General Physics Question 1
Question 1
- Three vectors $\mathbf{P}, \mathbf{Q}$ and $\mathbf{R}$ are shown in the figure. Let $S$ be any point on the vector $\mathbf{R}$. The distance between the points $P$ and $S$ is $b[\mathbf{R}]$. The general relation among vectors $\mathbf{P}, \mathbf{Q}$ and $\mathbf{S}$ is
(2017 Adv.)
(a) $\mathbf{S}=\left(1-b^{2}\right) \mathbf{P}+b \mathbf{Q}$
(b) $\mathbf{S}=(b-1) \mathbf{P}+b \mathbf{Q}$
(c) $\mathbf{S}=(1-b) \mathbf{P}+b \mathbf{Q}$
(d) $\mathbf{S}=(1-b) \mathbf{P}+b^{2} \mathbf{Q}$
Passage Based Questions
Paragraph
If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $z=x / y$. If the errors in $x, y$ and $z$ are $\Delta x, \Delta y$ and $\Delta z$ respectively, then
$$ z \pm \Delta z=\frac{x \pm \Delta x}{y \pm \Delta y}=\frac{x}{y} \quad 1 \pm \frac{\Delta x}{x} \quad 1 \pm \frac{\Delta y}{y} $$
The series expansion for $1 \pm \frac{\Delta y}{y}^{-1}$, to first power in $\Delta y / y$, is
$1 \mp(\Delta y / y)$. The relative errors in independent variables are always added. So, the error in $z$ will be
$$ \Delta z=z \frac{\Delta x}{x}+\frac{\Delta y}{y} $$
The above derivation makes the assumption that $\Delta x / x \ll 1, \Delta y / y \ll 1$. Therefore, the higher powers of these quantities are neglected.
(There are two questions based on PARAGRAPH " $A$ “, the question given below is one of them)
(2018 Adv.)
Show Answer
Answer:
Correct Answer: 2. (c)
Solution:
- $\mathbf{S}=\mathbf{P}+b \mathbf{R}=\mathbf{P}+b(\mathbf{Q}-\mathbf{P})=\mathbf{P}(1-b)+b \mathbf{Q}$
