Data Sufficiency - 49

Does the curve (x-a)^2 + (y-b)^2=16 intersect the y-axis ?

(1) a^2+b^2>16
(2) a=|b|+5

Answer: B

The given curve will intersect the y-axis when x=0
Thus we get a^2 + (y-b)^2 = 16
<=> a^2 + y^2 + b^2 - 2yb = 16
<=> y^2 - 2yb + a^2 + b^2 -16 = 0

In order to have real roots b^2 - 4ac >= 0

=> 4b^2 - 4(1)(a^2 + b^2 -16) >=0
=> a^2 <=16

From statement (1): Given that a^2 + b^2 > 16
No information about a^2 <=16 --- hence insufficient

From statement (2): Given a = |b|+5
=> |b| is positive or zero.
=> a is atleast equal to 5 and value of a^2 is atleast 25.
But we know that a^2 <=16 ====> does not intersect the y axis ---- hence sufficient

Hence the answer B

Data Sufficiency - 48

In the number line, are x and y on different sides of zero point?

1). The distance from x to zero is equal to the distance from y to 1
2). The sum of the distance from x to zero and the distance from y to 1 is less than 1

Answer: E

From statement (1): |x-0|=|y-1|
x = y-1
x = 1-y ....hence insufficient

From statement (2): |x-0|+|y-1| less than 1
= x+y-1 less than 1
= x+1-y less than 1
= -x+y-1 less than 1
= -x+1-y less than 1 ....hence insufficient

Taking statements (1) and (2) together: still insufficient

Data Sufficiency - 47

If Line K in the XY-Plane has equation y=mx+b, where m and b are constants, what is the slope of K?

1. K is parallel to the line with equation y=(1-m)x+(b+1)
2. K intersects the line with equation y=2x+3 at the point (2,7)

Answer: A

From statement (1): y=(1-m)x+(b+1) has the same slope as y=mx+b. (Parallel lines have same slope)
Thus 1-m = m
implies Slope of K=m=1/2 ---- Hence sufficient

From statement (2): just says line y=2x+3 is not parallel to K, these two lines can have any angle between them other than 0, 180, 360 degrees ---- hence insufficient

Hence answer A

Data Sufficiency - 40

In XY plane, does the line with equation y=3x+2 contain point (r,s)?

1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r - 6 - s)(3r + 2 - s) = 0

Answer: C

Given that y = 3x+2 implies that does 3x+2-y = 0 contains the point (r,s) implies is (3r+2-s) = 0 ?

From statement (1): (3r+2-s)(4r+9-s) = 0 implies either (3r+2-s) = 0 or (4r+9-s) = 0.
Now when (3r+2-s)...the line passes through (r,s)
When (4r+9-s) = 0 ...we cannot determine that whether the line passes through (r,s) or not.
Hence insufficient

From statement (2): (4r-6-s)(3r+2-s) = 0 implies either (4r-6-s) = 0 or (3r+2-s) = 0
Now when (4r-6-s) = 0 ... we cannot determine that whether the line passes through (r,s) or not
When (3r+2-s) = 0..the line passes through (r,s)
Hence insufficient

Taking statement (1) and (2) together: (3r+2-s)(4r+9-s) = 0 and (4r-6-s)(3r+2-s) =0... We cannot have both 4r+9-s=0 and 4r-6-s=0 so it is (3r+2-s) = 0 ... only this equation makes both the equations to be 0
Hence sufficient

Data Sufficiency - 34

In the xy-plane, at what two points does the graph of y=(x+a)(x+b) intersect the x-axis?

1). a+b= -1
2). The graph intersects the y-axis at (0,-6)

Answer: C

From Statement (1) -- a+b = -1...no information about a and b ...hence insufficient

From Statement (2) -- If x = 0 the y = -6 thus ab = 6....insufficient

Taking statements (1) and (2) together: (x+a)*(x+b)=0

x^2+(a+b)x+ab=0

Hence x=-3, x=2

Thus the answer C

DS Question - 17

In the xy-plane, the line k passes through the origin and through the point (a, b), where ab is not equal to 0. Is b positive ?

1). The slope of line k is negative.
2). a is less than b

Answer : C

It is given that the line passes through (0,0) and (a,b)
So it's slope = (b-0)/(a-0) = b/a

Statement (1) ---- Slope of line k is (-ve).
=> (b/a) less than 0 implies a and b are of opposite signs.
From the above we cannot conclude that b is less than 0.
Statement (2) ---- a is less than b ...again this alone cannot help us to infer that whether b is greater than 0 or less than 0 as b can take the value either way round.

Combining Statement (1) and (2) ---- we know that a and b are of opposite signs and that a is less than b.
Therefore, clearly a is negative and hence b is positive.

Hence the answer is C.