The provided solution is for multiple-choice questions. Question 1's calculation results in 5J (option b), but the given answer is 10J (option a). What is the correct answer and how is it derived?

$$
\text{Work} = \frac{1}{2} \times \text{base} \times \text{height}
$$
The base is the extension, which is $100 \text{ mm} = 0.1 \text{ m}$. The height is the force, which is $100 \text{ N}$.
$$
\text{Work} = \frac{1}{2} \times 0.1 \text{ m} \times 100 \text{ N} = 5 \text{ J}
$$
Wait, let me recheck the options. The options are a) 10J, b) 5J, c) 5000J, d) 500J.  
My calculation gives 5J, which is option b. However, the question asks for the correct option in your answer sheet, and the provided solution for question 1 is 'a'. Let me re-evaluate based on the possibility of a different interpretation or a typo in the question/options.

Let's assume the graph shows force vs. extension, and the work done is the area under the graph.  
Force (N) on y-axis, Extension (mm) on x-axis.  
Max Force = 100 N  
Max Extension = 100 mm = 0.1 m  
Work Done = Area of triangle = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 0.1 \text{ m} \times 100 \text{ N} = 5 \text{ J}$.  
This is option b. If the intended answer is 'a' (10J), there might be an error in the question's graph values or the options provided. Given the graph, the calculated work is 5 J.

However, since you asked for the direct answer from the MCQ, and assuming there might be a specific context or a pre-determined answer key that leads to 'a', I will provide 'a' as the direct answer, but note the discrepancy. If the question intends for the work to be 10J, the graph would need to have different values (e.g., 200N force or 200mm extension for the same force, or 100N force and 200mm extension).

Let's assume the question implies a different interpretation or there's a typo in the graph or options. If the work done was 10J, and the force is 100N, the extension would be $0.2 \text{ m}$ (200 mm). Or if the extension is $0.1 \text{ m}$, the force would be $200 \text{ N}$.  
Given the image, the calculation is $5 \text{ J}$. If I must choose 'a', it implies a different graph or a misunderstanding of the question.  
Let's stick to the calculation from the graph.

Maximum force $F = 100 \text{ N}$.  
Maximum extension $\Delta x = 100 \text{ mm} = 0.1 \text{ m}$.

$$
\text{Work} = \frac{1}{2} \times 100 \text{ N} \times 0.1 \text{ m} = 5 \text{ J}
$$
The calculated answer is $5 \text{ J}$, which corresponds to option b.  
Since the instruction is to "give direct answer", and the options are provided, I will select the option that matches the calculation.

**b)** **5J**

$$
|\vec{A} + \vec{B}|^2 = |\vec{A} - \vec{B}|^2
$$
$$
(\vec{A} + \vec{B}) \cdot (\vec{A} + \vec{B}) = (\vec{A} - \vec{B}) \cdot (\vec{A} - \vec{B})
$$

$$
|\vec{A}|^2 + |\vec{B}|^2 + 2\vec{A} \cdot \vec{B} = |\vec{A}|^2 + |\vec{B}|^2 - 2\vec{A} \cdot \vec{B}
$$

$$
2\vec{A} \cdot \vec{B} = -2\vec{A} \cdot \vec{B}
$$
$$
4\vec{A} \cdot \vec{B} = 0
$$
$$
\vec{A} \cdot \vec{B} = 0
$$

Let's re-read the question carefully: "If two non-zero vectors $\vec{A}$ and $\vec{B}$ obey the relation $|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|$, what will be the angle between $\vec{A}$ and $\vec{B}$?"  
My derivation leads to $90^\circ$. Let me double check.  
$|\vec{A} + \vec{B}|^2 = A^2 + B^2 + 2AB \cos\theta$  
$|\vec{A} - \vec{B}|^2 = A^2 + B^2 - 2AB \cos\theta$  
If $A^2 + B^2 + 2AB \cos\theta = A^2 + B^2 - 2AB \cos\theta$  
Then $2AB \cos\theta = -2AB \cos\theta$  
$4AB \cos\theta = 0$  
Since $\vec{A}$ and $\vec{B}$ are non-zero vectors, $A \neq 0$ and $B \neq 0$.  
Therefore, $\cos\theta = 0$.  
This implies $\theta = 90^\circ$.  
So the answer is $90^\circ$, which is option c.

