The angle between the lines whose direction cosines satisfy the e

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411.

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0, l2 + m2 - n2 = 0 is

  • π6

  • π4

  • π3

  • π2


C.

π3

Given, l + m + n = 0,  l = - m - n and l2 + m2 + n2 = 0 - m - n 2 + m2 - n2 = 0 2m2 +2mn = 0 2m(m + n) = 0 m = 0 or m + n = 0 If  m = 0, then l = - m l1- 1 = m10 = n11and if     m + n = 0  m = - n, then  l = 0   l20 = m2- 1 = n21ie,    l1, m1, n1 = - 1, 0, 1and l2, m2, n2 = 0, - 1, 1 cosθ = 0 + 0 + 11 + 0 + 10 + 1+ 1 = 12         θ = π3


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412.

If a = - i^ + j^ + 2k^, b = 2i^ - j^ - k^, and c = - 2i^ + j^ + 3k^, then the angle between 2a - c and a + b is

  • π4

  • π3

  • π2

  • 3π2


413.

Suppose a = λi^ - 7j^ + 3k^, b = λi^ + j^ + 2λk^. Ifthe angle between a and b is greater than 90°, then A satisfies the inequality

  • - 7 < λ < 1

  • λ > 1

  • 1 < λ < 7

  • - 5 < λ < 1


414.

The volume of the tetrahedron having the edgei^ + 2j^ - k^,  i^ + j^ + k^,  i^ - j^ + λk^ as coterminous, is 23cubic unit. Then λ = ?

  • 1

  • 2

  • 3

  • 4


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415.

The image of the point (3, 2, 1) in the plane 2x - y + 3z = 7 is

  • (1, 2, 3)

  • (2, 3, 1)

  • (3, 2, 1)

  • (2, 1, 3)


416.

A point moves in the xy-plane such that the sum of its distance from two mutually perpendicular lines is always equal to 5 units. The area(in sq units) enclosed by the locus of the point,is

  • 254

  • 25

  • 50

  • 100


417.

If the pair of lines given by x2 + y2cos2θ = xcosθ + ysinθ2 are perpendicular to each other, then θ is equal to

  • 0

  • π4

  • π3

  • 3π4


418.

If the foot of the perpendicular from (0, 0, 0) to a plane is (1, 2, 3), then the equation of the plane is

  • 2x + y + 3z = 14

  • x + 2y + 3z = 14

  • x + 2y + 3z + 14 = 0

  • x + 2y - 3z = 14


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419.

In any ABC, r1r2 + r2r3 + r3r1 = ?

  • 2r2

  • r

  • 2r

  • 2


420.

If in a ABC, 1a +c + 1b + c = 3a + b + c, thenC = ?

  • 30°

  • 45°

  • 60°

  • 90°


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