### Featured Posts

### Math Assignment Class XI Ch - 2 | Relations & Functions

- Get link
- Other Apps

**MATHEMATICS ASSIGNMENT**

**Chapter 2 Class -11**

**Relations and Functions**

**Extra questions of chapter 2 class 11 with answer and hints to the difficult questions. Important and useful math. assignment for the students of class 11**

**ASSIGNMENT ON RELATIONS & FUNCTIONS**

**LEVEL-1**

**Question 1**

**i) Find x and y if (x + 3, 5) = (6, 2x + y) **

**Ans (x = 3, y = - 1)**

**ii) If ordered pair (x, -1) and (5, y) belongs to the set {(a, b): b = 2a - 3}, find the value of x and y**

**iii) Find the value of a and b If **** **

**[Ans a = 2, b = 1]**

**(iv) If a ****∈**** {-1, 2, 3, 4, 5}, b ****∈**** {0, 3, 6}, write the set of all ordered pairs (a, b) such thata + b = 5Solution-1 + 6 = 5 **

**⇒**

**(-1, 6)**

**∈**

**(a, b)**

2 + 3 = 5

2 + 3 = 5

**⇒**

**(2, 3)**

**∈**

**(a, b)**

5 + 0 = 5

5 + 0 = 5

**⇒**

**(5, 0)**

**∈**

**(a, b)**

**⇒**

**R = {(-1, 6), (2, 3), (5, 0)}**

**Question 2**

**i) If A = {1, 3, 5, 6} and B = {2, 4}, find A ****×**** B and B ****×**** A**

**ii) If A = {1, 2, 3}, B = {3, 4} and C ={1, 3, 5}, find **

**(a) A × (B ∩ C) **

**(b) (A ****×**** B) ∩ (A ****×**** C)**

**iii) If A = {1, 3, 5} , B = {x, y} then represent A ****×**** B and B ****×**** A in arrow diagrams**

**Question 3**

**Find the domain and range of the relation R defined by R = {(x, x ^{3 }) : x is a prime number less than 10} **

**Ans [Domain R = {2, 3, 5, 7},**

Range R = {8, 27, 125, 343}]

Range R = {8, 27, 125, 343}]

