TMUA MioMath Paper 2: Questions & Worked Solutions

Question 1

1 mark

If

a + b

,

b + c

,

a - c

are all even\nWhat can be inferred about the parity of

a, b, c

?

A. $a$ is even, $b$ and $c$ are odd
B. $a$ and $b$ are even, $c$ is odd
C. $a$ is odd, $b$ is even, $c$ is odd
D.Only one of $a, b, c$ is even
E. $a, b, c$ are all odd or all even

Answer: E

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Question 2

1 mark

An arithmetic sequence

(S_n)

is defined by its first term

a_1

, number of terms

n

, and common difference

d

.\nConsider the following statement: "If

a_1

is odd, then all terms

S_n

are odd." Identify a counterexample for this statement from the options below.

A. $a_1 = 2, d = 2, n = 20$
B. $a_1 = 1, d = 3, n = 10$
C. $a_1 = 3, d = 6, n = 30$
D. $a_1 = 5, d = 4, n = 15$
E. $a_1 = 4, d = 5, n = 8$

Answer: D

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Question 3

1 mark

If the statement P: "For any

n

such that

n \not\equiv 1 \pmod{6}

,

n

is not prime" is false, then which of the following is true?

A.There exist $n \not\equiv 1 \pmod{6}$ and $n$ is prime.
B.There exist $n \equiv 1 \pmod{6}$ and $n$ is prime.
C.For any prime $n, n \equiv 1 \pmod{6}$
D.For any prime $n, n \not\equiv 1 \pmod{6}$

Answer: A

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Question 4

1 mark

Find the number of positive integer solutions for the equation

n

which satisfy

((n-1)!)! = (n-1)!

A.0
B.1
C.2
D.3
E.4
F.5
G.6

Answer: B

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Question 5

1 mark

Laud Polygon: For any interior angle A, there exists an angle B such that

B = 2A

or

B = \frac{A}{2}

.\nA polygon is quiet if it is not a laud polygon.\nWhich of the following is/are quiet?\nI:

30^{\circ}, 60^{\circ}, 90^{\circ}

\nII:

40^{\circ}, 80^{\circ}, 80^{\circ}, 160^{\circ}

\nIII:

60^{\circ}, 120^{\circ}, 120^{\circ}, 120^{\circ}, 120^{\circ}

A.Only I
B.Only II
C.Only III
D.I and II
E.I and III
F.II and III
G.All of them

Answer: D

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Question 6

1 mark

For positive integers

a, b

and

c

, define three least common multiples:

L = \text{lcm}(a,b)

,

M = \text{lcm}(b,c)

, and

N = \text{lcm}(a,c)

. Which of these is not possible?

A. $L < M < N$
B. $L < M = N$
C. $L = M < N$
D. $L = M = N$
E.All of them are possible

Answer: A

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Question 7

1 mark

Given that

0 \le f(x) \le 1

for

0 < x < 4

, and that:

\int_0^2 f(x) dx = 1 \text{ and } \int_1^3 f(x) dx = 1.5

\nWhich of the following are possible values for

\int_0^4 f(x) dx

?\nI: 2.4\nII: 2.8\nIII: 3.2

A.Only I
B.Only II
C.Only III
D.I and II
E.I and III
F.II and III
G.All of them

Answer: D

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Question 8

1 mark

Let P be the statement "

f(x)

is divisible by

(x+2)(x-3)

" and Q be the statement "

f(x)

is divisible by

4x^2 - 4x - 24

".\nDetermine what type of condition P is for Q.

A.A sufficient condition for Q
B.A sufficient but not necessary condition for Q
C.A necessary and sufficient condition for Q
D.Neither a sufficient nor a necessary condition for Q
E.Q is a necessary condition for P
F.P is a sufficient and necessary condition for Q

Answer: C

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Question 9

1 mark

Which of the following is a necessary but insufficient condition for

\sin x > \frac{1}{2}

?

A. $\cos x = 0$
B. $\frac{1}{2} \sin x$
C. $\tan \frac{x}{2} < 1$
D. $\cos x > \frac{\sqrt{3}}{2}$
E. $\sin^2 x > \frac{9}{4}$

Answer: D

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Question 10

1 mark

For a cubic function

ax^3 + bx^2 + cx + d

, there are two local extrema, one at

x > 0

and the other at

x < 0

. What is the necessary and sufficient condition for this to occur?

A. $b > 3a$
B. $b < 3a$
C. $ac > 0$
D. $ac < 0$
E. $b^2 > 3ac$
F. $b^2 > 4ac$

Answer: D

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Question 11

1 mark

Given the function

f(x) = x^3 + px^2 + q

, it has a local maximum at

x = 0

and 3 distinct roots. Find the conditions for

p

and

q

.

A. $\frac{8}{27} p^3 < q < 0$
B. $0 < q < -\frac{8}{27} p^3$
C. $\frac{4}{27} p^3 < q < 0$
D. $0 < q < -\frac{4}{27} p^3$

Answer: C

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Question 12

1 mark

Given a geometric sequence

(S_n)

with the parity of first term

a_1

, the parity of the number of terms

n

, and the parity (odd or even nature) of the common ratio

r

, is it possible to determine the parity of the terms in the sequence

(S_n)

?

