Given that P = 2i − 7j and Q = -5i - 9j and 2P − 3Q + r = 0, find |r|.

here are the solutions to the remaining questions.

Q6) Given that $\mathbf{P} = 2\mathbf{i} - 7\mathbf{j}$ and $\mathbf{Q} = -5\mathbf{i} - 9\mathbf{j}$ and $2\mathbf{P} - 3\mathbf{Q} + \mathbf{r} = 0$, find $|\mathbf{r}|$.

Step 1: Express $\mathbf{r}$ in terms of $\mathbf{P}$ and $\mathbf{Q}$.
Given the equation $2\mathbf{P} - 3\mathbf{Q} + \mathbf{r} = 0$, we can rearrange it to solve for $\mathbf{r}$:

$$\mathbf{r} = -2\mathbf{P} + 3\mathbf{Q}$$

Step 2: Substitute the given vectors $\mathbf{P}$ and $\mathbf{Q}$ into the expression for $\mathbf{r}$.

$$\mathbf{r} = -2(2\mathbf{i} - 7\mathbf{j}) + 3(-5\mathbf{i} - 9\mathbf{j})$$ $$\mathbf{r} = (-4\mathbf{i} + 14\mathbf{j}) + (-15\mathbf{i} - 27\mathbf{j})$$

Step 3: Combine the $\mathbf{i}$ and $\mathbf{j}$ components to find $\mathbf{r}$.

$$\mathbf{r} = (-4 - 15)\mathbf{i} + (14 - 27)\mathbf{j}$$ $$\mathbf{r} = -19\mathbf{i} - 13\mathbf{j}$$

Step 4: Calculate the magnitude of $\mathbf{r}$.
The magnitude of a vector $a\mathbf{i} + b\mathbf{j}$ is given by $\sqrt{a^2 + b^2}$.

$$|\mathbf{r}| = \sqrt{(-19)^2 + (-13)^2}$$ $$|\mathbf{r}| = \sqrt{361 + 169}$$ $$|\mathbf{r}| = \sqrt{530}$$

The magnitude of $\mathbf{r}$ is $\boxed{\sqrt{530}}$.

Q7) A body of mass $2.5 \text{ kg}$ is suspended by two inextensible strings $XP$ and $YP$ which are inclined at $40^\circ$ and $50^\circ$ respectively to the downward vertical. Find the tension in both strings. (Take $g = 10 \text{ ms}^{-2}$)

Step 1: Calculate the weight of the body.
The weight $W$ is the force due to gravity acting on the mass.

$$W = mg$$ $$$$