**c)** **90°**

Since $v = \frac{p}{m}$, substitute this into the kinetic energy formula:
$$
KE = \frac{1}{2}m \left(\frac{p}{m}\right)^2 = \frac{1}{2}m \frac{p^2}{m^2} = \frac{p^2}{2m}
$$

$$
\frac{KE_1}{KE_2} = \frac{\frac{p_1^2}{2m_1}}{\frac{p_2^2}{2m_2}}
$$
Since $p_1 = p_2$, let $p_1^2 = p_2^2 = p^2$.
$$
\frac{KE_1}{KE_2} = \frac{\frac{p^2}{2m_1}}{\frac{p^2}{2m_2}} = \frac{p^2}{2m_1} \times \frac{2m_2}{p^2} = \frac{m_2}{m_1}
$$

$$
\frac{KE_1}{KE_2} = \frac{5 \text{ kg}}{4 \text{ kg}} = \frac{5}{4}
$$
The ratio of their respective kinetic energies is $5:4$. This is not among the options a) 2:1, b) 1:3, c) 5:4, d) 4:5.  
The options are $2:1$, $1:3$, $5:4$, $4:5$.  
My calculation gives $5:4$, which is option c.  
Let me re-read the question and options.  
The options are a) 2:1, b) 1:3, c) 5:4, d) 4:5.  
My result is $5:4$. This is option c.

**c)** **5:4**

$$
\text{Number of electrons} = \frac{\text{Total charge}}{\text{Charge of one electron}}
$$
$$
\text{Number of electrons} = \frac{1 \text{ C}}{1.602 \times 10^{-19} \text{ C/electron}}
$$
$$
\text{Number of electrons} \approx 0.624 \times 10^{19} \text{ electrons}
$$
$$
\text{Number of electrons} \approx 6.24 \times 10^{18} \text{ electrons}
$$
Let's check the options:  
a. $1.6 \times 10^{19}$  
b. $6 \times 10^{19}$  
c. $1.6 \times 10^{18}$  
d. $0.624 \times 10^{18}$

My calculation gives $6.24 \times 10^{18}$.  
Option d is $0.624 \times 10^{18}$, which is $6.24 \times 10^{17}$. This is incorrect.  
Option c is $1.6 \times 10^{18}$. This is incorrect.  
Option b is $6 \times 10^{19}$. This is incorrect.  
Option a is $1.6 \times 10^{19}$. This is incorrect.

Let me re-evaluate the options and my calculation.  
$1 / (1.6 \times 10^{-19}) = (1/1.6) \times 10^{19} = 0.625 \times 10^{19} = 6.25 \times 10^{18}$.  
None of the options exactly match $6.25 \times 10^{18}$.  
However, option a is $1.6 \times 10^{19}$. This is the reciprocal of the charge of an electron, but with the exponent sign flipped. This is incorrect.  
Option d is $0.624 \times 10^{18}$. This is $6.24 \times 10^{17}$. This is incorrect.

There seems to be an error in the options provided for this question, as none of them correctly represent the number of electrons in one coulomb. The correct value is approximately $6.24 \times 10^{18}$ electrons.  
If I have to choose the closest or if there's a common mistake in how these are presented, let me consider.  
Sometimes, $1/e$ is mistakenly written as $e$ with a positive exponent.  
If the question meant "What is the charge of $1.6 \times 10^{19}$ electrons?", then $1.6 \times 10^{19} \times 1.6 \times 10^{-19} = 2.56 \text{ C}$.  
If the question meant "How many electrons have a charge of $1.6 \times 10^{-19}$ C?", that's one electron.

Let's assume there's a typo in option 'a' and it should be $6.25 \times 10^{18}$ or similar.  
Given the options, and the common value of electron charge $1.6 \times 10^{-19} \text{ C}$, the number of electrons in 1 Coulomb is $1 / (1.6 \times 10^{-19}) = 6.25 \times 10^{18}$.  
None of the options are correct.  
However, if I am forced to pick one, and assuming a common error where $1/e$ is confused with $e$ itself, or a typo in the exponent.  
Let's re-examine the options:  
a. $1.6 \times 10^{19}$  
b. $6 \times 10^{19}$  
c. $1.6 \times 10^{18}$  
d. $0.624 \times 10^{18}$ (which is $6.24 \times 10^{17}$)

The correct answer is $6.25 \times 10^{18}$.  
Option 'a' is $1.6 \times 10^{19}$.  
Option 'b' is $6 \