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# If a^2 + b^2 = 1, is (a + b) = 1? (1) ab = 0 (2) b = 0

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Joined: 02 Sep 2009
Posts: 60728
If a^2 + b^2 = 1, is (a + b) = 1? (1) ab = 0 (2) b = 0  [#permalink]

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30 Oct 2018, 04:16
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61% (01:32) correct 39% (01:08) wrong based on 71 sessions

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If a^2 + b^2 = 1, is (a + b) = 1?

(1) ab = 0
(2) b = 0

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Intern
Joined: 07 Jun 2018
Posts: 22
Location: India
Concentration: Entrepreneurship, Marketing
Schools: Wharton '21, Haas '21
Re: If a^2 + b^2 = 1, is (a + b) = 1? (1) ab = 0 (2) b = 0  [#permalink]

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30 Oct 2018, 04:28
let's say 1) is true:
Then a^2 + b^2 = a^2 + 2*ab + b^2 = (a + b)^2 = 1 (given)
Taking square root of both sides,
(a + b) = +-1
Hence, 1) alone is not sufficient.

With 2) alone as true,
a^2 + b^2 = a^2 = 1
Therefore, a = +-1
Hence, (a + b) = +-1
2) Alone is not sufficient.

If 1) and 2) are both true, then b = 0 and then a = +-1, again.
Therefore, we cannot say from the given two conditions whether (a+b) = 1 for sure.

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Joined: 26 Mar 2019
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Re: If a^2 + b^2 = 1, is (a + b) = 1? (1) ab = 0 (2) b = 0  [#permalink]

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05 Dec 2019, 09:40
Quote:
If a^2 + b^2 = 1, is (a + b) = 1?

(1) ab = 0
(2) b = 0

From the task description we have that $$a^2 + b^2 = 1$$. Note that in the task it is not specified that both a and b are integers. Since then, it can be possible that $$a =\frac{\sqrt{3}}{2}$$ and $$b=\frac{1}{2}$$. In this case $$a^2 + b^2 = (\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2 = \frac{1}{4} + \frac{3}{4} = 1$$
Even though both $$a^2$$ and $$b^2$$ are positive, it is not necessarily that $$a$$ and $$b$$ are also positive. It is possible that $$a = 0$$ and $$b = -1$$: $$a^2 + b^2 = (0)^2 + (-1)^2 = 0 + 1= 1$$
Let us look closer to each statement.

Statement 1:
1) $$ab = 0$$
Since $$ab = 0$$, we can conclude that either $$a$$ or $$b$$ is equal to $$0$$. However, if we assume that $$b = 0$$, $$a$$ can be both $$1$$ and $$-1$$. In case $$a = -1$$, $$a + b = (-1) + 0 = -1$$ and in case $$a = 1$$, $$a + b = 1 + 0 = 1$$.
The statement is clearly insufficient.

Statement 2:
2) $$b = 0$$
This statement states the same thing that the statement 1 and thus, it is also insufficient.

Both statements 1 and 2 together state the same thing and for this reason both of them are insufficient.

Re: If a^2 + b^2 = 1, is (a + b) = 1? (1) ab = 0 (2) b = 0   [#permalink] 05 Dec 2019, 09:40
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