Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

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).

Find the interval in which the following function are increasing or decreasing  f(x) = 5x³ − 15x² − 120x + 3 ?

Find the interval in which the following function are increasing or decreasing f(x) = x³ − 6x² − 36x + 2 ?

Show that the points whose position vectors are  \[- 2 \hat{i} + 3 \hat{j} , \hat{i} + 2 \hat{j} + 3 \hat{k}  \text{ and }  7 \text{ i}  - \text{ k} \]  are collinear.

Find the interval in which the following function are increasing or decreasing f(x) = 2x³ − 15x² + 36x + 1 ?

Find the interval in which the following function are increasing or decreasing f(x) = 2x³ + 9x² + 12x + 20  ?

Find the cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line  \[\frac{- x - 2}{1} = \frac{y + 3}{7} = \frac{2z - 6}{3} .\] 

The cartesian equation of a line are 3x + 1 = 6y − 2 = 1 − z. Find the fixed point through which it passes, its direction ratios and also its vector equation.