Let a, b and c be distinct non-negative numbers. If the vectors�

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 Multiple Choice QuestionsMultiple Choice Questions

21.

A particle has two velocities of equal magnitude inclined to each other at an angle θ. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then θ is

  • 90°

  • 120°

  • 45°

  • 45°

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22.

The values of a, for which the points A, B, C with position vectors 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma space straight i with hat on top minus 3 straight j with hat on top minus 5 straight k with hat on top space and space straight a straight i with hat on top minus 3 straight j with hat on top plus straight k with hat on toprespectively are the vertices of a right-angled triangle with C =π/2  are

  • 2 and 1

  • −2 and −1

  • −2 and 1

  • −2 and 1

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23.

ABC is a triangle. Forces straight P with rightwards arrow on top comma space straight Q with rightwards arrow on top comma space straight R with rightwards arrow on top space acting along IA, IB and IC respectively are in equilibrium, where I is incentre of ∆ABC. Then P : Q : R is

  • sin A : sin B : sin C

  • sin space straight A over 2 space colon space Sin space straight B over 2 space colon space sin space straight C over 2
  • cos space straight A over 2 colon space cos space straight B over 2 space colon space Cos space straight C over 2
  • cos space straight A over 2 colon space cos space straight B over 2 space colon space Cos space straight C over 2
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24.

If C is the mid -point of AB and P is any point outside AB, then

  • PA with rightwards arrow on top space plus space PB with rightwards arrow on top space equals space 2 PC with rightwards arrow on top
  • PA with rightwards arrow on top space plus space PB with rightwards arrow on top space equals space PC with rightwards arrow on top
  • PA with rightwards arrow on top space plus space PB with rightwards arrow on top space plus space 2 PC with rightwards arrow on top space equals space 0
  • PA with rightwards arrow on top space plus space PB with rightwards arrow on top space plus space 2 PC with rightwards arrow on top space equals space 0
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25.

For any vector straight a with rightwards arrow on top  the value of left parenthesis straight a with rightwards arrow on top space straight x straight i with hat on top right parenthesis squared space plus left parenthesis straight a with rightwards arrow on top space straight x straight j with hat on top right parenthesis squared space plus left parenthesis straight a with rightwards arrow on top space straight x straight k with hat on top right parenthesis squared is equal to

  • 3 straight a with rightwards arrow on top squared
  • straight a with rightwards arrow on top squared
  • 2 straight a with rightwards arrow on top squared
  • 2 straight a with rightwards arrow on top squared
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26.

Let a, b and c be distinct non-negative numbers. If the vectors straight a straight i with hat on top space plus straight a space straight j with hat on top space plus space straight c straight k with hat on top comma space space straight i with hat on top space plus straight k with hat on top space and space straight c straight i with hat on top space plus straight c straight j with hat on top space plus straight b straight k with hat on top space 

  • the Geometric Mean of a and b

  • the Arithmetic Mean of a and b

  • equal to zero

  • equal to zero


A.

the Geometric Mean of a and b

Vector space straight a space straight i with hat on top space plus straight a straight j with hat on top space plus straight c straight k with hat on top space comma space straight i with hat on top space plus straight k with hat on top space and space straight c straight i with hat on top space plus straight c straight j with hat on top space plus straight b straight k with hat on top space are space coplanar
open vertical bar table row straight a straight a straight c row 1 0 1 row straight c straight c straight b end table close vertical bar space equals space 0
rightwards double arrow space straight c squared space equals ab
therefore space straight a comma straight b comma straight c space are space in space straight G. straight P
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27.

If straight a with rightwards arrow on top comma straight b with rightwards arrow on top comma space straight c with rightwards arrow on top are non -coplanar vector λ is a real number then

open square brackets straight lambda left parenthesis straight a with rightwards arrow on top space plus straight b with rightwards arrow on top right parenthesis space straight lambda squared space straight b with rightwards arrow on top straight lambda space straight c with rightwards arrow on top close square brackets space equals space open square brackets straight a with rightwards arrow on top space straight b with rightwards arrow on top space plus straight c with rightwards arrow on top straight b with rightwards arrow on top close square brackets space for

  • exactly one value of λ

  • no value of λ

  • exactly three values of λ

  • exactly three values of λ

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28. Let space straight a with rightwards arrow on top space equals space straight i with hat on top space minus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight x straight i with hat on top space plus straight j with hat on top space plus left parenthesis 1 minus straight x right parenthesis space straight k with hat on top space and space
straight c with rightwards arrow on top space equals space straight y straight i with hat on top space plus straight x space straight j with hat on top space plus left parenthesis 1 plus straight x minus straight y right parenthesis straight k with hat on top space. Then space left square bracket straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top right square bracket depends on
  • only y

  • only x

  • both x and y

  • both x and y

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29.

The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one-third of the other force. The ratio of larger force to smaller one is 

  • 2 : 1

  • 3 colon square root of 2
  • 3:2

  • 3:2

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30.

Let straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on top be three non-zero vectors such that no two of these are collinear. If the vector straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top is collinear withstraight c with rightwards arrow on top space and space straight b with rightwards arrow on top space plus 3 straight c with rightwards arrow on top is collinear with straight a with rightwards arrow on top (λ being some non-zero scalar) then straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top space plus 6 straight c with rightwards arrow on top equals

  • λstraight a with rightwards arrow on top

  • λstraight b with rightwards arrow on top

  • λstraight c with rightwards arrow on top

  • λstraight c with rightwards arrow on top

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