**Question 4**

**(i) n(A) = 3, n(B) = 4, then find n(A ****×**** A ****×**** B) **

**Ans: 36 **

**(ii) If A = {1, 2, 4}, B = {2, 4, 5) and C = {2, 5}, write (A - C)**

**×**

**(B - C)**

**Ans : {(1, 4), (4, 4)} **

**(iii) If A = {1, 2} and B = {3, 4}. Find A**

**×**

**B and total number of subsets of A**

**×**

**B. Also find the total number of relations from A to B**

**[Ans: 2 ^{2 ✕ 2} = **2

^{4 }

**=16 ]**

**(iv) If A = {1, 2, 3, 5}, B = {4, 6, 9} and R be a relation from A to B defined by**

** R = {(x, y) : |x - y| is odd}. Write R in roster form.**

**Ans: R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)**

**Question 5**

**Express the following functions as set of ordered pair and determine their ranges**

**(i) f : A****➝**** R, f(x) = x ^{2 }+ 1, where A = {-1, 0, 2, 4} **

**Ans(i): Range = {1, 2, 5, 17} **** **

**(ii) g : A**

**➝**

**N, g(x) = 2x, where A = { x : x**

**∈**

**N, x ≤ 7}**

** ****Ans(ii) : Range = {2, 4, 6, 8, 10, 12, 14}**

**Question 6**

**Let f : R - {2} ****➝ ****R be defined by **

** **

** and g : R ****➝ R be ****defined by g(x) = x + 2. ****Find whether f = g or not.**

**Solution: **

Also g(x) = x + 2 **⇒**** f(x) = g(x)**

**But Domain of f(x) = R - {2} and the domain of g(x) = R and two functions **

**are equal only if their domains are equal.**

**Hence f(x) ≠ g(x)**

**Question 7**

**Find the domain for which the functions f(x) = 2x ^{2} - 1**

**and g(x) = 1 - 3x are equal**

**Solution: **

**Here f(x) = g(x)**

**2x ^{2} - 1 = 1 - 3x **

**⇒ 2x**

^{2}+ 3x – 2 = 0**⇒ x = -2, 1/2**

**⇒**** f(x) and g(x) are equal on the set {-2, 1 / 2 }**

**Question 8**

** If f, g, h are three functions defined from R to R as follows**

**(i) f(x) = ****x**^{2}** (ii) f(x) = sinx (iii) f(x) = ****x**^{2}** + 1**

**Find the range of each function**

**Solution**

**(i) For all values of x, f(x) takes only +ve value. Also if x = 0, then f(x) = 0 . So Range of f(x) = [0, ∞)**

**(ii) Since -1 ≤ sinx ≤ 1 for all values of x. So Range of f(x) = [-1,1]**

**(iii) Since ****x**^{2}** is ≥ 0 ****⇒**** x**^{2 }**+ 1 ≥ 1 ****⇒**** Range of f(x) = [1, ∞)**

**Question 9**

**If**then find f(f(x))

**Solution: **

**Question 10**

**Find the domain of the function f(x) defined by **

**Solution**

**f(x) is defined for all x satisfying**

**4 - x ≥ 0 and x ^{2 }- 1 > 0**

**x ≤ 4 and (x - 1)(x + 1) > 0For (x - 1)(x + 1) = 0, the critical points are x = 1 and x = -1(x - 1)(x + 1) > 0**

**x ****∈ ****(- ∞, -1) U (1, ∞), But x ≤ 4**

**Therefore : D _{f} = (- ∞, -1) U (1, 4]**

**Question 11**

**Find the domain and range of the functions **

**Solution**

**By using quadratic formula we get **

**Now x is defined (real) if 1 - 4y**^{2}** ≥ 0 and y ≠ 0**

**⇒**** ****4y**^{2} - 1 **≤ 0 ****⇒**** (2y + 1)(2y - 1) ****≤ 0**

**Critical points are y = -1/2 and 1/2**

**(2y + 1)(2y - 1) ****≤ 0 and y ≠ 0 ****⇒**** y ****∈**** [****-1/2, 1/2] - {0}**

**⇒****R _{f} = [**

**-1/2, 1/2] - {0}**

**(ii) Find the domain and range of the functions**

**Solution Hint: **

** **

**For range find x in terms of y we get**

**Range = (-∝, 0) ⋃ [3/2, ∝)**

**(iii) Find the domain and range of the functions**

**Solution (iii)**

**f(x) is defined for all real numbers except at x = 3**

**Therefore D _{f} = R - 3**

** y = f(x) = **

** But x = 3 ****∉**** D _{f} **

**⇒**

**y = 3 + 3 = 6**

**∉**

**R**

_{f}**⇒**

**R**

_{f}= R - {6}**Question 12**

**Find the domain and range of the following functions**

**i) f(x) = |x - 1| Ans : [D = R] R = [0, ∞)**

**ii) x**D

^{2}+ y^{2}= 25 ,_{f}= [-5, 5], R

_{f}= [0, 5]

**iii)**

Ans:

iv)

Ans: Domain = R - {2}, Range = R - {-1}

v)

Ans: Domain = [-4, 4], Range = [0, 4]

vi)

Ans: Domain = (-3, 3) Range = [1, ∞)

**Solution (vi)**

**f(x) is defined if **

⇒ **9y ^{2 }- x^{2}y^{2 }**= 9

⇒ x^{2}y^{2} = 9y^{2} - 9

x is defined if 9y^{2 }- 9 ≥ 0 and y ≠ 0

x is defined if y^{2 }- 1 ≥ 0 and y ≠ 0

** **x is defined if (y + 1)(y - 1) ≥ 0 and y ≠ 0

x is defined if y ≤ -1, y ≥ 1 and y ≠ 0

⇒ y ∈ (-∞, -1] ⋃ [1, ∞)

⇒ R_{f }= (-∞, -1] ⋃ [1, ∞)

**(vi) Find the domain of the function**

Ans: D_{f }= (-∞, -3] ⋃ (-1, 3]

**Question 13**

**Draw the graph of the following write its range**

**Question 14**

**If **** then evaluate f(-2) + f(2)**

**Ans: 4**

**Question 15**

**Find the domain of the following function**

Ans : D_{f }= R - {1, 4}

**Question 16:**

**Solve :**

Ans:

