A tetrahedron has vertices at 0(0, 0, 0), A(1, - 2, 1), B (-2, 1,

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 Multiple Choice QuestionsMultiple Choice Questions

41.

If the four points with position vectors - 2i^ + j^ + k^, i^ + j^ + k^, j^ - k^ and λj^ + k^ are coplanar, then λ is equal to

  • 1

  • 2

  • - 1

  • 0


42.

The equation zz + az + az + b = 0, represents a circle, if

  • a2 = b

  • a2 > b

  • a2 <b

  •  None of the above


43.

If a, b, c are non-coplanar unit vectors such that

a × b × c = b + c2, then the angle between a and b is

  • 3π4

  • π4

  • π2

  • π


44.

If a, b, c and a', b',c' form a reciprocal system of vectors, then a . a'+ b . b' + c · c' is equal to

  • 0

  • 1

  • 2

  • 3


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45.

If a, b, c are three non - zero vectors which are pairwise non-collinear. Also, a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is

  • c

  • 0

  • a + c

  • a


46.

If d = a x b + b x c + c x a is a non-zero vector and (d · c) (a x b) + (d · a) (b x c) + (d . b) (c x a) = 0, then

  • a + b + c = d

  • a = b = c

  • a, b and c are coplanar

  • None of the above


47.

Let u, v and w be such that u = 1, v = 2 and w = 3. If the projection of v along u is equal to that of w along u and vectors v and w are perpendicular to each other, then u - v + w equals

  • 2

  • 7

  • 14

  • 14


48.

If a and b are two vectors, such that a . b < 0 and a . b = a × b then the angle between vectors a and b is

  • 3π4

  • π4

  • π

  • 7π4


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49.

A tetrahedron has vertices at 0(0, 0, 0), A(1, - 2, 1), B (-2, 1, 1) and C (1, - 1, 2). Then, the angle between the faces OAB and ABC will be

  • cos-112

  • cos-1- 16

  • cos-1- 13

  • cos-114


C.

cos-1- 13

Vector perpendicular to face OAB = n1

                                                = OA × OB= i^ - 2j^ + k^ × - 2i^ + j^ + k^= i^j^k^1- 21- 211= - 2 - 1i^ + - 2 - 1j^ + 1 - 4k^= - 3i^ - 3j^ - 3k^

Vector perpendicular to face ABC = n2

                                                = AB × AC= - 3i^ + 3j^ × j^ + k^= i^j^k^- 330011= - 2 - 1i^ + - 2 - 1j^ + 1 - 4k^= 3i^ + 3j^ - 3k^

Since, angle between faces equals to angle between their normals.

 cosθ = n1 . n2n1n2                = - 33 + - 33 + - 3- 39 + 9 + 99 + 9 + 9                = - 9 - 9 + 92727 = - 927 = - 13         θ = cos-1- 13


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50.

If a, b and c are three non-coplanar vectors, then (a + b - c) . [(a - b) x (b - c)] equals

  • 0

  • a . b × c

  • a . c × b

  • 3a . b × c


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