PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Find the angle between the following pair of line:

\[\frac{x - 5}{1} = \frac{2y + 6}{- 2} = \frac{z - 3}{1} \text{  and  } \frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 6}{5}\]

Find the angle between the following pair of line:

\[\frac{- x + 2}{- 2} = \frac{y - 1}{7} = \frac{z + 3}{- 3} \text{  and  } \frac{x + 2}{- 1} = \frac{2y - 8}{4} = \frac{z - 5}{4}\]

Determine the values of x for which the function f(x) = x² − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x² − 6x + 9 where the normal is parallel to the line y = x + 5 ? 

Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?

Show that f(x) = e^2x is increasing on R.

Find the angle between the pairs of lines with direction ratios proportional to 5, −12, 13 and −3, 4, 5

Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?

Find the angle between the pairs of lines with direction ratios proportional to  2, 2, 1 and 4, 1, 8 .

Find the angle between the pairs of lines with direction ratios proportional to  1, 2, −2 and −2, 2, 1 .

Find the angle between the pairs of lines with direction ratios proportional to   a, b, c and b − c, c − a, a − b.

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?

Find the angle between two lines, one of which has direction ratios 2, 2, 1 while the  other one is obtained by joining the points (3, 1, 4) and (7, 2, 12).