For any two real numbers a and b, we define a R b if and only if

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181.

For any two real numbers a and b, we define a R b if and only if sin2(a) + cos2(b) = 1. The relation R is

  • reflexive but not symmetric

  • symmetric but not transitive

  • transitive but not reflexive

  • an equivalence relation


D.

an equivalence relation

Let the given relation defined as

R = A, B | sin2a + cos2b = 1For reflexivesin2a + cos2b = 1              sin2θ + cos2θ = 1,  θ  R aRa  a, a  R R is reflexive.

For symmetric                     sin2a + cos2b = 1 1 - cos2a + 1 - sin2b = 1                sin2b + cos2a = 1                bRa

Hence, R is symmetric.

For transitive

Let aRb, bRc

  sin2a + cos2b = 1          ...(i)and sin2b + cos2c = 1         ...(ii)On adding Eqs. (i) and (ii), we getsin2a + sin2b + cos2b + cos2c = 2                       sin2a + cos2c + 1 = 2                               sin2a + cos2c = 1

Hence, R is transitive also.

Therefore, relation R is an equivalence relation.


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182.

The total number of injections (one-one into mappings) from {a1, a2, a3, a4} to {b1, b2, b3, b4, b5, b6, b7} is

  • 400

  • 420

  • 800

  • 840


 Multiple Choice QuestionsShort Answer Type

183.

Let IR be the set of real numbers and f : IR ➔ IR be such that for all x, y ∈ IR, . Prove that f is a constant function.


 Multiple Choice QuestionsMultiple Choice Questions

184.

The even function of the following is

  • fx = ax + a- xax - a- x

  • fx = ax + 1ax - 1

  • fx = x . ax - 1ax + 1

  • fx = log2x + x2 + 1


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185.

If f(x + 2y, x - 2y) = xy, then f(x, y) is equal to

  • 14xy

  • 14x2 - y2

  • 18x2 - y2

  • 12x2 + y2


186.

Let R be the set of real numbers and the mapping f : R R and g : R  R be defined by f(x) = 5 - x2 and g(x) =3λ - 4, then the value of (fog) (- 1) is

  • - 44

  • - 54

  • - 32

  • - 64


187.

A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} are two sets, and function f : Aa B is defined by f(x) = x + 2  A, then the function!

  • bijective

  • onto

  • one-one

  • many-one


188.

The function f(x) = seclogx + 1 + x2

  • odd

  • even

  • neither odd nor even

  • constant


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189.

The domain of the function f(x) = cos-11 - x2 is

  • (- 3, 3)

  • [- 3, 3]

  • - , - 3  3, 

  • (- , - 3]  [3, )


190.

A mapping from N to N is defined as follows f : N N f(n) = (n + 5)2, n  N

(N is the set of natural numbers). Then,

  • f is not one to one

  • f is onto

  • f is both one to one and onto

  • f is one to one but not onto


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