Question

If a line x = -1 divides the area of region bounded by \(\{(x,y):1+x^2 \leq y \leq 3-x\}\) in the ratio \(\frac{m}{n}\) then (m + n) equal (where HCF of (m,n) = 1) :

• A. 25
• B. 26
• C. 27
• D. 28

Hint:

To find the area between two curves, first find their points of intersection to determine the limits of integration. Then, integrate the difference between the upper function and the lower function over this interval.

Answer 1

The Correct Option is C

Concept: At the distance of closest approach, the initial kinetic energy of the alpha particle is completely converted into electrostatic potential energy. Step 1: Energy conservation equation. \[ K = \frac{1}{4\pi\varepsilon_0} \frac{(Ze)(2e)}{r_{min}} \] Here \(K = 7.9\,\text{MeV}\), \(Z=79\) (Gold), and \(\alpha\)-charge is \(2e\).
Step 2: Solve for \(r_{min}\). Using \(\frac{e^2}{4\pi\varepsilon_0} \approx 1.44\,\text{MeV}\cdot\text{fm}\): \[ r_{min} = \frac{2 Z (1.44)}{K} \] \[ r_{min} = \frac{2(79)(1.44)}{7.9} \] \[ r_{min} = 2(10)(1.44) = 28.8\,\text{fm} \]
Step 3: Calculate Diameter. If the particle touches the boundary, \(r_{min}\) is the radius. \[ \text{Diameter} = 2 \times r_{min} = 2 \times 28.8 = 57.6\,\text{fm} \] \[ \boxed{\text{Diameter of nucleus} = 57.6\,\text{fm}} \]