HSC Science (Computer Science) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `[bar"u" xx bar"v"     bar"u" xx bar"w"     bar"v" xx bar"w"]`

If the points A(1, 2, 3), B(–1, 1, 2), C(2, 3, 4) and D(–1, x, 0) are coplanar find the value of x.

Find the volume of the parallelopiped whose vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 4). 

If `barc = 3bara - 2barb` and `[bara     barb + barc     bara + barb + barc]` = 0 then prove that `[bara  barb  barc]` = 0

Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.

`bara=-3/5hati+1/2hatj+1/3hatk,barb=5hati+4hatj+3hatk`

Determine whether `bara and barb` are orthogonal, parallel or neither.

`bara = -3/5hati + 1/2hatj +1/3 hatk, barb = 5hati + 4hatj +3hatk`

If a vector has direction angles 45ºand 60º find the third direction angle. 

Determine whether `bara and barb` is orthogonal, parallel or neither.

`bara = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`

If `u=hati -2hatj + hatk, barr=3hati + hatk and w=hatj, hatk`  are given vectors, then find `[baru + barw]. [(barw xx barr)xx(barr xx barw)]`

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).

Determine whether `\bb(bara and barb)` are orthogonal, parallel or neither.

`bara = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk `

Determine whether `bara and barb` are orthogonal, parallel or neither.

`bara = -3/5 hati + 1/2 hatj + 1/3 hatk,  barb = 5 hati + 4 hatj + 3 hatk`

If `baru = hati - 2hatj + hatk,  barv = 3hati + hatk   "and"  barw = hatj - hatk` are given vectors, then find `[baru + barw]·[(baru xx barv)xx(barv xx barw)]`

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). 

Prove that `[bar"a"  bar"b" + bar"c"  bar"a" + bar"b" + bar"c"] = 0`

Prove that `(bar"a" + 2bar"b" - bar"c"). [(bar"a" - bar"b") xx (bar"a" - bar"b" - bar"c")] = 3 [bar"a"  bar"b"  bar"c"]`.

If `bar"c" = 3bar"a" - 2bar"b"`, then prove that `[bar"a"  bar"b"  bar"c"] = 0`.

If `bar "a" = hat"i" + 2hat"j" + 3hat"k" , bar"b" = 3hat"i" + 2hat"j"` and `bar"c" = 2hat"i" + hat"j" + 3hat"k"`, then verify that `bar"a" xx (bar"b" xx bar"c") = (bar"a".bar"c")bar"b" - (bar"a".bar"b")bar"c"`

Show that `bar"a" xx (bar"b" xx bar"c") + bar"b" xx (bar"c" xx bar"a") + bar"c" xx (bar"a" xx bar"b") = bar"0"`

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.