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 Multiple Choice QuestionsMultiple Choice Questions

1.

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 two values of λ


B.

no value of λ

open square brackets straight lambda space left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis close square brackets space straight lambda squared space straight b with rightwards arrow on top space straight lambda space straight c with rightwards arrow on top right square bracket space equals space open square brackets straight a with rightwards arrow on top straight b with rightwards arrow on top plus straight c with rightwards arrow on top space straight b with rightwards arrow on top close square brackets
open vertical bar table row straight lambda straight lambda 0 row 0 cell straight lambda squared end cell 0 row 0 0 straight lambda end table close vertical bar open vertical bar table row 1 0 0 row 0 1 1 row 0 1 0 end table close vertical bar
rightwards double arrow space straight lambda to the power of 4 space space equals negative 1
Hence space no space real space value space of space straight lambda
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2.

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

  • 0


D.

0

left parenthesis straight a with rightwards arrow on top space plus 2 straight b with rightwards arrow on top space right parenthesis space equals space straight t subscript 1 straight c with rightwards arrow on top space.... space left parenthesis 1 right parenthesis
and space straight b with rightwards arrow on top space plus 3 straight c with rightwards arrow on top space equals space straight t subscript 2 straight a with rightwards arrow on top space..... space left parenthesis 2 right parenthesis
left parenthesis 1 right parenthesis space minus space 2 space straight x space left parenthesis 2 right parenthesis
rightwards double arrow space straight a with rightwards arrow on top space left parenthesis 1 space plus 2 straight t subscript 2 right parenthesis space plus space straight c with rightwards arrow on top space left parenthesis negative straight t subscript 1 minus 6 right parenthesis space equals space 0
rightwards double arrow space 1 plus space 2 straight t subscript 2 space equals 0
rightwards double arrow space straight t subscript 2 space equals space minus space minus 1 divided by 2 space & space straight t subscript 1 space equals space minus 6
since space straight a with rightwards arrow on top space and space straight c with rightwards arrow on top space are space space non minus collinear
Putting space the space value space of space straight t subscript 1 space and space straight t subscript 2 space in space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space we space get space 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 space space equals space space 0
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3.

A particle is acted upon by constant forces 4 space straight i with hat on top space plus straight j with hat on top space minus 3 straight k with hat on top space space and 3 space straight i with hat on top space plus straight j with hat on top space minus straight k with hat on top which displace it from a point straight i with hat on top space plus 2 straight j with hat on top space plus 3 straight k with hat on top space to space the space point space 5 straight i with hat on top space plus space 4 straight j with hat on top space plus straight k with hat on top. space The work done in standard units by the forces is given by 

  • 40

  • 30

  • 25

  • 15


A.

40

work space done space by space the space forces space stack straight F subscript 1 with rightwards arrow on top space and space stack straight F subscript 2 with rightwards arrow on top space is space left parenthesis straight F with rightwards arrow on top subscript 1 space plus space stack straight F subscript 2 with rightwards arrow on top right parenthesis. straight d with rightwards arrow on top comma space where space straight d with rightwards arrow on top space is space displacement
According space to space question space stack straight F subscript 1 with rightwards arrow on top space plus stack straight F subscript 2 with rightwards arrow on top space equals space left parenthesis 4 straight i with hat on top space plus straight j with hat on top minus 3 straight k with hat on top right parenthesis space plus space left parenthesis 3 straight i with hat on top space plus straight j with hat on top minus straight k with hat on top right parenthesis space equals space 7 straight i with hat on top space plus 2 straight j with hat on top space minus 4 straight k with hat on top
and space straight d with rightwards arrow on top space equals left parenthesis 5 straight i with hat on top space plus 4 straight j with hat on top space plus straight k with hat on top right parenthesis space minus space left parenthesis straight i with hat on top space plus 2 straight j with hat on top space plus 3 straight k with hat on top right parenthesis space equals space 4 straight i with hat on top space plus 2 straight j with hat on top space minus 2 straight k with hat on top.
Hence space left parenthesis stack straight F subscript 1 space with rightwards arrow on top space plus straight F with rightwards arrow on top subscript 2 right parenthesis. space straight d space is space 40
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4.

