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Question

If a and b are two numbers such that a2+b2=7 and a3+b3=10, then which is the wrong statement.


A

greatest value of a+b is 5

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B

greatest value of a+b is 4

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C

greatest value of a+b is 1

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D

least value of a+b is 1

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Solution

The correct option is C

greatest value of a+b is 1


Explanation for the correct option:

Step 1: Framing the equations:

Given, a2+b2=7 and a3+b3=10

We know,

a+b3=a3+b3+3ab(a+b)a+b3=10+3ab(a+b)a3+b3=1010=a+b3-3ab(a+b)10=a+ba+b2-3ab10=a+ba2+b2+2ab-3aba+b2=a2+b2+2ab10=a+b7-aba2+b2=7...(i)

We also know

a+b2=a2+b2+2ab7=a+b2-2ab2ab=a+b2-7ab=a+b2-72

Putting ab in (i)

a+b7-a+b2-72=10a+b21-a+b2=2021a+b-a+b3=20a+b3-21a+b+20=0...(ii)

Step 2: Finding the greatest and least value

Let a+b=x

Then the above equation becomes

x3-21x+20=0

Substituting x=1 in this equation,

13-211+20=00=0LHS=RHS

So x-1 is a factor of x3-21x+20=0

By long division of polynomials,

(x-1)(x2+x-20)=0(x-1)(x+5)(x-4)=0

Thus,

x-1=0x=1x+5=0x=-5x-4=0x=4

But, a+b=x

So,

Least value of a+b=-5

Greatest value of a+b=4

Greatest value of a+b=5

Least value of a+b=1

Hence, options C is the wrong statement.


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