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(4.53−3.07)2(3.07−2.15)(2.15−4.53)+(3.07−2.15)2(2.15−4.53)(4.53−3.07)+(2.15−4.53)2(4.53−3.07)(3.07−2.15)\frac{(4.53-3.07)^2}{(3.07-2.15)(2.15-4.53)} + \frac{(3.07-2.15)^2}{(2.15-4.53)(4.53-3.07)} + \frac{(2.15-4.53)^2}{(4.53-3.07)(3.07-2.15)} is simplified to

Solution

✅ Correct Option: 4

Let: x=4.53x=4.53, y=3.07y=3.07 and z=2.15z=2.15

The original expression becomes:

(x−y)2(y−z)(z−x)+(y−z)2(z−x)(x−y)+(z−x)2(x−y)(y−z)\dfrac{(x-y)^{2}}{(y-z)(z-x)}+\dfrac{(y-z)^{2}}{(z-x)(x-y)}+\dfrac{(z-x)^{2}}{(x-y)(y-z)}

=(x−y)3+(y−z)3+(z−x)3(x−y)(y−z)(z−x)= \dfrac{(x-y)^{3}+ (y-z)^{3} +(z -x)^{3}}{(x-y)(y-z)(z-x)}

Now, let (x−y)=a,(y−z)=b(x-y) = a , (y-z) = b and (z−x)=c(z-x) = c

We observe, (a+b+c)=x−y+y−z+z−x=0(a + b + c) = x-y+y-z+z-x = 0

Thus, a3+b3+c3=3abca^3 + b^3 + c^3 = 3abc (since a+b+c=0a+b+c = 0)

Thus, the expression becomes:

=a3+b3+c3abc=3abcabc=3= \dfrac{a^3 + b^3 +c^3}{abc}= \dfrac{3abc}{abc} = 3

Hence, the correct option is Option 4.

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