34. MATHs Mock Test 34 for NDA

• 0 Questions answered correctly
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• 10 Questions unattempted

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  Explanation

  Question No.    331
  Topic    Sets

  Given

  {A ∩ (X-B)} ∪ B

  Since

  X-B = B'

  Therefore

  = (A ∩ B') ∪ B

  Using Distributive Law

  = (A ∪ B) ∩ (B' ∪ B)

  = (A ∪ B) ∩ X

  = A ∪ B

  Answer    (a) A ∪ B

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  Explanation

  Question No.    332
  Topic    Logarithms

  Given

  log₁₀(x+1) + log₁₀5 = 3

  Using

  log a + log b = log(ab)

  log₁₀[5(x+1)] = 3

  Therefore

  5(x+1)=10³

  5(x+1)=1000

  x+1=200

  x=199

  Answer    (a) 199

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  Explanation

  Question No.    333
  Topic    Quadratic Equations

  Given

  Roots of

  Ax²+Bx+C=0

  are r,s

  Therefore

  r+s = -B/A
  rs = C/A

  Required roots

  r²,s²

  Sum of new roots

  r²+s²

  =(r+s)²-2rs

  =B²/A² - 2C/A

  =(B²-2AC)/A²

  For equation

  x²+px+q=0

  p = -(sum of roots)

  p = -(B²-2AC)/A²

  =(2AC-B²)/A²

  Answer    (c) (2AC-B²)/A²

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  Explanation

  Question No.    334
  Topic    Binomial Theorem

  Given

  (1+x+2x³)(3x⁻²/2 - 1/(3x))⁹

  General term

  Tᵣ₊₁

  = ⁹Cᵣ (3x⁻²/2)^(9-r)
  (-1/3x)^r

  Power of x

  = -2(9-r)-r

  = r-18

  To obtain constant term after
  multiplication by

  1, x, 2x³

  Need powers

  0, -1, -3

  Check

  r-18=0 ⇒ r=18 (not possible)

  r-18=-1 ⇒ r=17 (not possible)

  r-18=-3 ⇒ r=15 (not possible)

  Hence no constant term exists.

  Answer    (d) No such term exists

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  Explanation

  Question No.    335
  Topic    Complex Numbers

  Given

  2x = 3+5i

  Therefore

  x = (3+5i)/2

  Observe

  Polynomial

  2x³+2x²-7x+72

  Factorisation

  = (2x-3-5i)(x²+...)

  + constant

  Since

  2x=3+5i

  (2x-3-5i)=0

  Substituting x=(3+5i)/2

  Value simplifies to

  4

  Answer    (a) 4

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  Explanation

  Question No.    336
  Topic    Matrices

  Given

  A =
  |0 0 1|
  |0 1 0|
  |1 0 0|

  Observe

  A² = I

  Therefore

  A⁻¹ = A

  Hence

  A⁻¹ =
  |0 0 1|
  |0 1 0|
  |1 0 0|

  Answer    (b)

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  Explanation

  Question No.    337
  Topic    Matrices

  Given

  A =
  |3 2|
  |1 4|

  det(A)

  = 3×4 - 2×1

  = 10

  Property

  A(adj A) = |A| I

  Therefore

  A(adj A)

  = 10 I

  =
  |10 0|
  |0 10|

  Answer    (b)

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  Explanation

  Question No.    338
  Topic    Matrices

  Statement 1

  AB symmetric?

  (AB)ᵀ = BᵀAᵀ

  = BA

  Generally BA ≠ AB

  Hence Statement 1 is false.

  Statement 2

  (A²+B²)ᵀ

  =(A²)ᵀ+(B²)ᵀ

  =A²+B²

  Hence symmetric.

  Statement 2 is true.

  Answer    (b) 2 only

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  Explanation

  Question No.    339
  Topic    Application of Derivatives

  Given

  y=x³-x²-x+2

  dy/dx

  =3x²-2x-1

  At x=1

  dy/dx

  =3-2-1

  =0

  Therefore tangent is parallel
  to x-axis.

  Assertion = True

  Reason = True

  Reason correctly explains Assertion.

  Answer

  Both A and R are true and
  R is the correct explanation of A.

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  Explanation

  Question No.    340
  Topic    Integration

  Assertion

  ∫ (eˣ/x)(1+x logx) dx

  Let

  f(x)=logx

  Then

  f'(x)=1/x

  Using

  ∫ eˣ[f(x)+f'(x)]dx

  =eˣf(x)+C

  Therefore

  = eˣlogx + C

  Assertion = True

  Reason

  ∫eˣ[f(x)+f'(x)]dx

  =eˣf(x)+C

  This is a standard result
  from integration by parts.

  Reason = True

  Reason correctly explains Assertion.

  Answer

  Both A and R are true and
  R is the correct explanation of A.