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

Find the vector equation for the line which passes through the point (1, 2, 3) and parallel to the vector \[\hat{i} - 2 \hat{j} + 3 \hat{k} .\]  Reduce the corresponding equation in cartesian from.

Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?

Find the vector equation of a line passing through (2, −1, 1) and parallel to the line whose equations are \[\frac{x - 3}{2} = \frac{y + 1}{7} = \frac{z - 2}{- 3} .\]

Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?

The cartesian equations of a line are \[\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2} .\]  Find a vector equation for the line.

Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?

Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .

Find the cartesian equation of a line passing through (1, −1, 2) and parallel to the line whose equations are  \[\frac{x - 3}{1} = \frac{y - 1}{2} = \frac{z + 1}{- 2}\]  Also, reduce the equation obtained in vector form.

Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?

Find the direction cosines of the line  \[\frac{4 - x}{2} = \frac{y}{6} = \frac{1 - z}{3} .\]  Also, reduce it to vector form. 

The cartesian equations of a line are x = ay + b, z = cy + d. Find its direction ratios and reduce it to vector form. 

Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x²  ?

Find the interval in which the following function are increasing or decreasing  f(x) = x² + 2x − 5  ?

Find the interval in which the following function are increasing or decreasing  f(x) = 6 − 9x − x²  ?

Find the interval in which the following function are increasing or decreasing   f(x) = 2x³ − 12x² + 18x + 15 ?

Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x² − 2x³ ?

Find the vector equation of a line passing through the point with position vector  \[\hat{i} - 2 \hat{j} - 3 \hat{k}\]  and parallel to the line joining the points with position vectors  \[\hat{i} - \hat{j} + 4 \hat{k} \text{ and } 2 \hat{i} + \hat{j} + 2 \hat{k} .\] Also, find the cartesian equivalent of this equation.

Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x² − 2x³ ?

Find the points on the line \[\frac{x + 2}{3} = \frac{y + 1}{2} = \frac{z - 3}{2}\]  at a distance of 5 units from the point P (1, 3, 3).