Choose the correct options in the following questions: (i) If y = elogx then dy/dx is: (ii) Radius of a circle is increasing at the rate of 2 m/s. Rate of change of its circumference is:

i) Step 1: Simplify the expression for y. We know that e^ x = x. So, y = x. Step 2: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)(x) = 1 The correct option is (c). The final answer is 1. ii) Step 1: Identify the given rate and the formula for circumference. Let r be the radius of the circle and C be its circumference. Given rate of change of radius: (dr)/(dt) = 2 m/s. The formula for the circumference of a circle is C = 2 r. Step 2: Differentiate the circumference formula with respect to time t. (dC)/(dt) = (d)/(dt)(2 r) = 2 (dr)/(dt) Step 3: Substitute the given value of (dr)/(dt). (dC)/(dt) = 2 (2 m/s) = 4 m/s The correct option is (a). The final answer is 4 m/s. iii) Step 1: Let the integral be I. I = _/6^/3 sqrt( x)sqrt( x) + sqrt( x) dx Step 2: Apply the property of definite integrals _a^b f(x) dx = _a^b f(a+b-x) dx. Here, a = ()/(6) and b = ()/(3). So, a+b = ()/(6) + ()/(3) = ()/(2). I = _/6^/3 (sqrt(()/(2) - x))sqrt((()/(2) - x)) + sqrt((()/(2) - x)) dx Using the identities (()/(2) - x) = x and (()/(2) - x) = x: I = _/6^/3 sqrt( x)sqrt( x) + sqrt( x) dx Step 3: Add the original integral and the transformed integral. 2I = _/6^/3 sqrt( x)sqrt( x) + sqrt( x) dx + _/6^/3 sqrt( x)sqrt( x) + sqrt( x) dx 2I = _/6^/3 sqrt( x) + sqrt( x)sqrt( x) + sqrt( x) dx 2I = _/6^/3 1 \, dx Step 4: Evaluate the integral. 2I = [x]_/6^/3 2I = ()/(3) - ()/(6) 2I = (2)/(6) - ()/(6) 2I = ()/(6) I = ()/(12) The correct option is (c). The final answer is ()/(12). iv) Step 1: Recognize the standard integral form. The integral is _0^1 (dx)/(1+x^2). We know that (1)/(1+x^2) dx = x + C. Step 2: Evaluate the definite integral using the limits. _0^1 (dx)/(1+x^2) = [ x]_0^1 = (1) - (0) Step 3: Substitute the known values of . We know that (1) = ()/(4) and (0) = 0. = ()/(4) - 0 = ()/(4) The correct option is (b). The final answer is ()/(4).