If $$\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C} $$
where C is a constant of integration, then the function ƒ(x) is equal to

Suppose that the points (h,k), (1,2) and (–3,4) lie on the line L₁ . If a line L₂ passing through the points (h,k) and (4,3) is perpendicular to L₁ , then $$k \over h$$ equals :

Let ƒ : [–1,3] $$ \to $$ R be defined as

$$f(x) = \left\{ {\matrix{ {\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr {x + \left| x \right|} & , & {1 \le x < 2} \cr {x + \left[ x \right]} & , & {2 \le x \le 3} \cr } } \right.$$

where [t] denotes the greatest integer less than or equal to t. Then, ƒ is discontinuous at:

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :