Subjective Test |

*(i) All questions are compulsory.*

*(ii) Questions*

**1**to**7**are very short answer type questions. These questions carry one mark each.*(iii) Questions*

**8**to**10**are short answer type questions. These questions carry four marks each.*(iv) Question*

**11**is long answer type question. This question carries six marks.

Question 1 ( 1.0 marks)

Let

*f*:**R**→**R**be a function defined by*f*(*x*) = 2*x*+ 3. Find a function*g*:**R**→**R**satisfying the condition*gof*=*fog*=*I*_{R}.
Question 2 ( 1.0 marks)

A
function

Without finding the actual inverse of

*f*:**Z**→*A*, whereis defined as.Without finding the actual inverse of

*f*, show that*f*is bijective.
Question 3 ( 1.0 marks)

Let

Operations ‘*’, ‘’ and ‘Î”’ on

Which of the given operations is (are) binary operation(s)?

*X*= {1, 2, 3, 5, 6, 10, 15, 30}Operations ‘*’, ‘’ and ‘Î”’ on

*X*are defined as follows:*a***b*= LCM (*a, b*)*a**b*= HCF (*a, b*)*a*Î”*b*=Which of the given operations is (are) binary operation(s)?

Question 4 ( 1.0 marks)

If

*f:***N**→**N**is one-one and onto function, then find the value of the following expression.
Question 5 ( 1.0 marks)

Let

*f*be any non-zero real valued function and let*g*be a function given by*g*(*x*) = (*k*+ 2)*x*. Find the value of*k*such that*gof*= 2*f*.
Question 6 ( 1.0 marks)

If

*Y*= {*x: x*∈**N**and*x*≤ 4}, then show that the relation*R*on*Y*defined by*R*= {(*a − b*): |*a − b*| is a multiple of 3} is**not**a trivial relation.
Question 7 ( 1.0 marks)

A
function

Find

*f*:**R**→**R**is defined as*f*(*x*) = 3*x*−5.Find

*Î±*such that*f*(*Î±*) =*f*^{−1}(*Î±*).
Question 8 ( 4.0 marks)

A
function
is defined as

*f*(*x*) =*x*^{2}− 3*x*+ 2.- By
**not**calculating the actual inverse of*f*show that*f*is invertible.

- Find
the inverse of
*f*.

Question 9 ( 4.0 marks)

Let

Let the binary operation * on

(i) Find the identity element of

(ii) Show that set

Also, find the inverse of (−5, 3).

*A*=**R**− {0} and*B*=*A*×**R**.Let the binary operation * on

*B*be defined as (*a, b*) * (*c, d*) = .(i) Find the identity element of

*B*with respect to the operation *.(ii) Show that set

*B*is invertible with respect to the operation *.Also, find the inverse of (−5, 3).

Question 10 ( 4.0 marks)

Let

A relation

Show that

**Q**^{+}denote the set of all positive rational numbers.A relation

*R*on**Q**^{+}×**Q**^{+}is defined as .Show that

*R*is an equivalence relation.
Question 11 ( 6.0 marks)

Two
real valued functions

Show that the functions are one-to-one and onto, and thus, find their inverse.

*f*and*g*are defined as*f*(*x*) =*e*^{2}^{x}and*g*(*x*) =*x*− 3.Show that the functions are one-to-one and onto, and thus, find their inverse.

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