Let top enclose straight a space comma top enclose straight b space and space top enclose straight c space be non-zero vectors such that left parenthesis top enclose straight a space straight x space top enclose straight b right parenthesis space straight x space top enclose straight c space equals space 1 third space vertical line straight b with bar on top vertical line vertical line straight c with bar on top vertical line vertical line straight a with bar on top vertical line If θ is the acute angle between the vectors top enclose straight b space and space top enclose straight c comma then sin θ equals

  • 1/3

  • fraction numerator square root of 1 over denominator 3 end fraction
  • 2/3

  • fraction numerator 2 square root of 2 over denominator 3 end fraction

D.

fraction numerator 2 square root of 2 over denominator 3 end fraction
1 third space vertical line straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line vertical line straight a with rightwards arrow on top vertical line space equals left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x space straight c with rightwards arrow on top
we space know space that space
left parenthesis straight a with rightwards arrow on top space straight x straight b with rightwards arrow on top right parenthesis space straight x space straight c with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis straight b with rightwards arrow on top space minus space left parenthesis straight b with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis straight a with rightwards arrow on top
1 third space vertical line straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line straight a with rightwards arrow on top space equals space left parenthesis straight a with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis. straight b with rightwards arrow on top space minus space left parenthesis straight b with rightwards arrow on top space. space straight c with rightwards arrow on top right parenthesis. straight a with rightwards arrow on top
On space comparing comma space we space get
1 third space vertical line straight b with rightwards arrow on top vertical line vertical line straight c with rightwards arrow on top vertical line space equals negative space left parenthesis straight b with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis space and space space left parenthesis straight a with rightwards arrow on top. straight c with rightwards arrow on top right parenthesis
rightwards double arrow space 1 third bc space equals negative space bc space cos space straight theta
cos space space equals space fraction numerator negative 1 over denominator 3 end fraction
cos squared space straight theta space equals space 1 third
rightwards double arrow 1 minus sin squared space straight theta space equals space 1 over 9
sin squared space straight theta space equals space 1 minus 1 over 9 space equals 8 over 9 space rightwards double arrow space sin space straight theta space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction
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5.

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
  • cosA : cosB : cosC


C.

cos space straight A over 2 colon space cos space straight B over 2 space colon space Cos space straight C over 2

Using Lami’s Theorem

therefore space straight P space colon space straight Q space colon straight R space equals space cos space straight A over 2 space colon space cos space straight B over 2 colon space cos straight C over 2

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6. If space straight a with bar on top comma space straight b with bar on top comma straight c with bar on top space are non-coplanar vectors and λ is a real number, then the vectors straight a with bar on top space plus 2 straight b with bar on top space plus 3 straight c with bar on top space comma space straight lambda straight b with bar on top space plus 4 space straight c with bar on top space and space left parenthesis 2 straight lambda space minus 1 right parenthesis straight c with bar on top are non-coplanar for
  • all values of λ

  • all except one value of λ

  • all except two values of λ

  • no value of λ


C.

all except two values of λ

Condition for given three vectors to be coplanar is open vertical bar table row 1 2 3 row 0 straight lambda 4 row 0 0 cell 2 straight lambda minus 1 end cell end table close vertical bar =0
 ⇒ λ = 0, 1/2.
Hence given vectors will be non coplanar for all real values of λ except 0, 1/2.

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7.

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 PC with rightwards arrow on top space equals space 0

A.

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 AC with rightwards arrow on top space plus CP with rightwards arrow on top space equals space 0
PB with rightwards arrow on top space plus BC with rightwards arrow on top space plus space CP with rightwards arrow on top space equals space 0
Adding comma space we space get
PA with rightwards arrow on top plus PB with rightwards arrow on top space plus space AC with rightwards arrow on top space plus space BC with rightwards arrow on top space plus 2 CP with rightwards arrow on top space equals space 0
Since space AC with rightwards arrow on top space equals space minus space BC with rightwards arrow on top
& space CP with rightwards arrow on top space equals space minus space PC with rightwards arrow on top
rightwards double arrow space PA with rightwards arrow on top space plus PB with rightwards arrow on top space minus 2 PC with rightwards arrow on top space equals space 0 space
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8.

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
  • 4 straight a with rightwards arrow on top squared

C.

