Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Find the interval in which the following function are increasing or decreasing f(x) = 2x³ − 9x² + 12x − 5 ?

Show that the three lines with direction cosines \[\frac{12}{13}, \frac{- 3}{13}, \frac{- 4}{13}; \frac{4}{13}, \frac{12}{13}, \frac{3}{13}; \frac{3}{13}, \frac{- 4}{13}, \frac{12}{13}\] are mutually perpendicular. 

Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x² − 2x³ ?

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the through the points (0, 3, 2) and (3, 5, 6).

Find the interval in which the following function are increasing or decreasing  f(x) = 2x³ − 24x + 107  ?

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and, (1, 2, 5).

Find the interval in which the following function are increasing or decreasing f(x) = −2x³ − 9x² − 12x + 1  ?

Find the cartesian equation of the line which passes through the point (−2, 4, −5) and parallel to the line given by  \[\frac{x + 3}{3} = \frac{y - 4}{5} = \frac{z + 8}{6} .\]

Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)² ?

Show that the lines \[\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} \text { and }\frac{x}{1} = \frac{y}{2} = \frac{z}{3}\]  are perpendicular to each other. 

Find the interval in which the following function are increasing or decreasing f(x) = x³ − 12x² + 36x + 17 ?

Find the interval in which the following function are increasing or decreasing  f(x) = 2x³ − 24x + 7 ?

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1). 

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?

Find the equation of a line parallel to x-axis and passing through the origin.

Find the interval in which the following function are increasing or decreasing f(x) = x⁴ − 4x ?

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?

Find the angle between the following pair of line: 

\[\overrightarrow{r} = \left( 4 \hat{i} - \hat{j} \right) + \lambda\left( \hat{i} + 2 \hat{j} - 2 \hat{k} \right) \text{ and }\overrightarrow{r} = \hat{i} - \hat{j} + 2 \hat{k} - \mu\left( 2 \hat{i} + 4 \hat{j} - 4 \hat{k} \right)\]

Find the interval in which the following function are increasing or decreasing  f(x) = x⁴ − 4x³ + 4x² + 15 ?