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

If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 4 \hat{i} - 2 \hat{j} + 3 \hat{k} \text { and } \vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a vector of magnitude 6 units which is parallel to the vector \[2 \vec{a} - \vec{b} + 3 \vec{c .}\]

Find a vector of magnitude of 5 units parallel to the resultant of the vectors \[\vec{a} = 2 \hat{i} + 3 \hat{j} - \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} +\widehat{k} .\]

Find a vector \[\vec{r}\] of magnitude \[3\sqrt{2}\] units which makes an angle of \[\frac{\pi}{4}\] and \[\frac{\pi}{4}\] with y and z-axes respectively. 

A vector \[\vec{r}\] is inclined at equal angles to the three axes. If the magnitude of \[\vec{r}\] is \[2\sqrt{3}\], find \[\vec{r}\].

Evaluate the following integrals as limit of sums:

Find a vector in the direction of \[\overrightarrow{a} = 2 \hat{i} - \hat{j} + 2 \hat{k} ,\] which has magnitude of 6 units.

Write a vector in the direction of vector \[5 \hat{i} - \hat{j} + 2 \hat{k}\] which has magnitude of 8 unit.

Find a vector \[\overrightarrow{a}\] of magnitude \[5\sqrt{2}\], making an angle of \[\frac{\pi}{4}\] with x-axis, \[\frac{\pi}{2}\] with y-axis and an acute angle θ with z-axis. 

Find a vector in the direction of vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\] which has magnitude 21 units.

If in a ∆ABC, A = (0, 0), B = (3, 3 \[\sqrt{3}\]), C = (−3\[\sqrt{3}\], 3), then the vector of magnitude 2 \[\sqrt{2}\] units directed along AO, where O is the circumcentre of ∆ABC is 

Find the distance of the point  \[2 \hat{i} - \hat{j} - 4 \hat{k}\]  from the plane  \[\vec{r} \cdot \left( 3 \hat{i}  - 4 \hat{j}  + 12 \hat{k}  \right) - 9 = 0 .\]

Show that the points \[\hat{i}  - \hat{j}  + 3 \hat{k}  \text{ and }  3 \hat{i}  + 3 \hat{j}  + 3 \hat{k} \] are equidistant from the plane \[\vec{r} \cdot \left( 5 \hat{i}  + 2 \hat{j}  - 7 \hat{k}  \right) + 9 = 0 .\]