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51.

If `vec x` and `vec y` are two non-collinear vectors and ABC is a triangle with side lengths a,b and c satisfying (20a-15b)`vec x` + (15b-12c)`vec y` + (12c-20a)`vec x xx vec y` is:A. an acute angle triangleB. an obtuse angle triangleC. a right angle triangleD. an isosceles triangle

Answer» Correct Answer - C
Discussion
52.

If ` vec A O+ vec O B= vec B O+ vec O C`, prove that `A , B , C`are collinearpoints.

Answer» It is given that,
`vec(AO)+vec(OB) = vec(BO)+vec(OC)`
`=>vec(AB) = vec(BC)`
Adding `vec(BC)` both sides,
`=> vec(AB)+vec(BC) = 2vec(BC)`
`=>vec(AC) = 2vec(BC)`
As, these two vectors pass from same point and are in same direction and one is double of other, it means
`A`,`B` and `C` are collinear.
Discussion
53.

Five forces ` vec A B`, ` vec A C`, ` vec A D`, ` vec A E`and ` vec A F`act at the vertex of a regular hexagon `A B C D E Fdot`Prove that the resultant is `6 vec A O`, where `O`is the centre of heaagon.

Answer» `vec(AB)+vec(BO)=vec(AO)`
`vec(AC)+vec(CO)=vec(AO)`
`vec(AD)+vec(DO)=vec(AO)`
`vec(AE)+vec(EO)=vec(AD)`
`vec(AF)+vec(FO)=vec(AO)`
`(vec(AB)+vec(AC)+vec(AD)+vec(AB)+vec(AF))+(vec(BO)+vec(CO)+vec(DO)+vec(EO)+vec(FO))=5vec(AO)`
`x+vec(AO)=5vec(AO)`
`x=6vec(AO)`.
Discussion
54.

Show that thefour points `A ,B , Ca n dD`with positionvectors ` vec a , vec b , vec c`and ` vec d`respectivelyare coplanar if and only if `3 vec a-2 vec b+ vec c-2 vec d=0.`

Answer» Position vectors, `veca,vecb,vecc and vec d` are coplanar if,
`xveca+yvecb+zvecc+wvecd = vec0->(1)`
and
`x+y+z+w = 0`
If we put, ` x = 3, y = -2, z = 1 and w = -2`,
Then, `x+y+z+w = 0`
If we put these values in (1),
`3veca-2vecb+vecc-2vecd = vec0.`
Thus, these `4` vectors are coplanar if
`3veca-2vecb+vecc-2vecd = vec0.`
Discussion
55.

Suppose that `vec p , vec q and vec r ` non- coplanar vectors in `R^(3)` . Let the components of a vector `vec s " along " vec p , vec q and vec r ` be 4, 3 and 5 respectively. If the components of this vectors `vec s " along " -vec p + vec q + vec r , vec p - vec q + vec r and - vec p - vecq + vec r ` are x , y and z respectively, then the value of `2x-y+z,` isA. 7B. 8C. 9D. 6

Answer» Correct Answer - C
It is given that the components of vectors `vecs " along " vecp, vec q and vec r ` are 4, 3 and 5 respectively.
`therefore vecs = 4vec p + 3 vecq + 5 vec r " " ` …(i)
The components of `vec s ` along `-vec p + vec q + vec r , vec p - vec q + vec r and - vec p - vec q + vec r` and x, y and z respectively.
` therefore vec s = x (- vec p + vec q + vec r ) +y( vec p - vec q + vec r ) + z ( - vec p - vec q + vec r ) `
`rArr vec s = (-x+y-z) vec p + (x-y-z) vec q + (x+y+z) vec r " " ` ...(ii)
From (i) and (ii) , we obtain
`-x+y-z=4, x-y-z=3 and x+y+z=5`
Solving these equations, we obtain
`x=4, y=(9)/(2), z=(7)/(2)`
`therefore 2x+y+z=8+(9)/(2)-(7)/(2)=9.`
Discussion
56.

`veca , vec b , vec c ` are non-coplanar vectors and ` x vec a + y vec b + z vec c = vec 0` thenA. at least of one of x, y, z is zeroB. x, y, z are necessarily zeroC. none of them are zeroD. none of these

Answer» Correct Answer - B
Discussion
57.

Prove that a necessaryand sufficient condition for three vectors ` vec a , vec b`and ` vec c`to becoplanar is that there exist scalars `l , m , n`not all zerosimultaneously such that `l vec a+m vec b+n vec c= vec0dot`

Answer» Here, it is given that ,
`lveca+mvecb+nvecc = 0`
If we put, `l =-1 , m = x and n = y`, then,
`-veca +xvecb+yvecc = 0`
`=> veca = xvecb+yvecc`, which is the neccessary condition for all three vectors to be coplanar.
Now,
`lveca+mvecb+nvecc = 0`
`=>lveca = -mvecb-nvecc`
`=>veca = -m/lvecb-n/lvecc`, which is the sufficient condition for all three vectors to be coplanar.
Discussion
58.

Forces `3 O vec A , 5 O vec B ` act along OA and OB. If their resultant passes through C on AB, thenA. C is a mid-point of ABB. C divides AB in the ratio `2:1`C. `3 AC = 5 CB`D. `2 AC = 3 CB`

Answer» Correct Answer - C
Discussion