32. MATHs Mock Test 32 for NDA

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  Question No.    311
  Topic    Trigonometry

  Given

  Angle = 1000°

  Reduce angle

  1000° = 2×360° + 280°

  Therefore

  1000° ≡ 280°

  Now

  270° < 280° < 360°

  Hence terminal side lies in

  Fourth Quadrant

  Answer    (d) Fourth Quadrant

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  Question No.    312
  Topic    Trigonometry

  Given

  x = sinθ cosθ

  y = sinθ + cosθ

  Square y

  y²

  = sin²θ + cos²θ + 2sinθcosθ

  = 1 + 2x

  Therefore

  y² - 2x = 1

  Answer    (a) y² - 2x = 1

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  Explanation

  Question No.    313
  Topic    Heights & Distances

  Let

  Height of balloon = h km

  Initial distance = x km

  tan30° = h/x

  h = x/√3      ...(1)

  After walking 1 km

  Distance = x-1

  tan60° = h/(x-1)

  h = √3(x-1)   ...(2)

  Equating

  x/√3 = √3(x-1)

  x = 3x - 3

  2x = 3

  x = 3/2

  From (1)

  h = (3/2)/√3

  = √3/2 km

  Answer    (b) √3/2 km

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  Explanation

  Question No.    314
  Topic    3D Geometry

  Fact

  Three non-collinear points
  determine a unique plane.

  Therefore

  Number of planes

  = 1

  Answer    (c) 1

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  Explanation

  Question No.    315
  Topic    3D Geometry

  Line 1

  x+z=0 , y=0

  Direction ratios

  (1,0,-1)

  Line 2

  20x=15y=12z

  Direction ratios

  (3,4,5)

  Angle θ

  cosθ

  = |(1)(3)+(0)(4)+(-1)(5)|
   / [√2 × √50]

  = |-2|/10

  = 1/5

  Therefore

  θ = cos⁻¹(1/5)

  Answer    (a) cos⁻¹(1/5)

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  Question No.    316
  Topic    Conic Sections

  For ellipse

  Length of latus rectum

  = 2b²/a

  Minor axis

  = 2b

  Given

  2b²/a = (1/2)(2b)

  ⇒ b²/a = b/2

  ⇒ 2b = a

  ⇒ b = a/2

  Eccentricity

  e² = 1 - b²/a²

  = 1 - 1/4

  = 3/4

  e = √3/2

  Answer    (d) √3/2

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  Question No.    317
  Topic    Sphere

  Given

  x²+y²+z²+2ux+2vy+2wz+d=0

  Radius²

  = u²+v²+w²-d

  For real sphere

  Radius² > 0

  Therefore

  u²+v²+w²-d > 0

  u²+v²+w² > d

  Answer    (b) u² + v² + w² > d

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  Explanation

  Question No.    318
  Topic    Vectors

  Given

  |a+b| = |a-b|

  Squaring both sides

  |a+b|² = |a-b|²

  a²+b²+2a·b

  = a²+b²-2a·b

  4a·b = 0

  a·b = 0

  Hence

  a ⟂ b

  Answer    (b) a is perpendicular to b

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  Explanation

  Question No.    319
  Topic    Vector Algebra

  Given

  a×(b+c)+b×(c+a)+c×(a+b)

  Expand

  = a×b+a×c
  +b×c+b×a
  +c×a+c×b

  Using

  b×a = -a×b
  c×a = -a×c
  c×b = -b×c

  All terms cancel

  Result = 0

  Answer    (d) 0⃗

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  Question No.    320
  Topic    Probability

  Required

  A and B occur
  but C does not occur

  A occurs ⇒ A

  B occurs ⇒ B

  C does not occur ⇒ C̅

  Required event

  A ∩ B ∩ C̅

  Answer    (b) A ∩ B ∩ C̅