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

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.

If y = x^x, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]

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

Evaluate : \[\int\limits_0^\pi \frac{x \tan x}{\sec x \cdot cosec x}dx\] .

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.

Differentiate sin(log sin x) ?

If `x=a (cos t +t sint )and y= a(sint-cos t )`  Prove that `Sec^3 t/(at),0<t< pi/2` 

If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]

Differentiate `log [x+2+sqrt(x^2+4x+1)]`

Differentiate the following with respect to x: 

\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]

Solve the differential equation:  ` (dy)/(dx) = (x + y )/ (x - y )`

Find `int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `

Find `int_  sin ("x" - a)/(sin ("x" + a )) d"x"`