The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :

A natural number has prime factorization given by n = 2x3y5z, where y and z are such that y + z = 5 and y$$-$$1 + z$$-$$1 = {5 \over 6}, y > z. Then the number of odd divisions of n, including 1, is :

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $$\alpha$$ be the number of triangles having these points from different sides as vertices and $$\beta$$ be the number of quadrilaterals having these points from different sides as vertices. Then ($$\beta$$ $$-$$ $$\alpha$$) is equal to :

Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then n is equal to :

If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to :

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :

If $${}^n{P}_r={}^n{P}_{r+1}$$ and $${}^n{C}_r={}^n{C}_{r-1}$$, then the value of r is equal to :

Let P1, P2, ......, P15 be 15 points on a circle. The number of distinct triangles formed by points Pi, Pj, Pk such that i +j + k $$\ne$$ 15, is :

Let p and q be two positive numbers such that p + q = 2 and p4+q4 = 272. Then p and q are roots of the equation :

The integer 'k', for which the inequality x2 $$-$$ 2(3k $$-$$ 1)x + 8k2 $$-$$ 7 > 0 is valid for every x in R, is :

Let $$\alpha$$ and $$\beta$$ be the roots of x2 $$-$$ 6x $$-$$ 2 = 0. If an = $$\alpha$$n $$-$$ $$\beta$$n for n $$\ge$$ 1, then the value of {{{a_{10}} - 2{a_8}} \over {3{a_9}}} is :

The value of 4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}} is :

The value of 3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}} is equal to

If $$\alpha$$ and $$\beta$$ are the distinct roots of the equation $$x^2+(3{)}^{1/4}x+3^{1/2}=0$$, then the value of $${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$$ is equal to :

Let [x] denote the greatest integer less than or equal to x. Then, the values of x$$\in$$R satisfying the equation $$[e^x{]}^2+[e^x+1]-3=0$$ lie in the interval :

The number of real roots of the equation $$e^{6x}-e^{4x}-2e^{3x}-12e^{2x}+e^x+1=0$$ is :

Let $$\alpha =\underset{x\in R}{\max }\{8^{2\sin 3x}.4^{4\cos 3x}\}$$ and $$\beta =\underset{x\in R}{\min }\{8^{2\sin 3x}.4^{4\cos 3x}\}$$. If $$8x^2+bx+c=0$$ is a quadratic equation whose roots are $$\alpha$$1/5 and $$\beta$$1/5, then the value of c $$-$$ b is equal to :

If [x] be the greatest integer less than or equal to x, then \sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]} is equal to :

The number of real solutions of the equation, x2 $$-$$ |x| $$-$$ 12 = 0 is :