﻿ If a ε R and the equation - 3(x-[x]2 + 2(x-[x] +a2 = 0(where,[x] denotes the greatest integer ≤ x) has no  integral solution, then all possible value of  lie in the interval | Relations and Functions

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# Relations and Functions

#### Multiple Choice Questions

1.

If the function g(x) = is differntiable, then the value of k+m is

• 2

• 16/5

• 10/3

• 10/3

379 Views

# 2.If a ε R and the equation - 3(x-[x]2 + 2(x-[x] +a2 = 0(where,[x] denotes the greatest integer ≤ x) has no  integral solution, then all possible value of  lie in the interval (-1,0) ∪ (0,1) (1,2) (-2,-1) (-∞,-2) ∪ (2, ∞)

A.

(-1,0) ∪ (0,1)

Given a ε R and equation is
-3{x-[x]}2 + 2{x-[x] +a2 = 0

Let t = x - [x], then equation is
-3t2 +2t+ a2 = 0
⇒
∵ t = x - [x] = {X}
∴ 0≤ t≤1

Taking positive sign, we get

⇒

⇒ 1+3a2 <4
⇒ a2-1 <0
⇒(a+1)(a-1) <0

a ε (-1,1)
For no integral solution of a, we consider the interval (-1,0) ∪ (0,1)

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3.

Consider the following relations:
R = {(x, y)| x, y are real numbers and x = wy for some rational number w}; S = {(m/p, p/q)| m, n, p and q are integers such that n, q ≠ 0 and qm = pn}. Then

• R is an equivalence relation but S is not an equivalence relation

• neither R nor S is an equivalence relation

• S is an equivalence relation but R is not an equivalence relation

• S is an equivalence relation but R is not an equivalence relation

308 Views

4.

Let p(x) be a function defined on R such that  = 1, p'(x)  p'(1-x),for all x∈[0,1] p(0) = 1 and p(1) = 41. Then  equals

• √41

• 21

• 41

• 41

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5.

The function

• neither injective nor surjective.

• invertible

• injective but not surjective.

• injective but not surjective.

371 Views

6.

Let W denote the words in the English dictionary. Define the relation R by :
R = {(x, y) ∈ W × W | the words x and y have at least one letter in common}. Then R is

• not reflexive, symmetric and transitive

• reflexive, symmetric and not transitive

• reflexive, symmetric and transitive

• reflexive, symmetric and transitive

141 Views

7.

The graph of the function y = f(x) is symmetrical about the line x = 2, then

• f(x + 2)= f(x – 2)

• f(2 + x) = f(2 – x)

• f(x) = f(-x)

• f(x) = f(-x)

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8.

The domain of the function

• [2, 3]

• [2, 3)

• [1, 2]

• [1, 2]

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9.

is equal to

10.

If , then

• - 1 < x < 4

• 2 < x < 3

• 1 < x < 4

• 1 < x < 3