The value of $$\tan \left( {2{{\tan }^{ - 1}}\left( {{3 \over 5}} \right) + {{\sin }^{ - 1}}\left( {{5 \over {13}}} \right)} \right)$$ is equal to :

The lines x = ay $$-$$ 1 = z $$-$$ 2 and x = 3y $$-$$ 2 = bz $$-$$ 2, (ab $$\ne$$ 0) are coplanar, if :

If [x] denotes the greatest integer less than or equal to x, then the value of the integral $$\int_{ - \pi /2}^{\pi /2} {[[x] - \sin x]dx} $$ is equal to :

If the real part of the complex number $${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$$ is $${1 \over 5}$$ for $$\theta \in (0,\pi )$$, then the value of the integral $$\int_0^\theta {\sin x} dx$$ is equal to:

Let $$f:R - \left\{ {{\alpha \over 6}} \right\} \to R$$ be defined by $$f(x) = {{5x + 3} \over {6x - \alpha }}$$. Then the value of $$\alpha$$ for which (fof)(x) = x, for all $$x \in R - \left\{ {{\alpha \over 6}} \right\}$$, is :

If $$f:R \to R$$ is given by $$f(x) = x + 1$$, then the value of $$\mathop {\lim }\limits_{n \to \infty } {1 \over n}\left[ {f(0) + f\left( {{5 \over n}} \right) + f\left( {{{10} \over n}} \right) + ...... + f\left( {{{5(n - 1)} \over n}} \right)} \right]$$ is :

Let A, B and C be three events such that the probability that exactly one of A and B occurs is (1 $$-$$ k), the probability that exactly one of B and C occurs is (1 $$-$$ 2k), the probability that exactly one of C and A occurs is (1 $$-$$ k) and the probability of all A, B and C occur simultaneously is k², where 0 < k < 1. Then the probability that at least one of A, B and C occur is :

The sum of all the local minimum values of the twice differentiable function f : R $$\to$$ R defined by $$f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x + f''(1)$$ is :

Let y = y(x) satisfies the equation $${{dy} \over {dx}} - |A| = 0$$, for all x > 0, where $$A = \left[ {\matrix{ y & {\sin x} & 1 \cr 0 & { - 1} & 1 \cr 2 & 0 & {{1 \over x}} \cr } } \right]$$. If $$y(\pi ) = \pi + 2$$, then the value of $$y\left( {{\pi \over 2}} \right)$$ is :

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and $${{20} \over 3}$$, respectively, then the value of | a $$-$$ b | is equal to :

Let $$g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$$, where $$f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R$$. Then which one of the following is correct?

Let P be a variable point on the parabola $$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :

The value of k $$\in$$R, for which the following system of linear equations

3x $$-$$ y + 4z = 3,

x + 2y $$-$$ 3z = $$-$$2

6x + 5y + kz = $$-$$3,

has infinitely many solutions, is :

In a triangle ABC, if $$\left| {\overrightarrow {BC} } \right| = 3$$, $$\left| {\overrightarrow {CA} } \right| = 5$$ and $$\left| {\overrightarrow {BA} } \right| = 7$$, then the projection of the vector $$\overrightarrow {BA} $$ on $$\overrightarrow {BC} $$ is equal to :

The number of solutions of the equation

$${\log _{(x + 1)}}(2{x^2} + 7x + 5) + {\log _{(2x + 5)}}{(x + 1)^2} - 4 = 0$$, x > 0, is :

Let a curve y = y(x) be given by the solution of the differential equation $$\cos \left( {{1 \over 2}{{\cos }^{ - 1}}({e^{ - x}})} \right)dx = \sqrt {{e^{2x}} - 1} dy$$. If it intersects y-axis at y = $$-$$1, and the intersection point of the curve with x-axis is ($$\alpha$$, 0), then e^$$\alpha$$ is equal to __________________.

For p > 0, a vector $${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$$ is obtained by rotating the vector $${\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j$$ by an angle $$\theta$$ about origin in counter clockwise direction. If $$\tan \theta = {{\left( {\alpha \sqrt 3 - 2} \right)} \over {\left( {4\sqrt 3 + 3} \right)}}$$, then the value of $$\alpha$$ is equal to _____________.

Consider a triangle having vertices A($$-$$2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point $$\left( {0,{\alpha \over 2}} \right)$$, then the value of real number $$\alpha$$ is ________________.

If the point on the curve y² = 6x, nearest to the point $$\left( {3,{3 \over 2}} \right)$$ is ($$\alpha$$, $$\beta$$), then 2($$\alpha$$ + $$\beta$$) is equal to _____________.

Let a function g : [ 0, 4 ] $$\to$$ R be defined as

$$g(x) = \left\{ {\matrix{ {\mathop {\max }\limits_{0 \le t \le x} \{ {t^3} - 6{t^2} + 9t - 3),} & {0 \le x \le 3} \cr {4 - x,} & {3 < x \le 4} \cr } } \right.$$, then the number of points in the interval (0, 4) where g(x) is NOT differentiable, is ____________.

If $$\mathop {\lim }\limits_{x \to 0} {{\alpha x{e^x} - \beta {{\log }_e}(1 + x) + \gamma {x^2}{e^{ - x}}} \over {x{{\sin }^2}x}} = 10,\alpha ,\beta ,\gamma \in R$$, then the value of $$\alpha$$ + $$\beta$$ + $$\gamma$$ is _____________.