The area of the bounded region enclosed by the curve

$$y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|$$ and the x-axis is :

Let x = x(y) be the solution of the differential equation

$$2y\,{e^{x/{y^2}}}dx + \left( {{y^2} - 4x{e^{x/{y^2}}}} \right)dy = 0$$ such that x(1) = 0. Then, x(e) is equal to :

Let the slope of the tangent to a curve y = f(x) at (x, y) be given by 2 $$\tan x(\cos x - y)$$. If the curve passes through the point $$\left( {{\pi \over 4},0} \right)$$, then the value of $$\int\limits_0^{\pi /2} {y\,dx} $$ is equal to :

Let a triangle be bounded by the lines L₁ : 2x + 5y = 10; L₂ : $$-$$4x + 3y = 12 and the line L₃, which passes through the point P(2, 3), intersects L₂ at A and L₁ at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to :