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11th CBSE Mathematics Unit 2 Relations and Functions Important Question Paper | QB365

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11th CBSE Mathematics Unit 2 Relations and Functions Important Question Paper

Published on: 25/07/2019

Relations and Functions

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Questions:

Generic 2

17 x 2 = 34

Let R be a relation on the set Z of integers defined by R = {(x, y) : x, y \(\in\) Z, x² + y² = 25} Find R in roster form

Let R be a relation on set N of natural numbers defined by R = {(x, y) : x, y\(\in\)N, x + 3y = 12} Find Range of R

If A = {1, 2, 3}, B = {3, 4} and C = {1, 3, 5}, find A x (B \(\cap\) C)

If A = {1, 2, 3}, B = {3, 4} and C = {1, 3, 5}, find A x (B \(\cup\) C)

If A = {1, 2, 3}, B = {3, 4} and C = {1, 3, 5}, find (A x B) \(\cap\) (A x C)

Find the values of x and y if ( x³-x, y² - 5y + 6) = (0, 0)

Find the values of x and y if (x + 4, x + 2y) = (6, 8)

Let\(f:\left[ 2,\infty \right] \rightarrow R\) and \(g:\left[ -2,\infty \right] \rightarrow R\) be two real functions defined by \(f(x)=\sqrt { x-2 } \) and \(g(x)=\sqrt { x+2 } \). Find \(f+g\) and \(f-g\)

Find the domain and range of the relation R = {(x, y) : x + y = 8; x, \(y\in N\)}

10.

If U = {1,2,3,4} and R = {(x,y) : y > x for all, x, \(y\in U\)}, then find domain and range of R.

11.

Let \(f\) and \(g\) be real functions defines by \(f(x)=\sqrt { x+2 } \) and \(g\), then\(g(x)=\sqrt { 4-{ x }^{ 2 } } \) find the following function: \(\frac { f }{ g } \)

12.

Let \(f\) and \(g\) be real functions defines by \(f(x)=\sqrt { x+2 } \) and \(g\) ,then\(g(x)=\sqrt { 4-{ x }^{ 2 } } \) find the following function: f . g

13.

Let A = {a,b,c} and B={x: x\( \in \) N, x is a prime number less than 5}. Find A x B ans B x A.
Show that A x B \( \neq \) B x A.

14.

Find the cartesian product of three sets A = {1,2}, B = {3,4} and C = { x : x \(\in \) N and 4 \(\le \) x \(\le \)6}.

15.

Which of the following relations are functions?
\(\left\{ (2,0),(4,8),(2,1),(3,6) \right\} \)

16.

Which of the following relations are functions?
\( \left\{ (3,3),(4,2),(5,1),(6,0),(7,7) \right\} \)

17.

Find the values of a and b, if (a-3,b+7)=(3,7)

Generic 3

2 x 3 = 6

18.

Determine the domain and range of the relation R, where R = {(2x+3, x³)} : x is a prime number less than 10}.

19.

Let A and B be two sets, such that A x B consists of 6 elements. If 3 elements of A x B are (1, 4), (2, 6) and (3, 6), then find A x B and B x A.

Generic 5

2 x 5 = 10

20.

Determine the domain and range of the relation defined as: R = {(a, b) : a, b\(\in\)N, a < 5, b = 3a + 1}

21.

The cartesian product A x A has 9 elements among which are found (-1,0) and (0,1). Find the set A and the remaining elements of A x A.

Multiple Choice Question

5 x 1 = 5

22.

Let f(x) = Ix-1I then ______.

(a)

f(x²) = [f(x)]²

(b)

f(x +y) = f(x) f(y)

(c)

f(IxI) = I f(x) I

(d)

none of these

23.

Which one of the following is not a function

(a)

{(x, y) (:) x, y \(\in\) R, x² = y}

(b)

{(x, y) (:) x, y \(\in\) R, y² = x}

(c)

{(x, y) (:) x, y \(\in\) R, x = y³}

(d)

{(x, y) (:) x, y \(\in\) R, y = x³}

24.

If R is a relation from a finite set A having m elements to a finite set B having n elemen.ts, then the number of relations from A to B is ______.

