MATH

• 1

  

  A.

  a +

  B.

  a − ɑ

  C.

  s’^ab

  D.

  1

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• 2

  If l(x) is the least integer not less than x and g(x) is the greatest integer not greater than x, then

  A.

  9

  B.

  13

  C.

  1

  D.

  None of these

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• 3

  If 0 < a < 1, then the solution set of the inequation

  A.

  (1, 1/a)

  B.

  (0, a)

  C.

  (1, 1/a) ∪ (0, a)

  D.

  None of these

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• 4

  If the sum of the greatest integer less than or equal to x and the least integer greater than or equal to x is 5, then the solution set for x is

  A.

  (2, 3)

  B.

  (0, 5)

  C.

  [5, 6)

  D.

  None of these

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  We have, [𝑥] + (𝑥) = 5
  If 𝑥 ≤ 2, then [𝑥] + (𝑥) ≤ 2 + 2 < 5
  If 𝑥 ≥ 3, then [𝑥] + (𝑥) ≥ 3 + 3 > 5
  If 2 < 𝑥 < 3, then [𝑥] + (𝑥) = 2 + 3 = 5
  Hence, the solution set is (2, 3)
• 5

  A stick of length 20 units is to be divided into n parts so that the product of the lengths of the parts is greater than unity. The maximum possible value of n is

  A.

  18

  B.

  19

  C.

  20

  D.

  21

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• 6

  The number of ordered 4-tuples (x, y, z,w) where x, y, z,w ∈ [0,10] which satisfy the inequality
  2^{sin2 x} × 3>^{cos2 x} × 4^{sin2 z} + × 5>^{cos2 w} N ≥ 120, is

  A.

  81

  B.

  144

  C.

  0

  D.

  Infinite

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• 7

  

  A.

  Only positive values of s¥

  B.

  Only negative values of x

  C.

  All real numbers except zero

  D.

  Only for x > 1

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• 8

  Let x =

  A.

  s¥² ≤ 2

  B.

  x² < 2

  C.

  x² > 2

  D.

  x² ≥ 2

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• 9

  The equation

  A.

  No solution

  B.

  One solution

  C.

  Two solutions

  D.

  More than two solutions

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• 10

  The number of roots of the equation sin πx = |log|x||, is

  A.

  2

  B.

  4

  C.

  5

  D.

  6

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• 11

  The number of real solutions of 1 + |s’^x − 1| = s’^x(s’^x − 2), is

  A.

  1

  B.

  2

  C.

  3

  D.

  4

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• 12

  The number of positive integers satisfying the inequality n + 1c_n-2 − n + 1c_n-1 ≤ 50 is

  A.

  9

  B.

  8

  C.

  7

  D.

  6

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• 13

  The solution of the inequation log_1/3(x² + x + 1) + 1 < 0 is

  A.

  (−∞,−2) ∪ (1,∞)

  B.

  [−1, 2]

  C.

  (−2, 1)

  D.

  (−∞,∞)

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• 14

  If a, ɑ, s are sides of triangle, then

  A.

  [1, 2]

  B.

  [2, 3]

  C.

  [3, 4]

  D.

  [4, 5]

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• 15

  If 0 < x <

  A.

  √3

  B.

  

  C.

  

  D.

  1

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• 16

  Let y =

  A.

  −1 ≤ s¥ < 2 or s¥ ≥ 3

  B.

  −1 ≤ x < 3 or x > 2

  C.

  1 ≤ x < 2 or x ≥ 3

  D.

  None of these

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• 17

  If x, y, z are three real numbers such that x + y + z = 4 and x² + y² + z² = 6, then the exhaustive set of values of x, is

  A.

  [2/3, 2]

  B.

  [0, 2/3]

  C.

  [0, 2]

  D.

  [−1/3, 2/3]

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• 18

  Non- negative real numbers such that a₁ + a₂+. . .+a_n

  A.

  

  B.

  

  C.

  

  D.

  

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• 19

  Solution set of inequality log_e

  A.

  (2,∞)

  B.

  (−∞, 2)

  C.

  (−∞,∞)

  D.

  (3,∞)

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• 20

  (x − 1)(x² − 5x + 7) < (x − 1),then x belongs to

  A.

  (1,2) ∪ (3,∞)

  B.

  (2, 3)

  C.

  (−∞, 1) ∪ (2,3)

  D.

  None of these

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  (𝑥 − 1)(𝑥² − 5𝑥 + 7) < (𝑥 − 1)
  ⇒ (𝑥 − 1)(𝑥² − 5𝑥 + 6) < 0
  ⇒ (𝑥 − 1)(𝑥 − 2)(𝑥 − 3) < 0
  ∴ 𝑥 ∈ (−∞, 1) ∪ (2, 3)
• 21

  The value of ⁿP_r is equal to

  A.

  ^n-1P_r + r. ^n-1P_r-1

  B.

  n.^n-1 P_r + ^n-1P_r-1

  C.

  n( ^n-1P_r+ ^n-1P_r-1)

  D.

  ^n-1P_r-1 + ^n-1P_r

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• 22

  

  A.

  ^n+m+1C_n+1

  B.

  ^n+m+2C_n

  C.

  ^n+m+3C_n-1

  D.

  None of these

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• 23

  The number of straight lines can be formed out of 10 points of which 7 are collinear

  A.

  26

  B.

  21

  C.

  25

  D.

  None of these

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  If there were no three points collinear, we should
  have ¹⁰𝐶₂ lines; but since 7 points are collinear
  we must subtract ⁷𝐶₂ lines and add the one
  corresponding to the line of collinearity of the
  seven points. Thus, the required number of straight lines =
  ¹⁰𝐶₂ − ⁷𝐶₂ + 1 = 25
• 24

  A committee of 5 is to be formed from 9 ladies and 8 men. If the committee commands a lady majority, then the number of ways this can be done is

  A.

  2352

  B.

  1008

  C.

  3360

  D.

  3486

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  The required number is ⁹𝐶₅ + ⁹𝐶₄ × ⁸𝐶₁ +
  ⁹𝐶₃ × ⁸𝐶₂ = 3486
• 25

  Consider the following statements : 1.These are 12 points in a plane of which only 5 are collinear, then the number of straight lines obtained 3.Three letters can be posted in five letter boxes in 3⁵ways. Which of the statements given above is/are correct?

  A.

  Only (1)

  B.

  Only (2)

  C.

  Only(3)

  D.

  None of these

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