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

If the function f(x) = x³ − 9kx² + 27x + 30 is increasing on R, then

The function f(x) = x⁹ + 3x⁷ + 64 is increasing on

A vector makes an angle of \[\frac{\pi}{4}\] with each of x-axis and y-axis. Find the angle made by it with the z-axis.

A vector \[\vec{r}\] is inclined at equal acute angles to x-axis, y-axis and z-axis.
If |\[\vec{r}\]| = 6 units, find \[\vec{r}\].

A vector \[\vec{r}\] is inclined to -axis at 45° and y-axis at 60°.  If \[|\vec{r}|\] = 8 units, find \[\vec{r}\].

Find the direction cosines of the following vector:

`2hati + 2hatj - hatk`

Find the direction cosines of the following vectors:
\[6 \hat{i} - 2 \hat{j} - 3 \hat{k}\]

Find the direction cosines of the following vectors:
\[3 \hat{i} - 4 \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[\hat{i} - \hat{j} + \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[\hat{j} - \hat{k}\]

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[4 \hat{i} + 8 \hat{j} + \hat{k}\]

Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined with the axes OX, OY and OZ.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are \[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} .\]

If a unit vector \[\vec{a}\] makes an angle \[\frac{\pi}{3}\] with \[\hat{i} , \frac{\pi}{4}\] with \[\hat{j}\]  and an acute angle θ with \[\hat{k}\], then find θ and hence, the components of \[\vec{a}\].

Write the direction cosines of the vector \[\overrightarrow{r} = 6 \hat{i} - 2 \hat{j} + 3 \hat{k} .\]

A unit vector \[\overrightarrow{r}\] makes angles \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] with \[\hat{j}\text{ and }\hat{k}\]  respectively and an acute angle θ with \[\hat{i}\]. Find θ.

What is the cosine of the angle which the vector \[\sqrt{2} \hat{i} + \hat{j} + \hat{k}\] makes with y-axis?