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

The value of c in Rolle's theorem for the function f (x) = x³ − 3x in the interval [0,\[\sqrt{3}\]] is 

If f (x) = e^x sin x in [0, π], then c in Rolle's theorem is

Find the points on the curve x² + y² − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Find the angle between the line \[\vec{r} = \left( 2 \hat{i}+ 3 \hat {j}  + 9 \hat{k}  \right) + \lambda\left( 2 \hat{i} + 3 \hat{j}  + 4 \hat{k}  \right)\]  and the plane  \[\vec{r} \cdot \left( \hat{i}  + \hat{j}  + \hat{k}  \right) = 5 .\]

Find the angle between the line \[\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}\]  and the plane 2x + y − z = 4.

The line  \[\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j}  - 3 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( m \hat{i}  + 3 \hat{j}  + \hat{k}  \right) = 4 .\] Find m. 

Show that the line whose vector equation is \[\vec{r} = 2 \hat{i}  + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i}  + 3 \hat{j}  + 4 \hat{k}  \right)\] is parallel to the plane whose vector  \[\vec{r} \cdot \left( \hat{i} + \hat{j}  - \hat{k}  \right) = 7 .\]  Also, find the distance between them.

Find the angle between the line \[\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}\] and the plane

3x + 4y + z + 5 = 0.

State when the line \[\vec{r} = \vec{a} + \lambda \vec{b}\]  is parallel to the plane  \[\vec{r} \cdot \vec{n} = d .\]Show that the line  \[\vec{r} = \hat{i}  + \hat{j}  + \lambda\left( 3 \hat{i}  - \hat{j}  + 2 \hat{k}  \right)\]  is parallel to the plane  \[\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .\]   Also, find the distance between the line and the plane.

Show that the plane whose vector equation is \[\vec{r} \cdot \left( \hat{i}  + 2 \hat{j}  - \hat{k}  \right) = 1\] and the line whose vector equation is  \[\vec{r} = \left( - \hat{i}  + \hat{j} + \hat{k}  \right) + \lambda\left( 2 \hat{i}  + \hat{j}  + 4 \hat{k}  \right)\]   are parallel. Also, find the distance between them. 

Find the angle between the line

Write the angle between the line \[\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z + 3}{- 2}\]  and the plane x + y + 4 = 0. 

To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients-calcium, protein and calories and the person's diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below:
 

What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.

Show that  \[\begin{vmatrix}y + z & x & y \\ z + x & z & x \\ x + y & y & z\end{vmatrix} = \left( x + y + z \right) \left( x - z \right)^2\]

x + y = 1
x + z = − 6
x − y − 2z = 3

Find `"dy"/"dx"` if, ye^x + xe^y = 1 

If y = x^x, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`

Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`

Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`