Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions

Find the angle between the planes.

2x − y + z = 4 and x + y + 2z = 3

Find the angle between the planes.

x + y − 2z = 3 and 2x − 2y + z = 5

Find the angle between the planes.

 x − y + z = 5 and x + 2y + z = 9

Find the angle between the planes.
 2x − 3y + 4z = 1 and − x + y = 4

Find the angle between the planes.

Show that the following planes are at right angles.

\[\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k}  \right) = 5 \text{ and }  \vec{r} \cdot \left( - \hat{i}  - \hat{j} + \hat{k}  \right) = 3\]

Show that the following planes are at right angles.

The acute angle between the planes 2x − y + z = 6 and x + y + 2z = 3 is

\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]

\[\int \left( e^x + 1 \right)^2 e^x dx\]

\[\int \sin^3  \left( 2x + 1 \right)  \text{dx}\]

\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]

\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]

Let `veca` , `vecb` and `vecc` be three vectors such that `|veca| = 1,|vecb| = 2, |vecc| = 3.` If the projection of `vecb` along `veca` is equal to the projection of `vecc` along `veca`; and `vecb` , `vecc` are perpendicular to each other, then find `|3veca - 2vecb + 2vecc|`.

Find: `int (3x +5)/(x^2+3x-18)dx.`

The projection of vector `vec"a" = 2hat"i" - hat"j" + hat"k"` along `vec"b" = hat"i" + 2hat"j" + 2hat"k"` is ______.