2 straight a with rightwards arrow on top squared
Let space straight a with rightwards arrow on top space equals space straight x straight i with hat on top plus straight y straight j with hat on top plus straight z straight k with hat on top
straight a with rightwards arrow on top space straight x space straight i with hat on top space equals space straight z straight j with hat on top space minus space straight y straight k with hat on top
rightwards double arrow left parenthesis straight a with rightwards arrow on top space straight x space straight i with hat on top right parenthesis squared space equals space straight y squared plus straight z squared
rightwards double arrow left parenthesis straight a with rightwards arrow on top space straight x space straight j with hat on top right parenthesis space equals space straight x squared plus straight z squared
rightwards double arrow left parenthesis straight a with rightwards arrow on top space straight x space straight k with hat on top right parenthesis space equals space straight x squared plus straight y squared
similarly space left parenthesis straight a with rightwards arrow on top space straight x space straight j with hat on top right parenthesis squared space equals space straight x squared plus straight z squared
and space open parentheses straight a with rightwards arrow on top space straight x stack space straight k with hat on top close parentheses squared space equals space straight x squared plus straight y squared
rightwards double arrow space left parenthesis straight a with rightwards arrow on top space straight x space straight i with hat on top right parenthesis squared space plus space left parenthesis straight a with rightwards arrow on top space straight x straight j with hat on top right parenthesis squared space plus space left parenthesis straight a with hat on top space straight x straight k with hat on top right parenthesis squared space equals space 2 space left parenthesis straight x squared space plus straight y squared plus space straight z squared right parenthesis space equals space 2 straight a with rightwards arrow on top squared
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9.

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

  • the Harmonic Mean of a and b


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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10.

Let straight u with bar on top comma space straight v with bar on top space comma space straight w with bar on top be such that vertical line top enclose straight u vertical line space equals space vertical line top enclose straight v vertical line space equals space 2 comma space vertical line top enclose straight w vertical line space equals space 3 If the projection top enclose straight v space along space top enclose straight u  is equal to that of space top enclose straight w space along space top enclose top enclose straight u end enclose space and space top enclose straight v comma space top enclose straight w space are perpendicular to each other then vertical line top enclose straight u space minus top enclose straight v space plus top enclose straight w vertical line equals

  • 2

  • square root of 14
  • square root of 12
  • 12


B.

square root of 14
vertical line top enclose straight u vertical line space equals space 1 comma space vertical line top enclose straight v vertical line space equals space 2 comma space vertical line top enclose straight w vertical line space equals space 3
projection space of space top enclose straight v space along space straight u with rightwards arrow on top space equals fraction numerator straight v with rightwards arrow on top. straight u with rightwards arrow on top over denominator vertical line top enclose straight u vertical line end fraction

projection space of space top enclose straight u space along space straight w with rightwards arrow on top space equals space fraction numerator straight w with rightwards arrow on top. straight u with rightwards arrow on top over denominator vertical line straight u with rightwards arrow on top vertical line end fraction
rightwards double arrow stack straight v. with rightwards arrow on top straight u with rightwards arrow on top space equals space straight w with rightwards arrow on top. straight u with rightwards arrow on top
and space straight v with rightwards arrow on top comma straight w with rightwards arrow on top space are space perpendicular space to space each space other
therefore space straight v with rightwards arrow on top comma straight w with rightwards arrow on top space equals space 0
vertical line straight u with rightwards arrow on top minus straight v with rightwards arrow on top space plus straight w with rightwards arrow on top vertical line squared space equals space vertical line straight u with rightwards arrow on top vertical line squared space plus space vertical line straight v with rightwards arrow on top vertical line squared space plus vertical line straight w with rightwards arrow on top vertical line squared space minus 2 straight u with rightwards arrow on top. straight v with rightwards arrow on top space plus 2 straight u with rightwards arrow on top. straight w with rightwards arrow on top space minus space 2 straight v with rightwards arrow on top. straight w with rightwards arrow on top
vertical line straight u with rightwards arrow on top minus straight v with rightwards arrow on top space plus straight w with rightwards arrow on top vertical line squared space space equals space 1 plus 4 plus 9 space minus 2 straight u with rightwards arrow on top. straight v with rightwards arrow on top space plus 2 straight v with rightwards arrow on top space. straight u with rightwards arrow on top
vertical line straight u with rightwards arrow on top minus straight v with rightwards arrow on top space plus straight w with rightwards arrow on top vertical line squared space equals space 1 plus 4 plus 9
vertical line straight u with rightwards arrow on top minus straight v with rightwards arrow on top space plus straight w with rightwards arrow on top vertical line squared space equals space square root of 14
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