NCERT Solutions class-11 Maths Miscellaneous

Miscellaneous Exercise

1. The relation is defined by . The relation is defined by . Show that is a function and is not a function.

Ans. Given: and

At and

It is observed that takes unique value at each point in its domain [0, 10]. Therefore, is a function.

Now, and

At and

Therefore, does not have unique value at

Hence, is not a function.

---

2. If find

Ans. Given:

At

and

---

3. Find the domain of the function

Ans. Given:

is a rational function of

assumes real values of all except for those values of for which

Domain of function = R – {2, 6}

---

4. Find the domain and range of the real function defined by

Ans. Given: assumes real values if

Domain of

For

Range of = all real numbers 0 =

---

5. Find the domain and range of the real function defined by

Ans. Given:

The function is defined for all values of

Domain of = R

When ,

When ,

When ,

Range of = All real numbers 0 =

---

6. Let be a function from R into R. Determine the range of

Ans. Here

Putting

Now, will be real if

Range of

---

7. Let be defined respectively by Find and

Ans. Given: and

Now,

And

And

---

8. Let be a function from Z to Z defined by for some integers Determine

Ans. Given: and

Now

……….(i)

And

……….(ii)

Solving eq. (i) and (ii), we get and

---

9. Let R be a relation from N to N defined by R = Are the following true:

(i) R for all N

(ii) R implies R

(iii) R, R implies R

Ans. Given: R =

(i) No, (3, 3) R because

(ii) No, (9, 3) R but (3, 9) R

(iii) No, (81, 9) R but (81, 3) R

---

10. Let A = {1, 2, 3, 4}, B = {1, 5 9, 11, 15, 16} and = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true:

(i) is a relation from A to B.

(ii) is a function from A to B.

Justify your answer in each case.

Ans. (i) Here A = {1, 2, 3, 4} and B = {1, 5, 9, 11, 15, 16}

= {(1, 1), (1, 5), (1, 9), (1, 11), (1, 15), (1, 16), (2, 1), (2, 5), (2, 9), (2, 11),

(2, 15), (2, 16), (3, 1), (3, 5), (3, 9), (3, 11), (3, 15), (3, 16), (4, 1), (4, 5),

(4, 9), (4, 11), (4, 15), (4, 16)}

= {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

Now, (1, 5), (2, 9), (3, 1), (4, 5), (2, 11)

is a relation from A to B.

---

11. Let be a subset of defined by Is a function from Z to Z? Justify your answer.

Ans. We observed that = 4 and = 4

and

(4, 5) and (4, 4)

It shows that is not a function from Z to Z.

---

12. Let A = {9, 10, 11, 12 13} and let be defined by the highest prime factor of Find the range of

Ans. Here A = {9, 10, 11, 12, 13}

For

[and 3 is highest prime factor of 9]

For

[]

For

[]

For

[]

For

[]

Range of = {5, 11, 3, 13}

= {3, 5, 11, 13}