HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Important Questions for Mathematics and Statistics

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

If the vectors `-3hati+4hatj-2hatk, hati+2hatk, hati-phatj` are coplanar, then the value of of p is

(A) -2

(B) 1

(C) -1

(D) 2

If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment  AB.

Find the volume of the parallelopiped whose coterminus edges are given by vectors

`2hati+3hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`

(A) 100

(B) 101

(C) 110

(D) 109

If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.

If the vectors `2hati-qhatj+3hatk and 4hati-5hatj+6hatk` are collinear, then value of q is

(A) 5

(B) 10

(C) 5/2

(D) 5/4

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

if `bara = 3hati - 2hatj+7hatk`, `barb  = 5hati + hatj -2hatk`and `barc = hati + hatj - hatk` then find `bara.(barbxxbarc)`

Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`

Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.                               

 Show that the points A (-7 , 4 , -2),B (-2 , 1 , 0)and C (3 ,-2 ,2) are collinear.

Find the value of p, if the vectors `hat"i" - 2hat"j" + hat"k", 2hat"i" -5hat"j"+"p" hat "k" , 5hat"i" -9hat"j" + 4 hat"k"` are coplanar.

If a vector has direction angles 45° and 60°, find the third direction angle.

If a line has the direction ratios 4, −12, 18, then find its direction cosines

If the vectors `- 3hati + 4hatj - 2hatk , hati + 2hatk` and `hati - phatj` are coplanar, then find the value of p.

If `bara = hati - 2hatj`, `barb = hati + 2hatj, barc = 2hati + hatj - 2hatk`, then find (i) `bara xx (barb xx barc)` (ii) `(bara xx barb) xx barc`. Are the results same? Justify.

Using properties of scalar triple product, prove that `[(bara + barb,  barb + barc,  barc + bara)] = 2[(bara, barb, barc)]`.

If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`

Prove that the volume of a tetrahedron with coterminus edges `overlinea, overlineb` and `overlinec` is `1/6[(overlinea, overlineb, overlinec)]`.

Hence, find the volume of tetrahedron whose coterminus edges are `overlinea = hati + 2hatj + 3hatk, overlineb = -hati + hatj + 2hatk` and `overlinec = 2hati + hatj + 4hatk`.