Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

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 ______.

Projection vector of `vec"a"` on `vec"b"` is ______.

Find: `int (sin2x)/sqrt(9 - cos^4x) dx`

The scalar projection of the vector `3hati - hatj - 2hatk` on the vector `hati + 2hatj - 3hatk` is ______.

If `veca` and `vecb` are two vectors such that `|veca + vecb| = |vecb|`, then prove that `(veca + 2vecb)` is perpendicular to `veca`.

If `veca` and `vecb` are unit vectors and θ is the angle between them, then prove that `sin  θ/2 = 1/2 |veca  - vecb|`.