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

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 

Evaluate : `int_0^3dx/(9+x^2)`

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result: