If x + y = -10 and xy = 16, which of the following statements must be true?(I)x andndash; y = -6(II)x2 + y2 = 68(III)(x-y)2 leaves a remainder of 1 when divided by 7a)I onlyb)II onlyc)III onlyd)I and IIe)II and IIICorrect answer is option 'E'. Can you explain this answer?

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If x + y = -10 and xy = 16, which of the following statements must be true?

(I) x – y = -6

(II) x² + y² = 68

(III) (x-y)² leaves a remainder of 1 when divided by 7

a)
I only
b)
II only
c)
III only
d)
I and II
e)
II and III

Correct answer is option 'E'. Can you explain this answer?

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Test: Quadratic Equations- 1

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If x + y = -10 and xy = 16, which of the following statements must be ...

Step 1: Question statement and Inferences

It is given that

x+y=−10 ....(i)

xy=16 ....(ii)

We need to find which of the given statements must be true.

Step 2 & 3: Finding required values and calculating the final answer

Squaring (i) on both sides, we get:

(x+y)²=100

x²+y²+2xy=100

....(iii)

We know that

(x−y)²=x²+y²−2xy

(x−y)²=x²+y²+2xy−4xy

(x−y)²=(x+y)²−4xy

Using (ii) and (iii) we get:

(x−y)²=100−64

(x−y)²=36

⇒(x−y)=±6

Therefore (I) is not a must be true statement.

Now let us look at (II).

(iii) – 2*(ii) gives us:

(x²+y²+2xy)−2xy=100−32

⇒x²+y²=68

Therefore statement II is true.

We already observed above that

(x−y)²=36

We know that

36=35+136=7∗5+1

Therefore

(x−y)²

leaves a remainder of 1 when divided by 7.

Answer: Option (E)

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If x + y = -10 and xy = 16, which of the following statements must be true?(I)x – y = -6(II)x2 + y2 = 68(III)(x-y)2 leaves a remainder of 1 when divided by 7a)I onlyb)II onlyc)III onlyd)I and IIe)II and IIICorrect answer is option 'E'. Can you explain this answer?

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