**Solution: Let |x| = y **

** Critical points are y = 5, y = 3**

**y = |x| = 5 and y = |x| = 3 ⇒ x = åœŸ 5, x = ****åœŸ 3**

**Question 17:**

**Question 18 (DAV Final Paper, 2023)**

**Question 19 (DAV Final Paper, 2023)**

X |
-3 |
-2 |
-1 |
0 |
1 |
2 |

y |
-1 |
-1 |
-1 |
1 |
3 |
5 |

**Question 20** : **(DAV SP 2023)**

Find the domain of

**Solution **

**Questions deleted from CBSE syllabus**

**Question 1**

**Find the inverse relation R ^{-1} in each of the following**

**(i) R = {(1,2), (1,3), (2, 3), (3, 2), (5, 6) **

**Ans {(2,1), (3,1), (3,2), (2,3), (6,5)}**

**(ii) R = {(x, y) : x, y ****∈**** N, x + 2y = 8 **

**Ans {(3, 2), (2, 4), (1,6)**

**SolutionIf x = 2, y = 3 **

**⇒**

**2 + 2**

**×**

**3 = 8**

**⇒**

**(2, 3)**

**∈**

**R**

**If x = 4, y = 2 ****⇒**** 4 + 2 ****×**** 2 = 8 ****⇒**** (4, 2) ****∈**** R**

**If x = 6, y = 1 ****⇒**** 6 + 2 ****×**** 1 = 8 ****⇒**** (6, 1) ****∈**** R**

**R = {(2, 3), (4, 2), (6, 1)} ****⇒**** {(3, 2), (2, 4), (1, 6)}**

**Question 2**

**Let f : R****➝****R be a function given by f(x) = x ^{2 }**

**+ 1. Find f**

^{-1}**{10, 37}**

**Solution**

**If f(x) = y then x = f ^{-1}(y)**

**f ^{-1}(10) = x **

**⇒**

**f(x) = 10**

**⇒**

**x**

^{2 }+ 1 = 10**⇒**** x ^{2 }= 9 **

**⇒**

**x =**

**åœŸ**

**3**

** f ^{-1}(37) = x **

**⇒**

**f(x) = 37**

**⇒**

**x**

^{2 }+ 1 = 37**⇒**** x ^{2 }= 36 **

**⇒**

**x =**

**åœŸ**

**6**

** f ^{-1}{10, 37} = {-3, 3, - 6, 6}**

**Question 3**

** Let A = {-2, -1, 0, 1, 2} and f : A ****➝ Z be a function defined by f(x) = x ^{2 }- 2x - 3. Find**

**(i) Range of f i.e. f(A) (ii) Pre - image of 6, - 3 and 5**

**Solution**

**(i) f(A) = {f(-2), f(-1), f(0), f(1), f(2) } = {5, 0, - 3, - 4, - 3} = {- 4,- 3, 0, 5}**

**(ii) Let Pre - image of 6 = x ****⇒**** f **^{-1}**(6) = x ****⇒**** f(x) = 6**

**⇒**** x ^{2}**

**- 2x - 3 = 6**

**⇒**

**x**

^{2 }**- 2x - 9= 0**

**There is no real value of x which satisfies this equation. **

**So Pre - image of 6 = Î¦**

**Let Pre - image of - 3 = x ****⇒**** f **^{-1}**(-3) = x ****⇒**** f(x) = - 3**

**⇒**** x ^{2}**

**- 2x - 3 = - 3**

**⇒**

**x**

^{2}**- 2x = 0**

**⇒**

**x = {0, 2}**

**Let Pre - image of 5 = x ****⇒**** f **^{-1}**(5) = x ****⇒**** f(x) = 5**

**⇒**** x ^{2}**

**-2x - 3 = 5**

**⇒**

**x**

^{2}**- 2x - 8 = 0**

**⇒**

**x = {-2, 4}**

**Question 4**

**(i) f(x) = 3x ^{4 }- 5x^{2 }+ 9, find f(x - 1) **

**Ans [3x ^{4 }- 12x^{3 }+ 13x^{2} - 2x + 7]**

**(ii) Write the domain of f(x) = x**

^{2 }+ 1 and draw its graph. Also find the value of x for which f(x) = f(x + 1)**(iii) If f(x) = x**

^{3 }- 3x + 4, then find the value of x such that f(x) = f(2x + 1)** [Ans x = -1, 2/3]**

**(iv) F(x) = 4x - x**

^{2}, x**∈**

**R, then find f(a + 1) – f(a - 1)**

**Thanks for your visit**

**Please comment below **

- Get link
- Other Apps

## Comments

## Post a Comment