HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

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.

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

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

If `barc= 3bara - 2barb  and [bara  barb+barc  bara+barb+barc] = "then proved" [bara barb barc] = 0`

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

If `overlinec = 3overlinea - 2overlineb` and `[overlinea         overlineb + overlinec         overlinea + overlineb + overlinec]` = 0 then prove that `[overlinea  overlineb  overlinec]` = 0

Show that the volume of the parallelopiped whose coterminus edges are `bara barb barc` is `[(bara, barb, barc)].`

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

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

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