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NDA (Held On: 18 Apr 2021) Maths Previous Year paper

Option 4 : m = 4n

Electric charges and coulomb's law (Basic)

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10 Questions
10 Marks
10 Mins

**Concept:**

For any n numbers \(\rm x_1 ,x_2 ,x_3 ,..........,x_n\)

GM = \(\rm \sqrt[n]{x_1\times x_2\times x_3\times ..........\times x_n}\)

HM = \(\rm {\left[{1\over x_1}+{1\over x_2}+{1\over x_3}+..........+{1\over x_n}\over n\right]}^{-1}\)

**Calculation:**

GM of m and n = y

\(\rm \sqrt{mn}\) = y

HM of m and n = x

\(\rm {\left[{1\over m}+{1\over n}\over 2\right]}^{-1}\) = x

\(\rm {2mn\over m+ n}\) = x

Now given 5x = 4y

\(\rm {10mn\over m+ n}\) = \(\rm 4\sqrt{mn}\)

\(\rm 10\sqrt{mn}\) = 4(m + n)

100mn = 16(m^{2} + n^{2} + 2mn)

16m2 + 16n2 - 68mn = 0

\(\rm16{m\over n}+16{n\over m} -68= 0\)

Let \(\rm m\over n\) = t

16t + \(\rm 16\over t\) - 68 = 0

16t^{2} - 68t + 16 = 0

t = \(\rm {68 \pm \sqrt{(-68)^2-4(16)(16)} \over 2\times16}\)

t = \(\rm {68 \pm \sqrt{4624-1024} \over 32}\)

t = \(\rm {68 \pm60\over 32}\) = 4 or \(\rm 1\over 4\)

\(\rm m\over n\) = 4 or \(\rm 1\over 4\)

∴ **m = 4n** or n = 4m