PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions

If \[\vec{a,} \vec{b}\] represent the diagonals of a rhombus, then

Vectors \[\vec{a} \text{ and }  \vec{b}\] are inclined at angle θ = 120°. If \[\left| \vec{a} \right| = 1, \left| \vec{b} \right| = 2,\] then  \[\left[ \left( \vec{a} + 3 \vec{b} \right) \times \left( 3 \vec{a} - \vec{b} \right) \right]^2\]  is equal to 

If \[\vec{a} = \hat{ i }  + \hat{ j }  - \hat{ k }  , \vec{b} = - \hat{ i }  + 2\hat{ j }  + 2 \hat{ k }  \text{ and }  \vec{c} = - \hat{ i } + 2 \hat{ j }  - \hat{ k }  ,\]  then a unit vector normal to the vectors \[\vec{a} + \vec{b} \text{ and }  \vec{b} - \vec{c}\]  is

A unit vector perpendicular to both \[\hat{ i }  + \hat{ j } \text{ and }  \hat{ j } + \hat{ k } \] is

If \[\vec{a} = 2 \hat{ i }  - 3 \hat{ j }  - \hat{ k }  \text{ and }  \vec{b} = \hat{ i } + 4 \hat{ j }  - 2 \hat{ k 
} , \text{ then } \vec{a} \times \vec{b}\]  is

If \[\hat{ i }  , \hat{ j }  , \hat{ k } \] are unit vectors, then

If θ is the angle between the vectors \[2 \hat{ i }  - 2 \hat{ j}  + 4 \hat{ k }  \text{ and } 3 \hat{ i }  + \hat { j }  + 2 \hat{ k }  ,\]  then sin θ =

If \[\left| \vec{a} \times \vec{b} \right| = 4, \left| \vec{a} \cdot \vec{b} \right| = 2, \text{ then }  \left| \vec{a} \right|^2 \left| \vec{b} \right|^2 =\]

The value of \[\left( \vec{a} \times \vec{b} \right)^2\] is 

The value of  \[\hat{ i }  \cdot \left( \hat{ j }  \times \hat{ k }  \right) + \hat{ j }  \cdot \left( \hat{ i }  \times \hat{ k }  \right) + \hat{ k }  \cdot \left( \hat{ i }  \times \hat{ j }  \right),\]  is 

If θ is the angle between any two vectors `bara` and `barb` and `|bara · barb| = |bara xx barb|` then θ is equal to ______.

Solve the following.

`int_2^3x/((x+2)(x+3))dx`

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

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

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

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

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 

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

For what value of λ are the vectors \[\vec{a} \text{ and  }\vec{b}\] perpendicular to each other if \[\vec{a} = \lambda \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{b} = 4\hat{i} - 9 \hat{j} + 2\hat{k}\]