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

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

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

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|`.

Write the projection of the vector `(vecb + vecc)` on the vector `veca`, where `veca = 2hati - 2hatj + hatk, vecb = hati + 2hatj - 2hatk` and `vecc = 2hati - hatj + 4hatk`.

Projection of vector `2hati + 3hatj` on the vector `3hati - 2hatj` is ______.

A unit vector `hata` makes equal but acute angles on the coordinate axes. The projection of the vector `hata` on the vector `vecb = 5hati + 7hatj - hatk` is ______.

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, show that the vector `vecc* vecd = 15` is equally inclined to `veca, vecb "and"  vecc.` 

Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane `vec r.(hati+hatj+hatk)=2`

Find the vector equation of the plane which contains the line of intersection of the planes `vecr (hati+2hatj+3hatk)-4=0` and `vec r (2hati+hatj-hatk)+5=0` which is perpendicular to the plane.`vecr(5hati+3hatj-6hatk)+8=0`

Find the vector equation of the plane passing through three points with position vectors ` hati+hatj-2hatk , 2hati-hatj+hatk and hati+2hatj+hatk` . Also find the coordinates of the point of intersection of this plane and the line `vecr=3hati-hatj-hatk lambda +(2hati-2hatj+hatk)`

Find the vector equation of the plane with intercepts 3, –4 and 2 on x, y and z-axis respectively.