Subject

Mathematics

Class

JEE Class 12

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JEE Mathematics 2005 Exam Questions

Multiple Choice Questions

61.

Three houses are available in a locality. Three persons apply for the houses. Each applies to one house without consulting others. The probability that all the three apply for the same house is 

  • 2/9

  • 1/9

  • 8/9

  • 7/9


B.

1/9

For a particular house being selected

Probability = 1/3

Prob(all the persons apply for the same house) =open parentheses 1 third space straight x 1 third space straight x 1 third close parentheses 3 space equals space 1 over 9

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62.

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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63.

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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64.

The angle between the lines 2x = 3y = − z and 6x = − y = − 4z is

  • 0o

  • 90o

  • 45o

  • 30o


B.

90o

Angle between the lines 2x = 3y = - z & 6x = -y = -4z is 90°
Since a1a2 + b1b2 + c1c2 = 0

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65.

If the plane 2ax − 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres

x2 + y2 + z2 + 6x − 8y − 2z = 13 and x2 + y2 + z2 − 10x + 4y − 2z = 8, then a equals

  • -1

  • 1

  • -2

  • 2


C.

-2

Plane 2ax – 3ay + 4az + 6 = 0 passes through the mid point of the centre of spheres x2 + y2 + z2 + 6x – 8y – 2z = 13 and x2 + y2 + z2 – 10x + 4y – 2z = 8 respectively centre of spheres are (-3, 4, 1) & (5, - 2, 1) Mid point of centre is (1, 1, 1) Satisfying this in the equation of plane,
we get 2a – 3a + 4a + 6 = 0
⇒ a = -2.

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66.

The distance between the line straight r with rightwards arrow on top space equals space 2 straight i with hat on top space minus 2 straight j with hat on top space plus 3 straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top minus straight j with hat on top space plus 4 straight k with hat on top right parenthesis and the plane straight r with rightwards arrow on top. left parenthesis straight i with hat on top space plus 5 straight j with hat on top space plus straight k with hat on top right parenthesis space equals space 5 space is

  • 10/9

  • fraction numerator 10 over denominator 3 square root of 3 end fraction
  • 3/10

  • 10/3


B.

fraction numerator 10 over denominator 3 square root of 3 end fraction

Distance between the line 

straight r with rightwards arrow on top space equals space 2 straight i with hat on top space minus 2 space straight j with hat on top space plus 3 straight k with hat on top space plus straight lambda space left parenthesis straight i with hat on top space minus straight j with hat on top space plus 4 straight k with hat on top right parenthesis space and space the space plane space straight r with rightwards arrow on top. space left parenthesis straight i with hat on top space plus 5 straight j with hat on top space plus straight k with hat on top right parenthesis space equals space 5
equation of plane is x + 5y + z = 5 ∴ Distance of line from this plane = perpendicular distance of point (2, -2, 3) from the plane
straight i. straight e space open vertical bar fraction numerator 2 minus 10 plus 3 minus 5 over denominator square root of 1 plus 5 squared plus 1 end root end fraction close vertical bar space equals space fraction numerator 10 over denominator 3 square root of 3 end fraction

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67.

If a vertex of a triangle is (1, 1) and the mid-points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is

  • (-1, 7/3)

  • (-1/3, 7/3)

  • (1, 7/3)

  • (1/3, 7/3)


C.

(1, 7/3)

Vertex of triangle is (1, 1) and midpoint of sides through this vertex is (-1, 2) and (3, 2) ⇒ vertex B and C come out to be (-3, 3) and (5, 3)
 therefore centroid  is fraction numerator 1 minus 3 plus 5 over denominator 3 end fraction comma fraction numerator 1 plus 3 plus 3 over denominator 3 end fraction
rightwards double arrow space left parenthesis 1 comma 7 divided by 3 right parenthesis 

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68.

The value of integral subscript negative straight pi end subscript superscript straight pi space fraction numerator cos squared space straight x over denominator 1 plus space straight a to the power of straight x end fraction space dx comma space straight a greater than 0 space is

  • a π

  • π/2

  • π/a


B.

π/2

integral subscript negative straight pi end subscript superscript straight pi space fraction numerator cos squared over denominator 1 plus straight a to the power of straight x end fraction space dx space equals space integral subscript 0 superscript straight pi space cos squared space straight x space dx space equals space straight pi over 2
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69. 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

  • neither x nor y


D.

neither x nor y

straight a with rightwards arrow on top space equals space straight i with hat on top minus straight k with hat on top comma space straight b with rightwards arrow on top space equals space straight x straight i with rightwards arrow on top space plus straight j with rightwards arrow on top space plus left parenthesis 1 minus straight x right parenthesis straight k with hat on top space and space straight c with rightwards arrow on top space equals straight y straight i with hat on top space plus straight x 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
left square bracket straight a with rightwards arrow on top straight b with rightwards arrow on top straight c with rightwards arrow on top right square bracket space equals space straight a with rightwards arrow on top. space left parenthesis straight b with rightwards arrow on top straight x straight c with rightwards arrow on top right parenthesis
straight b with rightwards arrow on top space straight x straight c with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row straight x 1 cell 1 minus straight x end cell row straight y straight x cell 1 plus straight x minus straight y end cell end table close vertical bar
space equals space straight i with hat on top space left parenthesis 1 plus space straight x minus straight x minus straight x squared right parenthesis minus space straight j with hat on top space left parenthesis straight x plus straight x squared minus xy minus straight y plus xy right parenthesis space plus straight k with hat on top space left parenthesis straight x squared minus straight y right parenthesis
straight a with rightwards arrow on top. left parenthesis straight b with rightwards arrow on top straight x straight c with rightwards arrow on top right parenthesis space equals space 1
which does not depend on x and y.
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70.

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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