There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is :

Consider three boxes, each containing, 10 balls labelled 1, 2, … , 10. Suppose one ball is randomly drawn from each of the boxes. Denote by ni, the label of the ball drawn from the ith box, (i = 1, 2, 3). Then, the number of ways in which the balls can be chosen such that n1 < n2 < n3 is :

If $$\sum _{r=0}^{25}\left\{{}^{50}{C}_r.{}^{50-r}{C}_{25-r}\right\}=K\left({}^{50}{C}_{25}\right),$$ then K is equal to :

The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repitition of digits allowed) is equal to :

Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is :

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same team, is :

If $$\alpha$$, $$\beta$$ and $$\gamma$$ are three consecutive terms of a non-constant G.P. such that the equations $$\alpha$$x 2 + 2$$\beta$$x + $$\gamma$$ = 0 and x2 + x – 1 = 0 have a common root, then $$\alpha$$($$\beta$$ + $$\gamma$$) is equal to :

The number of real roots of the equation 5 + |2x – 1| = 2x (2x – 2) is

If $$\alpha$$ and $$\beta$$ are the roots of the quadratic equation, x2 + x sin $$\theta$$ - 2 sin $$\theta$$ = 0, \theta \in \left( {0,{\pi \over 2}} \right), then {{{\alpha ^{12}} + {\beta ^{12}}} \over {\left( {{\alpha ^{ - 12}} + {\beta ^{ - 12}}} \right).{{\left( {\alpha - \beta } \right)}^{24}}}} is equal to :

All the pairs (x, y) that satisfy the inequality {2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}}}} \le 1 also satisfy the equation

If m is chosen in the quadratic equation (m2 + 1) x2 – 3x + (m2 + 1)2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-

Let $$\alpha$$ and $$\beta$$ be the roots of the quadratic equation x2 sin $$\theta$$ – x(sin $$\theta$$ cos $$\theta$$ + 1) + cos $$\theta$$ = 0 (0 < $$\theta$$ < 45o), and $$\alpha$$ < $$\beta$$. Then \sum\limits_{n = 0}^\infty {\left( {{\alpha ^n} + {{{{\left( { - 1} \right)}^n}} \over {{\beta ^n}}}} \right)} is equal to :

Let p, q $$\in$$ R. If 2 - $$\sqrt{3}$$ is a root of the quadratic equation, x2 + px + q = 0, then :

The number of integral values of m for which the equation (1 + m2 )x2 – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is :

The number of integral values of m for which the quadratic expression, (1 + 2m)x2 – 2(1 + 3m)x + 4(1 + m), x $$\in$$ R, is always positive, is :

If $$\lambda$$ be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m – 4)x + 2 = 0, then the least value of m for which \lambda + {1 \over \lambda } = 1, is

If one real root of the quadratic equation 81x2 + kx + 256 = 0 is cube of the other root, then a value of k is

The value of $$\lambda$$ such that sum of the squares of the roots of the quadratic equation, x2 + (3 – $$\lambda$$)x + 2 = $$\lambda$$ has the least value is -

Consider the quadratic equation (c – 5)x2 – 2cx + (c – 4) = 0, c $$\ne$$ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2, 3). Then the number of elements in S is -

The number of all possible positive integral values of $$\alpha$$ for which the roots of the quadratic equation, 6x2 $$-$$ 11x + $$\alpha$$ = 0 are rational numbers is :