If the sum of first 11 terms of an A.P.,
a₁, a₂, a₃, .... is 0 (a $$ \ne $$ 0), then the sum of the A.P.,
a₁ , a₃ , a₅ ,....., a₂₃ is ka₁ , where k is equal to :

If a curve y = f(x), passing through the point (1, 2), is the solution of the differential equation,
2x²dy= (2xy + y²)dx, then $$f\left( {{1 \over 2}} \right)$$ is equal to :

Let the position vectors of points 'A' and 'B' be
$$\widehat i + \widehat j + \widehat k$$ and $$2\widehat i + \widehat j + 3\widehat k$$, respectively. A point 'P' divides the line segment AB internally in the ratio $$\lambda $$ : 1 ( $$\lambda $$ > 0). If O is the origin and
$$\overrightarrow {OB} .\overrightarrow {OP} - 3{\left| {\overrightarrow {OA} \times \overrightarrow {OP} } \right|^2} = 6$$, then $$\lambda $$ is equal to______.

Let [t] denote the greatest integer less than or equal to t.
Then the value of $$\int\limits_1^2 {\left| {2x - \left[ {3x} \right]} \right|dx} $$ is ______.