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

Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.

Evaluate: `int_0^(π/4) log(1 + tanx)dx`.

Find the projection of the vector `hati+3hatj+7hatk`  on the vector `2hati-3hatj+6hatk`

If `veca ` and `vecb` are two unit vectors such that `veca+vecb` is also a  unit vector, then find the angle between `veca` and `vecb`

Vectors `veca,vecb and vecc ` are such that `veca+vecb+vecc=0 and |veca| =3,|vecb|=5 and |vecc|=7 ` Find the angle between `veca and vecb`

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

Find the coordinates of the point, where the line `(x-2)/3=(y+1)/4=(z-2)/2` intersects the plane x − y + z − 5 = 0. Also find the angle between the line and the plane.

Find the acute angle between the plane 5x − 4y + 7z − 13 = 0 and the y-axis.

Show that the vectors `veca, vecb` are coplanar if `veca+vecb, vecb+vecc ` are coplanar.

If  `vec a, vec b, vec c`  are unit vectors such that `veca+vecb+vecc=0`, then write the value of  `vec a.vecb+vecb.vecc+vecc.vec a`.

If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`

The scalar product of the vector `veca=hati+hatj+hatk` with a unit vector along the sum of vectors `vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk` is equal to one. Find the value of λ and hence, find the unit vector along `vecb +vecc`

Find the angle between the planes whose vector equations are `vecr.(2hati + 2hatj - 3hatk) = 5 and hatr.(3hati - 3hatj  + 5hatk) = 3`

Show that each of the given three vectors is a unit vector:

`1/7 (2hati + 3hatj + 6hatj), 1/7(3hati - 6hatj + 2hatk), 1/7(6hati + 2hatj - 3hatk)`

The scalar product of the vector `hati + hatj + hatk` with a unit vector along the sum of vectors `2hati + 4hatj - 5hatk` and  `lambdahati + 2hatj +  3hatk` is equal to one. Find the value of `lambda`.

Prove that `(veca + vecb).(veca + vecb)` = `|veca|^2 + |vecb|^2` if and only if `veca . vecb` are perpendicular, given `veca != vec0, vecb != vec0.`

Find the magnitude of each of two vectors `veca` and `vecb` having the same magnitude such that the angle between them is 60° and their scalar product is `9/2`

Find \[\vec{a} \cdot \vec{b}\] when

 \[\vec{a} =\hat{i} - 2\hat{j} + \hat{k}\text{ and } \vec{b} = 4 \hat{i} - 4\hat{j} + 7 \hat{k}\]

Find \[\vec{a} \cdot \vec{b}\] when

\[\vec{a} = \hat{j} + 2 \hat{k}  \text{ and } \vec{b} = 2 \hat{i} + \hat{k}\]

Find \[\vec{a} \cdot \vec{b}\] when 

\[\vec{a} = \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + 3 \hat{j} - 2 \hat{k}\]