Let y = y(x) be the solution of the differential equation \sin x{{dy} \over {dx}} + y\cos x = 4x, $$x\in \left(0,\pi \right)$$. If y\left( {{\pi \over 2}} \right) = 0, then y\left( {{\pi \over 6}} \right) is equal to :

The differential equation representing the family of ellipse having foci eith on the x-axis or on the $$y$$-axis, center at the origin and passing through the point (0, 3) is :

A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is :

The sides of a rhombus ABCD are parallel to the lines, x $$-$$ y + 2 = 0 and 7x $$-$$ y + 3 = 0. If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is :

The foot of the perpendicular drawn from the origin, on the line, 3x + y = $$\lambda$$ ($$\lambda$$ $$\ne$$ 0) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :

In a triangle ABC, coordinates of A are (1, 2) and the equations of the medians through B and C are respectively, x + y = 5 and x = 4. Then area of $$\Delta$$ ABC (in sq. units) is :

Let $$\overrightarrow{A}$$ = $$\left(\hat{i}+\hat{j}\right)$$ and, $$\overrightarrow{B}=\left(2\hat{i}-\hat{j}\right).$$ The magnitude of a coplanar vector $$\overrightarrow{C}$$ such that $$\overrightarrow{A}.\overrightarrow{C}=\overrightarrow{B}.\overrightarrow{C}=\overrightarrow{A}.\overrightarrow{B},$$ is given by :

PQR is a triangular park with PQ = PR = 200 m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively 45$${}^{\circ }$$, 30$${}^{\circ }$$ and 30$${}^{\circ }$$, then the height of the tower (in m) is :

An aeroplane flying at a constant speed, parallel to the horizontal ground, $$\sqrt{3}$$ kmabove it, is obsered at an elevation of $$60^o$$ from a point on the ground. If, after five seconds, its elevation from the same point, is $$30^o$$, then the speed (in km / hr) of the aeroplane, is :

A tower T1 of height 60 m is located exactly opposite to a tower T2 of height 80 m on a straight road. Fromthe top of T1, if the angle of depression of the foot of T2 is twice the angle of elevation of the top of T2, then the width (in m) of the road between the feetof the towers T1 and T2 is :

A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min. for the angle of depression of the car to change from 30o to 45o ; then after this, the time taken (in min.) by the car to reach the foot of the tower, is :

If n is the degree of the polynomial, {\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} + {\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8} and m is the coefficient of xn in it, then the ordered pair (n, m) is equal to :

The sum of the co-efficients of all odd degree terms in the expansion of $${\left(x+\sqrt{x^3-1}\right)}^5+{\left(x-\sqrt{x^3-1}\right)}^5$$, $$\left(x>1\right)$$ is

The coefficient of x2 in the expansion of the product (2$$-$$x2) .((1 + 2x + 3x2)6 + (1 $$-$$ 4x2)6) is :

The coefficien of x10 in the expansion of (1 + x)2(1 + x2)3(1 + x3)4 is equal to :

If \int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \right) + C (where C is a constant of integration), then the ordered pair (A, B) is equal to :

If \int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}} \left( {{{K\,\tan x + 1} \over {\sqrt A }}} \right) + C,(C\,\, is a constant of integration) then the ordered pair (K, A) is equal to :

If f\left( {{{x - 4} \over {x + 2}}} \right) = 2x + 1, (x $$\in$$ R $$-$${1, $$-$$ 2}), then $$\int f\left(x\right)dx$$ is equal to : (where C is a constant of integration)

The integral \int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx is equal to

Let f : A $$\to$$ B be a function defined as f(x) = {{x - 1} \over {x - 2}}, Where A = R $$-$$ {2} and B = R $$-$$ {1}. Then f is :