28. MATHs Mock Test 28 for NDA

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• 0 Questions answered incorrect
• 10 Questions unattempted

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  Explanation

  Question No.    271
  Topic    Probability

  Given    4 coins are tossed

  Total Outcomes    2⁴ = 16

  Favourable Outcomes
  (Exactly 2 Heads)

  = ⁴C₂
  = 6

  Probability
  = 6/16
  = 3/8

  Answer    (d) 3/8

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  Explanation

  Question No.    272
  Topic    Straight Line

  Given Lines

  x - 3y + 1 = 0
  2x + 5y - 9 = 0

  Intersection Point

  x = 2
  y = 1

  Required Line passes through (2,1)

  Check Options

  (a) 2x-y-5=0
  At (2,1) ⇒ 4-1-5=-2 ✗

  (b) 2x+y+5=0
  At (2,1) ⇒ 10 ✗

  (c) 2x+y-5=0
  At (2,1) ⇒ 0 ✓

  Distance from Origin

  = |−5|/√(2²+1²)
  = 5/√5
  = √5 ✓

  Answer    (c) 2x + y - 5 = 0

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  Explanation

  Question No.    273
  Topic    Determinants

  Given

  | x   x²   1+x³ |
  | y   y²   1+y³ |
  | z   z²   1+z³ |

  and xyz = -1

  Apply

  C₃ → C₃ - xyz·C₁

  Since xyz = -1

  C₃ = C₃ + C₁

  ⇒ Third column becomes

  1+x³+x
  1+y³+y
  1+z³+z

  Using factorization and condition xyz=-1,
  the determinant reduces to a square factor,
  hence determinant is always positive.

  Answer    (b) Positive

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  Explanation

  Question No.    274
  Topic    Logarithms

  Given

  aˣ = bʸ = cᶻ = k

  Therefore

  a = k^(1/x)
  b = k^(1/y)
  c = k^(1/z)

  Also

  log_b a = log_c b

  ⇒ (1/x)/(1/y) = (1/y)/(1/z)

  ⇒ y/x = z/y

  ⇒ y² = xz

  Answer    (a) y² = xz

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  Explanation

  Question No.    275
  Topic    Functions

  Given

  f(n)=(n−1)/2 , n odd
  f(n)=−n/2 , n even

  Values

  f(1)=0
  f(2)=−1
  f(3)=1
  f(4)=−2
  f(5)=2
  f(6)=−3

  Range

  {0, ±1, ±2, ±3, ...}

  = Set of Integers

  Hence Onto

  Different natural numbers produce
  different integers

  Hence One-One

  Answer    (c) One-One and Onto

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  Explanation

  Question No.    276
  Topic    Series

  S = 1/2! + 1/3! + 1/4! + ...

  Using

  e = 1 + 1 + 1/2! + 1/3! + ...

  Therefore

  S = e - 2

  Since option (e−2) is absent,
  the intended answer in the paper is

  Answer    (None of the given options)

  Actual Sum = e − 2

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  Explanation

  Question No.    277
  Topic    Differential Equations

  Given

  dy/dx + y/x = y²

  Divide by y²

  (1/y²)(dy/dx) + 1/(xy) = 1

  Let

  v = 1/y

  Then

  dv/dx = -(1/y²)(dy/dx)

  Equation becomes

  dv/dx - v/x = -1

  Linear Differential Equation

  Integrating Factor

  IF = e^(∫(-1/x)dx)
    = 1/x

  After solving

  v = x log(C/x)

  Since v = 1/y

  1/y = x log(C/x)

  ⇒ xy log(C/x) = 1

  Answer    (d) xy log(C/x) = 1

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  Explanation

  Question No.    278
  Topic    Sets

  A = {a,b,c,d,e}
  B = {d,e,f,g}

  A-B = {a,b,c}

  B-A = {f,g}

  (A-B) ∩ (B-A)

  = {a,b,c} ∩ {f,g}

  = ϕ

  Answer    (d) ϕ

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  Explanation

  Question No.    279
  Topic    Probability

  Two dice thrown

  Total Outcomes

  = 36

  Required

  Sum ≥ 4

  Complement

  Sum < 4

  Possible sums

  2 : (1,1) → 1 outcome

  3 : (1,2),(2,1) → 2 outcomes

  Total = 3 outcomes

  Required outcomes

  = 36 - 3
  = 33

  Probability

  = 33/36
  = 11/12

  Answer    (c) 11/12

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  Explanation

  Question No.    280
  Topic    Statistics

  Given

  Mean = 5
  SD = 2

  New Mean

  = 5 + 5
  = 10

  New SD

  = 2
  (Addition of constant does not change SD)

  Coefficient of Variation

  = (SD/Mean) × 100

  = (2/10) × 100

  = 20%

  Answer    (b) 20