + (\text{maximum of } c_1, ..., c_{100})

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: D

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 17

1 mark

A student answered the following question:

a

and

b

are non-zero real numbers.
Prove that the equation

x^3 + ax^2 + b = 0

has three distinct real roots if

27b\left(b + \frac{4a^3}{27}\right) < 0

Here is the student's solution:
I We differentiate

y = x^3 + ax^2 + b

to get

\frac{dy}{dx} = 3x^2 + 2ax = x(3x + 2a)

Solving

\frac{dy}{dx} = 0

shows that the stationary points are at

(0,b)

and

\left(-\frac{2a}{3}, b + \frac{4a^3}{27}\right)

II If

27b\left(b + \frac{4a^3}{27}\right) < 0

, then

b

and

b + \frac{4a^3}{27}

must have opposite signs, and so one of the stationary points is above the

x

-axis and one is below.
III If the cubic has three distinct real roots, then one of the stationary points is above the

x

-axis and one is below.
IV Hence if

27b\left(b + \frac{4a^3}{27}\right) < 0

, then the equation has three distinct real roots.
Which one of the following options best describes the student's solution?

A.It is a completely correct solution.
B.The student has instead proved the converse of the statement in the question.
C.The solution is wrong, because the student should have stated step II after step III.
D.The solution is wrong, because the student should have shown the converse of the result in step II.
E.The solution is wrong, because the student should have shown the converse of the result in step III.

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 18

1 mark

P, Q, R and S show the graphs of

y = (\cos x)^{\cos x}

,

y = (\sin x)^{\sin x}

,

y = (\cos x)^{\sin x}

and

y = (\sin x)^{\cos x}

for

0 < x < \frac{\pi}{2}

in some order.

Which row in the following table correctly identifies the graphs?

A.y = (cosx)^cosx: P, y = (sinx)^sinx: Q, y = (cosx)^sinx: R, y = (sinx)^cosx: S
B.y = (cosx)^cosx: P, y = (sinx)^sinx: Q, y = (cosx)^sinx: S, y = (sinx)^cosx: R
C.y = (cosx)^cosx: Q, y = (sinx)^sinx: P, y = (cosx)^sinx: R, y = (sinx)^cosx: S
D.y = (cosx)^cosx: Q, y = (sinx)^sinx: P, y = (cosx)^sinx: S, y = (sinx)^cosx: R
E.y = (cosx)^cosx: R, y = (sinx)^sinx: S, y = (cosx)^sinx: P, y = (sinx)^cosx: Q
F.y = (cosx)^cosx: R, y = (sinx)^sinx: S, y = (cosx)^sinx: Q, y = (sinx)^cosx: P
G.y = (cosx)^cosx: S, y = (sinx)^sinx: R, y = (cosx)^sinx: P, y = (sinx)^cosx: Q
H.y = (cosx)^cosx: S, y = (sinx)^sinx: R, y = (cosx)^sinx: Q, y = (sinx)^cosx: P

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 19

1 mark

A polygon has

n

vertices, where

n \ge 3

. It has the following properties:
* Every vertex of the polygon lies on the circumference of a circle

C

.
* The centre of the circle

C

is inside the polygon.
* The radii from the centre of the circle

C

to the vertices of the polygon cut the polygon into

n

triangles of equal area.
For which values of

n

are these properties sufficient to deduce that the polygon is regular?

A.no values of $n$
B. $n = 3$ only
C. $n = 3$ and $n = 4$ only
D. $n = 3$ and $n \ge 5$ only
E.all values of $n$

Answer: B

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →

Question 20

1 mark

The functions

f_1

to

f_5

are defined on the real numbers by

f_1(x) = \cos x

f_2(x) = \sin(\cos x)

f_3(x) = \cos(\sin(\cos x))

f_4(x) = \sin(\cos(\sin(\cos x)))

f_5(x) = \cos(\sin(\cos(\sin(\cos x))))

where all numbers are taken to be in radians.
These functions have maximum values

m_1, m_2, m_3, m_4

and

m_5

, respectively.
Which one of the following statements is true?

A. $m_1, m_2, m_3, m_4$ and $m_5$ are all equal to 1
B. $0<m_5<m_4<m_3 <m_2 < m_1 = 1$
C. $m_1 = m_3 = m_5 = 1$ and $0 < m_2 = m_4 <1$
D. $m_1 = m_3 = m_5 = 1$ and $0<m_4<m_2<1$
E. $m_1 = m_3 = 1$ and $0<m_2 = m_4<1$ and $0<m_5<1$
F. $m_1 = m_3 = 1$ and $0<m_4<m_2<1$ and $0<m_5<1$

Answer: E

Full step-by-step worked solution

Locked. Available with a free account.

Unlock worked solutions →