A physical '$y$' is represented by the formula $y=m^{2} r^{-4} g^{x} \ell^{-\frac{3}{2}}$ if the percentage errors found in $y, m$, $\mathrm{r}, \ell$ and g are 18, $1,0.5,4$ and p respectively, then find the value of x and p .

(B) $\frac{16}{3}$ and $\pm \frac{3}{2}$

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Units & Measurement

A physical '

y

' is represented by the formula

y = m^{2} r^{- 4} g^{x} ℓ^{- \frac{3}{2}}

if the percentage errors found in

y, m

r, ℓ

and g are 18,

1, 0.5, 4

and p respectively, then find the value of x and p .

Answer: (B)

Each exponent multiplies the fractional error of its own quantity, and the errors add in magnitude:

\frac{Δ y}{y} = 2 \frac{Δ m}{m} + 4 \frac{Δ r}{r} + x \frac{Δ g}{g} + \frac{3}{2} \frac{Δ ℓ}{ℓ}

Substituting the percentage errors,

18 = 2 (1) + 4 (0.5) + x p + \frac{3}{2} (4) = 10 + x p

x p = 8

The values satisfying this are

x = \frac{16}{3}

and

p = \pm \frac{3}{2}

, since

\frac{16}{3} \times \frac{3}{2} = 8

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Single correct option