दो नल A तथा B एक टंकी को क्रमशः 6 मिनट तथा 7 मिनट में भर सकते है A से आरम्भ करते हुए प्रत्येक नल को बारी-बारी एक मिनट के लिए खोला जाता है टंकी को भरने में कितना समय लगेगा?

 (a) 5 मिनट

 (b) 5 2/3 मिनट

 (c) 6 3/7 मिनट

 (d) 6 1/4 मिनट

Using double integration find the area of the parallelogram whose vertices are A(1,0), B(3,1), C(2,2), D(0,1) 

Solve the differential equation           (D²- 4D + 3)y = 0

Verify Cauchy' s mean value theorem          and find c, if f(x) = e^x , g(x) = e^-x ,          x ∈ [a,b]

Find the surface area formed by revolving four - cusped hypocycloid (asteroid) x^2/3 + y^2/3 = a^2/3 about the     x-axis 

Find the surface generated by revolving the portion of the curve y² = 4 + x cut off by the straight line x = 2, about the        x - axis

Using Rolle's theorem, prove that there is no real a for which the equation        x² - 3x + a = 0 two different roots in [-1,1].

Find c of Lagrange's mean value theorem for the following function          x³ + x in [1,2].

Solve (1 + y²)dx = (tan^-1y - x)dy

Identify the nature of singular points of the differential equation                          
x²(x - 2)y" + (x - 1)y' + 2xy = 0 

For any two real numbers a and b         (a < b) prove that a²+ ab+ b² = 3c²,for some c ∈ (a,b) using Lagrange's mean value theorem.