The differential equation of the family of curves, x2 = 4b(y + b), b $$\in$$ R, is :

If y = y(x) is the solution of the differential equation {{5 + {e^x}} \over {2 + y}}.{{dy} \over {dx}} + {e^x} = 0 satisfying y(0) = 1, then a value of y(loge13) is :

The solution of the differential equation {{dy} \over {dx}} - {{y + 3x} \over {{{\log }_e}\left( {y + 3x} \right)}} + 3 = 0 is: (where c is a constant of integration)

Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x > 0. if y ($$\pi$$) = $$\pi$$ then y''\left( {{\pi \over 2}} \right) + y\left( {{\pi \over 2}} \right) is equal to :

If x3dy + xy dx = x2dy + 2y dx; y(2) = e and x > 1, then y(4) is equal to :

The solution curve of the differential equation, (1 + e-x)(1 + y2){{dy} \over {dx}} = y2, which passes through the point (0, 1), is :

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation, 2x2dy= (2xy + y2)dx, then f\left( {{1 \over 2}} \right) is equal to :

Let y = y(x) be the solution of the differential equation, {{2 + \sin x} \over {y + 1}}.{{dy} \over {dx}} = - \cos x, y > 0,y(0) = 1. If y($$\pi$$) = a and {{dy} \over {dx}} at x = $$\pi$$ is b, then the ordered pair (a, b) is equal to :

If {{dy} \over {dx}} = {{xy} \over {{x^2} + {y^2}}}; y(1) = 1; then a value of x satisfying y(x) = e is :

Let y = y(x) be a solution of the differential equation, \sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0, |x| < 1. If y\left( {{1 \over 2}} \right) = {{\sqrt 3 } \over 2}, then y\left( { - {1 \over {\sqrt 2 }}} \right) is equal to :

Let y = y(x) be the solution curve of the differential equation, \left( {{y^2} - x} \right){{dy} \over {dx}} = 1, satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is :

If y = y(x) is the solution of the differential equation, {e^y}\left( {{{dy} \over {dx}} - 1} \right) = {e^x} such that y(0) = 0, then y(1) is equal to:

If a $$\Delta$$ABC has vertices A(–1, 7), B(–7, 1) and C(5, –5), then its orthocentre has coordinates :

Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point (–1, –4) in this line is :

A ray of light coming from the point (2, $$2\sqrt{3}$$) is incident at an angle 30o on the line x = 1 at the point A. The ray gets reflected on the line x = 1 and meets x-axis at the point B. Then, the line AB passes through the point :

If the perpendicular bisector of the line segment joining the points P(1 ,4) and Q(k, 3) has y-intercept equal to –4, then a value of k is :

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is :

The set of all possible values of $$\theta$$ in the interval (0, $$\pi$$) for which the points (1, 2) and (sin $$\theta$$, cos $$\theta$$) lie on the same side of the line x + y = 1 is :

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the point :

Let two points be A(1, –1) and B(0, 2). If a point P(x', y') be such that the area of $$\Delta$$PAB = 5 sq. units and it lies on the line, 3x + y – 4$$\lambda$$ = 0, then a value of $$\lambda$$ is :