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

Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .

Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .

Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]

Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].

Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]

Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]

Evaluate:  \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]

The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is

The value of \[\int\frac{1}{x + x \log x} dx\] is

\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]

Find the vector and cartesian equations of the line through the point (5, 2, −4) and which is parallel to the vector  \[3 \hat{i} + 2 \hat{j} - 8 \hat{k} .\]

Find the vector equation of the line passing through the points (−1, 0, 2) and (3, 4, 6).

Find the vector equation of a line which is parallel to the vector \[2 \hat{i} - \hat{j} + 3 \hat{k}\]  and which passes through the point (5, −2, 4). Also, reduce it to cartesian form.

A line passes through the point with position vector \[2 \hat{i} - 3 \hat{j} + 4 \hat{k} \] and is in the direction of  \[3 \hat{i} + 4 \hat{j} - 5 \hat{k} .\] Find equations of the line in vector and cartesian form. 

ABCD is a parallelogram. The position vectors of the points A, B and C are respectively, \[4 \hat{ i} + 5 \hat{j} -10 \hat{k} , 2 \hat{i} - 3 \hat{j} + 4 \hat{k}  \text{ and } - \hat{i} + 2 \hat{j} + \hat{k} .\]  Find the vector equation of the line BD. Also, reduce it to cartesian form.