If $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$, then the system of equations

x + (cos $$\gamma$$)y + (cos $$\beta$$)z = 0

(cos $$\gamma$$)x + y + (cos $$\alpha$$)z = 0

(cos $$\beta$$)x + (cos $$\alpha$$)y + z = 0

has :

The domain of the function

$$f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x + 1}}} \right)$$ is :

Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g(1) is :

Let f : N $$\to$$ N be a function such that f(m + n) = f(m) + f(n) for every m, n$$\in$$N. If f(6) = 18, then f(2) . f(3) is equal to :

If $$\alpha = \mathop {\lim }\limits_{x \to {\pi \over 4}} {{{{\tan }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}$$ and $$\beta = \mathop {\lim }\limits_{x \to 0 } {(\cos x)^{\cot x}}$$ are the roots of the equation, ax² + bx $$-$$ 4 = 0, then the ordered pair (a, b) is :

The locus of mid-points of the line segments joining ($$-$$3, $$-$$5) and the points on the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 9} = 1$$ is :

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, $$\phi$$ > 0, and y(1) = $$-$$1, then $$\phi \left( {{{{y^2}} \over 4}} \right)$$ is equal to :

The sum of the roots of the equation

$$x + 1 - 2{\log _2}(3 + {2^x}) + 2{\log _4}(10 - {2^{ - x}}) = 0$$, is :

If z is a complex number such that $${{z - i} \over {z - 1}}$$ is purely imaginary, then the minimum value of | z $$-$$ (3 + 3i) | is :

Let a₁, a₂, a₃, ..... be an A.P. If $${{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}$$, p $$\ne$$ 10, then $${{{a_{11}}} \over {{a_{10}}}}$$ is equal to :

Let A be the set of all points ($$\alpha$$, $$\beta$$) such that the area of triangle formed by the points (5, 6), (3, 2) and ($$\alpha$$, $$\beta$$) is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is :

Let f be any continuous function on [0, 2] and twice differentiable on (0, 2). If f(0) = 0, f(1) = 1 and f(2) = 2, then

If [x] is the greatest integer $$\le$$ x, then

$${\pi ^2}\int\limits_0^2 {\left( {\sin {{\pi x} \over 2}} \right)(x - [x]} {)^{[x]}}dx$$ is equal to :

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

If the coefficient of a⁷b⁸ in the expansion of (a + 2b + 4ab)¹⁰ is K.2¹⁶, then K is equal to _____________.

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____________.

If $$\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = } $$

$$\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma {\tan ^{ - 1}}\left( {{{2\tan x - 1} \over {\sqrt 3 }}} \right) + C$$, when C is constant of integration, then the value of $$18(\alpha + \beta + {\gamma ^2})$$ is ______________.

The number of elements in the set $$\left\{ {A = \left( {\matrix{ a & b \cr 0 & d \cr } } \right):a,b,d \in \{ - 1,0,1\} \,and\,{{(I - A)}^3} = I - {A^3}} \right\}$$, where I is 2 $$\times$$ 2 identity matrix, is :

If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = $${3 \over 2}$$ and the curve y = 1 + 4x $$-$$ x², then 12 m is equal to _____________.

Let f(x) be a cubic polynomial with f(1) = $$-$$10, f($$-$$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $$-$$1. Then f(3) is equal to ____________.