24. MATHs Mock Test 24 for NDA

• 0 Questions answered correctly
• 0 Questions answered incorrect
• 10 Questions unattempted

• 1

  

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  Explanation

  Question:

  Find the area bounded by

  xy = c

  and x-axis between

  x = 1 and x = 4

  Explanation:

  Given:

  y = c/x

  Required area

  A = ∫₁⁴ (c/x) dx

   = c ∫₁⁴ dx/x

   = c [ln x]₁⁴

   = c(ln4 − ln1)

   = c ln4

   = 2c ln2

  Answer:
  (c) 2c log2

• 2

  

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  Explanation

  Question:

  In = dⁿ/dxⁿ (xⁿ log x)

  Find:

  In − nIn−1

  Explanation:

  Differentiate once:

  d/dx(xⁿ logx)

  = nxⁿ⁻¹logx + xⁿ⁻¹

  Applying (n−1)th derivative:

  In

  = nIn−1 + dⁿ⁻¹/dxⁿ⁻¹(xⁿ⁻¹)

  But

  dⁿ⁻¹/dxⁿ⁻¹(xⁿ⁻¹)

  = (n−1)!

  Hence

  In − nIn−1

  = (n−1)!

  Answer:
  (d) (n−1)!

• 3

  

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  Explanation

  Question:

  y = (sinx)^(sinx)^(sinx)...∞

  Find dy/dx

  Explanation:

  Since infinite power tower:

  y = (sinx)^y

  Taking logarithm:

  logy = y log(sinx)

  Differentiate:

  (1/y)dy/dx

  = (dy/dx)log(sinx)
   + y(cotx)

  Collect dy/dx terms:

  dy/dx [1/y − log(sinx)]

  = y cotx

  dy/dx

  = y² cotx
   /[1 − ylog(sinx)]

  Answer:
  (a)

• 4

  

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  Explanation

  Question:

  If A and B are mutually exclusive events

  then ?

  Explanation:

  Mutually exclusive means

  A ∩ B = ϕ

  No fixed relation exists between

  P(A) and P(B)

  Examples:

  P(A)=0.1, P(B)=0.5

  or

  P(A)=0.7, P(B)=0.1

  Both possible.

  Therefore none of the inequalities
  must always hold.

  Answer:
  (d) None of these

• 5

  

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  Explanation

  Question:

  In a non-leap year,

  Probability of getting

  53 Sundays or
  53 Tuesdays or
  53 Thursdays

  Explanation:

  365 days

  = 52 weeks + 1 day

  Only one weekday occurs 53 times.

  Extra day may be:

  Sun, Mon, Tue,
  Wed, Thu, Fri, Sat

  (7 equally likely cases)

  Favourable days:

  Sunday
  Tuesday
  Thursday

  Total favourable = 3

  Probability

  = 3/7

  Answer:
  (c) 3/7

• 6

  

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  Explanation

  Question:

  f(x)= (√2 cosx −1)/(cotx −1)

  x ≠ π/4

  f(π/4)=k

  For continuity find k.

  Explanation:

  At x=π/4

  Numerator:

  √2 cos(π/4)−1

  =1−1

  =0

  Denominator:

  cot(π/4)−1

  =0

  Apply L'Hospital Rule:

  k

  = lim [−√2 sinx]
       /[−cosec²x]

  at x=π/4

  = √2 sin(π/4)
   /cosec²(π/4)

  = 1/2

  Answer:
  (a) 1/2

• 7

  

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  Explanation

  Question:

  A = [3  1  −1
      0  1   2]

  Find AA'

  Explanation:

  For any matrix A,

  (AA')'

  = (A')'A'

  = AA'

  Therefore

  AA' is always symmetric.

  Also AA' is not generally
  orthogonal or skew-symmetric.

  Answer:
  (a) Symmetric matrix

• 8

  

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  Explanation

  Question:

  If A⁻¹ = A

  Find A²

  Explanation:

  Given:

  A⁻¹ = A

  Multiply both sides by A:

  AA = I

  A² = I

  Hence

  A² equals identity matrix.

  Answer:
  (c)

  [1 0 0
  0 1 0
  0 0 1]

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  Explanation

  Question:

  x + 2y + 3z = 1

  x − y + 4z = 0

  2x + y + 7z = 1

  Find nature of solutions.

  Explanation:

  From second equation:

  x = y − 4z

  Substitute in first:

  (y−4z)+2y+3z=1

  3y−z=1 ...(i)

  Substitute in third:

  2(y−4z)+y+7z=1

  3y−z=1 ...(ii)

  Both equations are identical.

  Thus only two independent equations
  for three unknowns.

  Hence infinitely many solutions.

  Answer:
  (d) Infinitely many solutions

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  Explanation

  Question:

  ∫₀^(π/2)

  2^(sinx)
  ---------------- dx
  2^(sinx)+2^(cosx)

  Explanation:

  Let

  I = ∫₀^(π/2)
     2^(sinx)
    ---------------- dx
    2^(sinx)+2^(cosx)

  Using x → π/2 − x

  I = ∫₀^(π/2)
     2^(cosx)
    ---------------- dx
    2^(sinx)+2^(cosx)

  Add both:

  2I

  = ∫₀^(π/2) 1 dx

  = π/2

  Therefore

  I = π/4

  Answer:
  (c) π/4