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A box contains 4 white and 3 black balls. Another box contains 3 white and 4 black balls. A die is thrown. If it exhibits a number `<=`3, the ball is drawn from the first box. Otherwise, a ball is drawn from the second box. A ball drawn is found to be black. The probability that it has been drawn from the second box is :

Sum of roots of the equation sin x `xx` tan x – sin x + tanx – 1 = 0 over [0, `2pi`] is :

If BC is latus rectum of parabola `y^2` = 4ax and A is the vertex. Then the minimum legnth of the projection of BC on a tangent drawn in the portion BAC is :

The function ƒ : `[a, oo) -> R` where R denotes the range corresponding to the given domain, with rule ƒ(x) =` 2x^3 – 3x^2 + 6,` will have an inverse provided

`L_1=Lim_(x->4) (x-6)^x and L_2=Lim_(x->4) (x-6)^4`

The value of `tan(1/2cot^-1(3))` equal

If `pi/4<x<pi/2`, then `int (sinx-cosx)/sqrt(1-sin2x)e^(sinx)` cosxdx equal

The locus of P(x, y) such that

`sqrt(x^2+y^2+8y+16)-sqrt(x^2+y^2-6x+9)=5` is

Let ƒ be a real valued function such that

`f(x)+2f(2002/x)`=3x for all x>0. the value of f(2)is

P and Q are points on the ellipse `x^2/a^2+y^2/b^2=1` whose centre is C. The accentric angles of P and Q differe by a right angle. If `/_PCQ` is maximum, then eccentric angle of P can be :

Let g:`R->R` be a differentiable function such that g(2)=-40 and g'(2)=-2 then

`Lim_(x->oo)(g(2-x^2)/g(2))x^(4/2)`is equal to :

The solution of the equation log `dy/dx`= ax + by is

If ƒ(x) is continuous and `int_0^9 f(x)dx=4` then value of the integral `int_0^3x xx f(x^2)dx` is

The number of points of discontinuity for the function ƒ(x) = (log|x|)sgn`(x^2–1)`, x `!=` 0, is/are :

Let i= `sqrt(-1)`. Define a sequence of complex number by `z_1=0 , z_(n+1)=z_n^2+1` for n `>=` 1. In the complex plane, then arg(`z_(111)`) is :

An isosceles triangle ABC is inscribed in the circle whose equation is `x^2 + y^2`=9 with vertex at A(3, 0) with base angles B and C each equal to 75º, then the product of the ordinates of B and C is :

If ƒ(x) = `a + bx + cx^2`, whre c > 0 and `b^2` – 4ac < 0, then the area enclosed by the coordinate axes, the line x = 2 and the curve y = ƒ(x) is given by :

The term independent of x in the expansion of `(9x-1/(3sqrtx))^18`, x > 0, is `alpha` time the corresponding binomial coefficient. Then `alpha` is :

If the curves y = `x^3` + ax and y = `bx^2` + c pass thorugh the point (–1, 0) and have a common tangent line at this point, then the value of a + b + c is

A line makes some angle `theta` with each of the x-axis and z-axis. If angle `beta`, which it makes with y-axis, is such that `sin^2beta = 3sin^2theta`, then `cos^2theta` equals

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