A.Possible
B.Impossible, need value of $a_1$
C.Impossible, need value of $n$
D.Impossible, need value of $r$
E.All of them are incorrect

Answer: A

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Question 13

1 mark

Given that

f'(x) < 0

for

0 \le x \le 1

, which of the following statements must be true?\nI.

\int_0^{\frac{1}{2}} f(x) dx > \int_{\frac{1}{2}}^1 f(x) dx

\nII.

\int_0^{\frac{1}{2}} f(x) dx < \int_0^{\frac{1}{2}} f(x^2) dx

\nIII.

f(0) < f(1)

A.Only I
B.Only II
C.Only III
D.I and II
E.I and III
F.II and III
G.All of them

Answer: C

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Question 14

1 mark

Given that

f(x) = ax^2 + bx + c

, which of the following is a sufficient condition for

f(|x|) = k

to have 4 different real number solutions?

A. $b^2 - 4ac < 0$
B. $b = 0$
C. $ab < 0$
D. $a > 0$
E. $b^2 > 4ac$

Answer: C

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Question 15

1 mark

A car starts from depot and moves repeatedly between 4 stations in the following pattern: 1, 2, 3, 4, 3, 2, 1, 2, 3, ...It follows the below statements:\nStatement 1: If the car has stopped at Station 1 a total of 10 times, it must have stopped at Station 2 exactly 20 times.\nStatement 2: If the car has stopped at Station 3 a total of 10 times, it must have stopped at Station 4 exactly 5 times.\nStatement 3: If the car has stopped at Station 2 a total of 10 times, it must have stopped at Station 3 exactly 10 times.\nDetermine which of the following statements are correct.

A.Only statement 1
B.Only statement 2
C.Only statement 3
D.Statement 2 and 3
E.Statement 1 and 2
F.Statement 1 and 3
G.All statements

Answer: D

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Question 16

1 mark

Given the sequence

u_{n+1} = 3u_n - 6

with

u_n \in \mathbb{Z}

, which of the following statements is true?
1. If

6 | u_6

, then

3 | u_3

.
2. If

10 | u_{10}

, then

30 | u_{30}

.

A.Only statement 1
B.Only statement 2
C.Both statements
D.Neither statement

Answer: C

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Question 17

1 mark

Given the function

f(x) = \frac{a}{x} + bx^2

, determine which of the following conditions are true if

f(x)

has a local minimum and a positive root.

A. $a > 0, b > 0$
B. $a > 0, b < 0$
C. $a < 0, b > 0$
D. $a < 0, b < 0$

Answer: D

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Question 18

1 mark

The expansion of the expression

(x^p + \frac{1}{x^q})^n

has a constant term.\nWhich of the following is/are necessary and sufficient conditions for this statement for some integer

k

?\nI:

kp + kq = n

\nII:

kp + kq = nq

\nIII:

kp + kq = np

A.Only I
B.Only II
C.Only III
D.I and II
E.I and III
F.II and III
G.All of them

Answer: F

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Question 19

1 mark

Define the left branch of a quadratic function as the portion to the left of its vertex, and similarly, define the right branch. Let

f(x)

and

g(x)

be two quadratic functions.\nStatement P: If

g(x)

is a translation of

f(x)

, then the left branch of

g(x)

and the left branch of

f(x)

intersect at one point, and the right branch of

g(x)

and the right branch of

f(x)

intersect at one point.\nWhich of the following is correct?

A.Statement P is correct
B.The converse of Statement P is correct
C.The contrapositive of Statement P is correct
D.None of them are correct

Answer: D

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Question 20

1 mark

Consider the following attempt to solve an equation. The steps have been numbered for reference.\nEquation:

3 \cos x = \sqrt{x}

, where

x

is in radians.\nSolving Steps:
(1) Start with the equation:

3 \cos x = \sqrt{x}

.
(2) Square both sides of the equation:

9 \cos^2 x = x

.
(3) Use the identity

\cos^2 x = \frac{1}{2}(1 + \cos 2x)

to rewrite the equation as

9 \cdot \frac{1}{2}(1 + \cos 2x) = x

.
(4) Graph the functions

y = \frac{1}{2}(1 + \cos 2x)

and

y = \frac{x}{9}

.
(5) From the graph, observe that at

x = \pi, y = \pi/9 \approx 0.35

is less than

y = \frac{1}{2}(1 + \cos 2\pi) = 1

, and at

x = 2\pi, y = 2\pi/9 \approx 0.70

is less than 1. The graphs intersect at five points.
(6) Conclude that the equation has five solutions.\nWhich one of the following statements is true?

A.All five solutions are solutions of the original equation.
B.None of the five solutions is a solution of the original equation.
C.Only three solutions are valid, and the incorrect solutions arise as a result of step (1).
D.Only three solutions are valid, and the incorrect solutions arise as a result of step (2).
E.Only three solutions are valid, and the incorrect solutions arise as a result of step (3).

Answer: UNSOLVABLE

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