Arts (English Medium) कक्षा १२ - CBSE Important Questions for Mathematics

Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.

If `veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk` ,then find ` |veca xx vecb|`

Find the angle between the vectors `hati-hatj and hatj-hatk`

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

A line passing through the point A with position vector `veca=4hati+2hatj+2hatk` is parallel to the vector `vecb=2hati+3hatj+6hatk` . Find the length of the perpendicular drawn on this line from a point P with vector `vecr_1=hati+2hatj+3hatk`

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 `|vecaxxvecb|^2+|veca.vecb|^2=400 ` and `|vec a| = 5` , then write the value of `|vecb|`

Find `veca.(vecbxxvecc), " if " veca=2hati+hatj+3hatk, vecb=-hati+2hatj+hatk  " and " vecc=3hati+hatj+2hatk`

If `veca and vecb` are perpendicular vectors, `|veca+vecb| = 13 and |veca| = 5` ,find the value of `|vecb|.`

Find a vector `veca` of magnitude `5sqrt2` , making an angle of `π/4` with x-axis, `π/2` with y-axis and an acute angle θ with z-axis. 

Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.

The two vectors \[\hat{j} + \hat{k}\] and \[3 \hat{i} - \hat{j} + 4 \hat{k}\] represents the sides \[\overrightarrow{AB}\] and \[\overrightarrow{AC}\] respectively of a triangle ABC. Find the length of the median through A.

Find the direction cosines of the following vector:

`2hati + 2hatj - hatk`

Find a vector in the direction of vector \[2 \hat{i} - 3 \hat{j} + 6 \hat{k}\] which has magnitude 21 units.

Write the projection of \[\hat{i} + \hat{j} + \hat{k}\] along the vector \[\hat{j}\] 

If \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors, \[\left| \vec{a} + \vec{b} \right| = 13\] and \[\left| \vec{a} \right| = 5\] find the value of \[\left| \vec{b} \right|\]

Find the image P' of the point P having position vector `hati+ 3hatj+ 4hatk` in the plane `vecr. (2hati - hatj + hatk) + 3 = 0 .` Hence find the length of PP'.

If \[\vec{a}  \times  \vec{b}  =  \vec{c}  \times  \vec{d}   \text { and }   \vec{a}  \times  \vec{c}  =  \vec{b}  \times  \vec{d}\] , show that \[\vec{a}  -  \vec{d}\] is parallel to \[\vec{b} - \vec{c}\] where \[\vec{a} \neq \vec{d} \text { and } \vec{b} \neq \vec{c}\] .

Find the area of a parallelogram whose adjacent sides are represented by the vectors\[2 \hat{i} - 3 \hat{k} \text { and } 4 \hat{j} + 2 \hat{k} .\]

Prove that, for any three vectors \[\vec{a} , \vec{b} , \vec{c}\] \[\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]\].