Arts (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

The vector \[\vec{b} = 3 \hat { i }+ 4 \hat {k }\] is to be written as the sum of a vector \[\vec{\alpha}\] parallel to \[\vec{a} = \hat {i} + \hat {j}\] and a vector \[\vec{\beta}\] perpendicular to \[\vec{a}\]. Then \[\vec{\alpha} =\]

The unit vector perpendicular to the plane passing through points \[P\left( \hat{ i } - \hat{ j }  + 2 \hat{ k }  \right), Q\left( 2 \hat{ i } - \hat{ k } \right) \text{ and }  R\left( 2 \hat{ j }  + \hat{ k }  \right)\]  is 

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 ______.

Find a unit vector perpendicular to both the vectors `veca and vecb` , where `veca = hat i - 7 hatj +7hatk`  and  `vecb = 3hati - 2hatj + 2hatk` . 

Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx`  and hence evaluate   `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .   

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Write the coordinates of the point which is the reflection of the point (α, β,  γ) in the XZ-plane.

Evaluate: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.

Find : `int_  (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.

Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`