please solve my doubts

[studygirl]

If a and b are integers, is b even?
(1) 3a + 4b is even.
(2) 3a + 5b is even.
A. Statement (1)

ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

Q33:
In the decimal representation of

x, where 0 < x < 1, is the tenths digit of x nonzero?
(1) 16x is an integer.
(2) 8x is an integer.
A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

Q32:
If

xyz ≠ 0, is x (y + z) ≥0?
(1) │y + z│ = │y│ + │z│

(2)

│x + y│ = │x│ + │y│

A. Statement (1)

ALONE is sufficient, but statement (2) alone is not sufficient.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

[clock60]

q1
my answer C
(1) 3a+4b=even
4b-even so 3a-must be even and it means that a-even but no information about b insufficient
(2) 3a+5b=even two cases exist
even+even=even, and a,b must be both even
odd+odd=even and a,b must be odd so 2 st insufficient
together
from first a-even and b must be even so C for me

3a+4b=2k
3a+5b=2m
b=2(m-k) so b-even

[clock60]

q2 my pick for B
(1) 16x-integer
if x=1/2=0,5 16*0,5=8-integer and unit digit 5
if x=1/16=0,0625, 1/16*16=1-integer but unit digit 0
so 1 st insufficient
(2) 8x-integer
x=1/2=0,5, 8*0,5=4-integer unit digit 5
x=1/4=0,25 8*0,25=2-integer unit digit 2
x=1/8=0,125, 0,125*8=1-integer unit digit 1
so B as in all cases unit digit is not 0

[clock60]

q3 my pick for C (twice thinking)

[studygirl]

Originally Posted by clock60

q2 my pick for B
(1) 16x-integer
if x=1/2=0,5 16*0,5=8-integer and unit digit 5
if x=1/16=0,0625, 1/16*16=1-integer but unit digit 0
so 1 st insufficient
(2) 8x-integer
x=1/2=0,5, 8*0,5=4-integer unit digit 5
x=1/4=0,25 8*0,25=2-integer unit digit 2
x=1/8=0,125, 0,125*8=1-integer unit digit 1
so B as in all cases unit digit is not 0

ANSWER GIVEN IS A

[xperience]

Originally Posted by clock60

if x=1/2=0,5 16*0,5=8-integer and unit digit 5

This is incorrect. The quesitons says the tenth's digit of the decimal. 0.5 is just the unit's digit. It does not have any tenths digit after the decimal.

The only option is that since 16x is an integer, so x must be 1/16 = 0.0625
If you consider any other decimal numberx, such that 16x is an integer, you will always get the ten's digit as non zero.

So 1 is suffucuent

[xperience]

Originally Posted by studygirl

Q32:
If xyz ≠ 0, is x (y + z) ≥0?
(1) │y + z│ = │y│ + │z│

(2)

│x + y│ = │x│ + │y│

A. Statement (1)

ALONE is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE is sufficient.

E. Statements (1) and (2) TOGETHER are NOT sufficient.

The answer for this should be (C)

From (1), we see that for │y + z│ = │y│ + │z│ , the numbers y and z need to be positive integers.
Hence y and z both are greater than zero. However since we do not have any information about x, we cannot say if x (y + z) ≥0 is valid or not

from (2), we can see that for │x + y│ = │x│ + │y│ , both of x and y need to be positive integers and only then this equation holds true. However since we do not know whether z is positive or not, we cannot say this is sufficient.

Using both, we can say that all the 3 variables x, y and z are positiver and hence the given question can be answered.
hence (C) should be the answer

[junoon]

Xperience: To the right of decimal point in a number, we start with the tenth digit, unit digit doesnt exist.

[clock60]

we see that for │y + z│ = │y│ + │z│ , the numbers y and z need to be positive integers.

what about y=-3, z=-5
|-3-5|=|-8|=8
|-3|=3, |-5|=5, 3+5=8...

[achar_varun]

Clock60 is right. both A and B hold true for negative values. But the answer still remains C because both cases are true only if both variables have the same sign, hence all the variables have the same sign. So the calculations remain the same.

Ex, |x+y| = |x| + |y|

x=5, y= -3

|x+y|= 2
and |x| + |y|= 8.
Hence the case fails for these values.

Clock60 is right about Q2. Answer should be B. The first digit after the decimal is called tenths. Clock60 got confused with the names, but the answer is still correct.

studygirl, could you re-check the answer for Q2?