\(∫\frac{dx}{(x2+1) (x2+4)} =\)
\(∫\frac{dx}{(x-1)^{34} (x+2)^{\frac54}}=\)
If order and degree of the differential equation corresponding to the family of curves y2 = 4a(x+a)(a is parameter) are m and n respectively, then m+n2 =
If l,m,n and a,b,c are direction cosines of two lines then
On differentiation if we get f (x,y)dy - g(x,y)dx = 0 from 2x2-3xy+y2+x+2y-8 = 0 then g(2,2)/f(1,1) =
If ∫ \(\frac{x^{49} Tan^{-1} (x^{50})}{(1+x^{100})}\)dx = k(Tan-1 (x50))2 + c, then k =
The general solution of the differential equation (x2 + 2)dy +2xydx = ex(x2+2)dx is
If the parametric equations of the circle passing through the points (3,4), (3,2) and (1,4) is x = a + r cosθ, y = b + r sinθ then ba ra =
If the line x cos α + y sin α = 2√3 is tangent to the ellipse \(\frac{x^2}{16} + \frac{y^2}{8} = 1\) and α is an acute angle then α =
A straight line parallel to the line y = √3 x passes through Q(2,3) and cuts the line 2x + 4y - 27 = 0 at P. Then the length of the line segment PQ is
If the radical center of the given three circles x2 + y2 = 1, x2 + y2 -2x - 3 =0 and x2 + y2 -2y - 3 = 0 is C(α,β) and r is the sum of the radii of the given circles, then the circle with C(α,β) as center and r as radius is
A random variable X has the following probability distribution
For the events E = {x/x is a prime number} and F = {x/x <4} then P(E ∪ F)
If the angle between the pair of tangents drawn to the circle $ x^2 + y^2 - 2x + 4y + 3 = 0 $ from the point $(6, -5)$ is \(\theta\) than \(\cot \theta\) =
The radius of a circle touching all the four circles (x ± λ)2 + (y ± λ)2 = λ2 is
If (h,k) is the image of the point (3,4) with respect to the line 2x - 3y -5 = 0 and (l,m) is the foot of the perpendicular from (h,k) on the line 3x + 2y + 12 = 0, then lh + mk + 1 = 2x - 3y - 5 = 0.
The angle between the circles \(x^2+y^2−4x−6y−3=0\), \(x^2+y^2+8x−4y+11=0\) is \(\frac{\pi}{2}\), then the value of K is?
If the angle between the asymptotes of a hyperbola is 30° then its eccentricity is
A tangent PT is drawn to the circle x2 + y2 = 4 at the point P(√3, 1). If a straight line L which is perpendicular to PT is a tangent to the circle (x- 3)2 + y2 = 1, then a possible equation of L is
If a point P moves so that the distance from (0,2) to P is \(\frac{1}{√2 }\) times the distance of P from (-1,0), then the locus of the point P is
Let d be the distance between the parallel lines 3x - 2y + 5 = 0 and 3x - 2y + 5 + 2√13 = 0. Let L1 = 3x - 2y + k1 = 0 (k1 > 0) and L2 = 3x - 2y + k2 = 0 (k2 > 0) be two lines that are at the distance of \(\frac{4d}{√13}\) and \(\frac{3d}{√13}\) from the line 3x - 2y + 5y = 0. Then the combined equation of the lines L1 = 0 and L2 = 0 is:
5 persons entered a lift cabin in the cellar of a 7-floor building apart from cellar. If each of the independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probability of all the 5 persons leaving the cabin at different floors is
If sin y = sin 3t and x = sin t, then \(\frac{dy}{dx}\) =
The number of diagonals of a polygon is 35. If A, B are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having AB as one of its sides is:
In △ABC, if a : b : c = 4 : 5 : 6, then the ratio of the circumference to its in radius is