Consider a system of two vectors
`overset(rarr)a=3hat I +4 hat j, overset(rarr)b=hat i+hatj` ,

Options

A. `overset(rarr)r=sqrt(2)overset(rarr)a+5overset(rarr)b` will be making equal angles with and `overset(rarr)a` and `overset(rarr)b`
B. The angle made by `overset(rarr)r=overset(rarr)a+overset(rarr)b` with `overset(rarr)a` will be smaller as compared to angle made by with `overset(rarr)r` with `overset(rarr)b`
C. Angle made by `overset(rarr)r=overset(rarr)a+overset(rarr)b` with be greater as compared to angle made by with `overset(rarr)b`
D. Incomplete information to predict any of the above options

Explanation

To solve the problem, we will analyze the two vectors given and find their resultant. We will also determine the angles made by the resultant vector with each of the original vectors.

### Step-by-Step Solution:

1. **Identify the Vectors:**
- Given vectors are:
\[ \vec{a} = 3\hat{i} + 4\hat{j} \]
\[ \vec{b} = \hat{i} + \hat{j} \]

2. **Calculate the Magnitude of Vector \( \vec{a} \):**
- The magnitude of vector \( \vec{a} \) is calculated using the formula:
\[ |\vec{a}| = \sqrt{x^2 + y^2} \]
where \( x = 3 \) and \( y = 4 \):
\[ |\vec{a}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Calculate the Magnitude of Vector \( \vec{b} \):**
- The magnitude of vector \( \vec{b} \) is:
\[ |\vec{b}| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]

4. **Determine the Angles with the X-axis:**
- For vector \( \vec{a} \):
\[ \theta_1 = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \]
- For vector \( \vec{b} \):
\[ \theta_2 = \tan^{-1}\left(\frac{1}{1}\right) = 45^\circ \]

5. **Calculate the Resultant Vector \( \vec{R} = 2\vec{a} + 5\vec{b} \):**
- First, calculate \( 2\vec{a} \):
\[ 2\vec{a} = 2(3\hat{i} + 4\hat{j}) = 6\hat{i} + 8\hat{j} \]
- Next, calculate \( 5\vec{b} \):
\[ 5\vec{b} = 5(\hat{i} + \hat{j}) = 5\hat{i} + 5\hat{j} \]
- Now, add these two results:
\[ \vec{R} = (6\hat{i} + 8\hat{j}) + (5\hat{i} + 5\hat{j}) = (6 + 5)\hat{i} + (8 + 5)\hat{j} = 11\hat{i} + 13\hat{j} \]

6. **Calculate the Magnitude of the Resultant Vector \( \vec{R} \):**
- The magnitude is:
\[ |\vec{R}| = \sqrt{11^2 + 13^2} = \sqrt{121 + 169} = \sqrt{290} \]

7. **Determine the Angle of Resultant Vector \( \vec{R} \) with the X-axis:**
- The angle \( \alpha \) is given by:
\[ \alpha = \tan^{-1}\left(\frac{13}{11}\right) \approx 49.76^\circ \]

8. **Calculate the Angles Between Vectors:**
- Angle between \( \vec{a} \) and \( \vec{R} \):
\[ \text{Angle} = \theta_1 - \alpha \approx 53.13^\circ - 49.76^\circ \approx 3.37^\circ \]
- Angle between \( \vec{b} \) and \( \vec{R} \):
\[ \text{Angle} = \alpha - \theta_2 \approx 49.76^\circ - 45^\circ \approx 4.76^\circ \]

9. **Conclusion:**
- The angles made by the resultant vector with \( \vec{a} \) and \( \vec{b} \) are not equal, confirming that the angle made by \( \vec{R} \) with \( \vec{a} \) is smaller than that with \( \vec{b} \).