I think answer is wrong as

am=-1 when two lines are perpendicular.

Pls Comment

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The proof of understanding is the ability to explain it.

B is the odd man out. B can't coexist with the rest of the options.

a=1; m=1; b,d=0

All are true but B.
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~fluke

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This solution could be best explained with a figure;

When do you think the slopes of two lines are equal; either when they are parallel to each other or when they overlap.


Let's consider the case when they coincide:
So; the slopes are equal. "2". and both the lines intercept y-axis at same point. Thus, b=d. With these conditions; all of the
given options but B are possible.

a=m=2
am=-1. Not possible because=2*2=4; not -1
a^2=m^2=2^2=4
|2|=\sqrt{m^2}=\sqrt{2^2}=2
am=2*2>0

For parallel lines as well; all of the above hold true; because parallel lines have same slope

Let's consider two parallel lines with -ve slope = -3. The only difference in parallel lines and coinciding lines is that they intercept y-axis at different points. Here; we don't care about b or d(y-intercept).
a=m=-3
am=-3*-3=9!=-1
a^2=m^2=(-3)^2=9
|a|=\sqrt{m^2}=3
am=-3*-3>0

When the lines are not parallel; their slopes will not be equal. For unequal slopes;
Say a=1; m=2 or anything other than 1.
A. a!=m
B. am!=-1
C. a^2!=m^2

Thus; these options don't hold good for two lines with different slopes.

What if these two lines are perpendicular;
a=-1; m=1

a!=m
am=-1. Possible
a^2=m^2=1. Possible
|1|=\sqrt{m^2}=\sqrt{(-1)^2}=1 Possible
am=-1<0. Not possible.
We see that A and E both aren't true for the perpendicular lines.

Thus; we can conclude that the given equations are equation of parallel lines or overlapping lines, where option B is the odd man out.
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~fluke

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Please check the options;

option E. am>0; but when the lines are perpendicular
a=1 m=-1
a*m=1*-1=-1
-1<0 not greater than 0.
So option E is ruled out.

am=-1 means option B holds true or B is possible.
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~fluke

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