Question 10

Let A = R – {3} and B = R – {1}. Consider the function *f* : A → B defined by

. Is f one-one and onto? Justify your answer.

Answer

A = **R** - {3}, B = **R** - {1}

*f*: A → B is defined as.

.

∴ *f* is one-one.

Let *y* ∈B = **R** - {1}. Then, *y* ≠ 1.

The function *f* is onto if there exists *x* ∈A such that *f*(*x*) = *y.*

Now,

Thus, for any *y* ∈ B, there existssuch that

Hence, function *f* is one-one and onto.

- Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by

(i) R = { (a,b) : |a - b| is a multiple of 4}

(ii) R = {(a,b):a = b} is an equivalence relation.

Find the set of all elements related to 1 in each case. - Q:-
Check the injectivity and surjectivity of the following functions:

(i)

*f*:**N → N**given by*f(x*) = x^{2}(ii)

*f*:**Z → Z**given by*f(x)*= x^{2}(iii)

*f*:**R → R**given by*f(x)*= x^{2}(iv)

*f*:**N → N**given by*f(x)*= x^{3}(v)

*f*:**Z → Z**given by*f(x)*= x^{3 } - Q:-
Prove that the Greatest Integer Function

*f*: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. - Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
- Q:-
Show that the Modulus Function

*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.

- Q:-
Let

*f*: {1, 3, 4} → {1, 2, 5} and*g*: {1, 2, 5} → {1, 3} be given by*f*= {(1, 2), (3, 5), (4, 1)} and*g*= {(1, 3), (2, 3), (5, 1)}. Write down gof. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- Check whether the relation R in R defined as R = {(a, b): a ≤ b
^{3}} is reflexive, symmetric or transitive. - Q:-
State with reason whether following functions have inverse

(i)

*f*: {1, 2, 3, 4} → {10} with*f*= {(1, 10), (2, 10), (3, 10), (4, 10)}(ii)

*g*: {5, 6, 7, 8} → {1, 2, 3, 4} with*g*= {(5, 4), (6, 3), (7, 4), (8, 2)}(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with

*h*= {(2, 7), (3, 9), (4, 11), (5, 13)} - Q:-
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

- Q:- If \begin{align} \frac{d}{dx} f(x) = 4x^3 - \frac{3}{x^4}\end{align} such that f(2) = 0 , then f(x) is

\begin{align} (A) x^4 + \frac {1}{x^3} - \frac{129}{8} \;\;\;\;(B) x^3 + \frac{1}{x^4} + \frac{129}{8}\end{align}

\begin{align} (c) x^3 + \frac {1}{x^4} + \frac{129}{8} \;\;\;\;(D) x^3 + \frac{1}{x^4} - \frac{129}{8}\end{align} - Q:-
Prove that the Greatest Integer Function

*f*: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. - Q:-
Let

*f*: X → Y be an invertible function. Show that*f*has unique inverse.(Hint: suppose g1 and g2 are two inverses of f. Then for all

*y ∈ Y, fog1(y) = 1Y(y) = fog2(y)*. Use one-one ness of*f*). - Q:- Find the value of if \begin{align} sin^{-1} x= y,\end{align}

then

\begin{align} (A) 0 ≤ y ≤ x \;\;\;\;\; (B) -\frac{\pi}{2} ≤ y ≤ \frac{\pi}{2} \end{align}

\begin{align} (C)\; 0 < y < \pi \;\;\;\;\; (D) -\frac{\pi}{2} < y < \frac{\pi}{2} \end{align} - Q:-
Answer the following as true or false.

\begin{align}(i) \overrightarrow{a}\; and\; \overrightarrow{-a}\; are\; collinear.\end{align}

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

- NCERT Chapter

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