Two quantities x and y are measured to be $3 \pm Δ x$ and $2 \pm Δ y .$ Three quantities A, B and C are calculated as $A = x + y, B = x - y a n d C = x y .$
Column – I
Column – II
A) Uncertainty in A is P) higher than that in B
B) Uncertainty in B is Q) lower than that in C
C) Uncertainty in $\frac{A}{B}$ is R) higher than that in A
D) Uncertainty in AB is S) higher than that in C

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A – Q; B-Q; C-PRS; D-PRS

A – S; B-P; C-RS; D-PQRS

A – P; B-Q; C-R; D-S

A – PQ; B-RS; C-PQR; D-PRS

answer is A.

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Detailed Solution

$A = x + y \Rightarrow d A = d x + d y B = x - y \Rightarrow d B = d x + d y d [\frac{A}{B}] = d [\frac{x + y}{x - y}] = \frac{(x - y) (d x + d y) + (x + y) (d x - d y)}{{(x - y)}^{2}} = \frac{(d x) (2 x) + d y (- 2 y)}{{(x - y)}^{2}} = \frac{2 (x d x + y d y)}{{(x - y)}^{2}} = \frac{2 [3 d x + 2 d y]}{{(3 - 2)}^{2}} = 6 d x + 4 d y d (A B) = B (d A) + A (d B) = 1 (d x + d y) + 5 (d x + d y) = 6 d x + 6 d y d C = d (x y) = x d y + y d x = 2 d x + 3 d y C l e a r l y, d A = d B < d C < d (\frac{A}{B}) < d (A B)$

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Column – I		Column – II
A)	Uncertainty in A is	P)	higher than that in B
B)	Uncertainty in B is	Q)	lower than that in C
C)	Uncertainty in $\frac{A}{B}$ is	R)	higher than that in A
D)	Uncertainty in AB is	S)	higher than that in C