Let y = y(x) be the solution of the differential equation (x $$-$$ x3)dy = (y + yx2 $$-$$ 3x4)dx, x > 2. If y(3) = 3, then y(4) is equal to :

Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y($$\pi$$) = 0, then y\left( {{\pi \over 2}} \right) is equal to :

Let y = y(x) be solution of the differential equation {\log _{}}\left( {{{dy} \over {dx}}} \right) = 3x + 4y, with y(0) = 0. If y\left( { - {2 \over 3}{{\log }_e}2} \right) = \alpha {\log _e}2, then the value of $$\alpha$$ is equal to :

Let y = y(x) be a solution curve of the differential equation $$(y+1){\tan }^{2}xdx+\tan xdy+ydx=0$$, x \in \left( {0,{\pi \over 2}} \right). If $$\underset{x\to 0+}{\lim }xy(x)=1$$, then the value of y\left( {{\pi \over 4}} \right) is :

Let y(x) be the solution of the differential equation 2x2 dy + (ey $$-$$ 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :

Let y = y(x) be the solution of the differential equation {{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x such that y(0) = 7. Then y($$\pi$$) is equal to :

Let us consider a curve, y = f(x) passing through the point ($$-$$2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x2. Then :

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, $$-$$3) from the line 3x + 4y = 5, is given by :

If the solution curve of the differential equation (2x $$-$$ 10y3)dy + ydx = 0, passes through the points (0, 1) and (2, $$\beta$$), then $$\beta$$ is a root of the equation :

If {{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}, y(0) = 1, then y(1) is equal to :

If {{dy} \over {dx}} = {{{2^x}y + {2^y}{{.2}^x}} \over {{2^x} + {2^{x + y}}{{\log }_e}2}}, y(0) = 0, then for y = 1, the value of x lies in the interval :

If y{{dy} \over {dx}} = x\left[ {{{{y^2}} \over {{x^2}}} + {{\phi \left( {{{{y^2}} \over {{x^2}}}} \right)} \over {\phi '\left( {{{{y^2}} \over {{x^2}}}} \right)}}} \right], x > 0, $$\varphi$$ > 0, and y(1) = $$-$$1, then \phi \left( {{{{y^2}} \over 4}} \right) is equal to :

If y = y(x) is the solution curve of the differential equation {x^2}dy + \left( {y - {1 \over x}} \right)dx = 0 ; x > 0 and y(1) = 1, then y\left( {{1 \over 2}} \right) is equal to :

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is {1 \over 4}. Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then, which of these stones is / are on the path of the man?

The image of the point (3, 5) in the line x $$-$$ y + 1 = 0, lies on :

The intersection of three lines x $$-$$ y = 0, x + 2y = 3 and 2x + y = 6 is a :

Let A($$-$$1, 1), B(3, 4) and C(2, 0) be given three points. A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A1 and A2 be the areas of $$\Delta$$ABC and $$\Delta$$PQC respectively, such that A1 = 3A2, then the value of m is equal to :

In a triangle PQR, the co-ordinates of the points P and Q are ($$-$$2, 4) and (4, $$-$$2) respectively. If the equation of the perpendicular bisector of PR is 2x $$-$$ y + 2 = 0, then the centre of the circumcircle of the $$\Delta$$PQR is :

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

The equation of one of the straight lines which passes through the point (1, 3) and makes an angles $${\tan }^{-1}\left(\sqrt{2}\right)$$ with the straight line, y + 1 = 3$$\sqrt{2}$$ x is :