(a)

2^mn

(b)

2^mn-1

(c)

2mn

(d)

mⁿ

25.

Let R be a relation from a set A to B, then ______.

(a)

R=A\(\cup\)B

(b)

R =A\(\cap\)B

(c)

R\(\subset\)A x B

(d)

R \(\subset\) B x A

26.

If the set A has m elements, B has n elements then the number of elements in A x B is ______.

(a)

m + n

(b)

m + n + 1

(c)

(d)

n²

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Answers Key:

Generic 2

17 x 2 = 34

R= {(0,5), (0, -5), (3, 4), (-3, 4), (3, -4), (-3, -4), (4, 3), (-4, 3) (4, -3), (-4, -3), (5, 0), (-5, O)}

Range of R = {3, 2, 1}

{(1, 3), (2, 3), (3, 3)}

{(1, 1), (1, 3), (1, 4), (1, 5), (2, 1), (2, 3) (2, 4), (2, 5), (3, 1), (3, 3,) (3, 4), (3, 5)}

{(1, 3) (2, 3), (3, 3)}

x = 0, -1, 1;Y = 2,3

x = 2, y = 3

Domain \(f=\left[ 2,\infty \right] ={ D }_{ 1 }\) [say]
Domain g= \(\left[ -2,\infty \right] ={ D }_{ 2 }\) [say];
Ans \(\left( \sqrt { x-2 } +\sqrt { x+2 } \right) \)and\(\left( \sqrt { x-2 } -\sqrt { x+2 } \right) \)

Domain (R) = {1, 2, 3, 4, 5, 6, 7}
Range (R) = {7, 6, 5, 4, 3, 2, 1}

10.

Domain( R ) = {1,2,3,4}, Range (R) = {2,3,4}

11.

\(f(x)=\sqrt { x+2 } ,g(x)=\sqrt { 4-{ x }^{ 2 } } \)
\(f\)(x) is defined for \(x+2\ge 0\Rightarrow \ge -2\)
Domain(\(f\)) = \([-2,\infty )\)
\(g(x)\)is defined for \(4-{ x }^{ 2 }\ge =0\Rightarrow { x }^{ 2 }-4\le 0\)
\(\Rightarrow (x-2)(x+2)\le 0\Rightarrow x\in \left[ -2,2 \right] ;Domain(g)=\left[ -2,2 \right] \)
Domain(\(f\)) \(\cap \) domain(\(g\)) = [-2,2]
\(1/\sqrt { 2-x } \)

12.

\(f(x)=\sqrt { x+2 } ,g(x)=\sqrt { 4-{ x }^{ 2 } } \)
\(f\)(x) is defined for \(x+2\ge 0\Rightarrow \ge -2\)
Domain(\(f\))=\([-2,\infty )\)
\(g(x)\)is defined for \(4-{ x }^{ 2 }\ge =0\Rightarrow { x }^{ 2 }-4\le 0\)
\(\Rightarrow (x-2)(x+2)\le 0\Rightarrow x\in \left[ -2,2 \right] ;Domain(g)=\left[ -2,2 \right] \)
Domain(\(f\)) \(\cap \) domain(\(g\)) = [-2,2]
\(\left( x+2 \right) \sqrt { 2-x } \)

13.

Given sets are A={a,b,c} and B={x: x \( \in \)N, x is a prime number leaa than 5}={2,3}
For element a of set A, All ordered pairs are (a,2), (a,3).
For element b of set A, All ordered pairs are (b,2), (b,3).
For element c of set A, All ordered pairs are (c,2), (c,3).
\( \therefore \) A X B = {(a,2)(a,3),(b,2)(b,3)(c,2)(c,3)}
For element 2 of set B, all ordered pairs are (2,a), (2,b),(2,c).
For element 3 of set B, all ordered pairs are (3,a), (3,b), (3,C).
\( \therefore \)B X A = {(2,a), (2,b), (2,c), (3,a), (3,b), (3,c)}
Since, (a,2) \( \neq \) (2,a)
A X B \( \neq \) B X A

14.

We have, A = {1,2},B = {3,4} and C = {x : x \(\in \) N and 4 \(\le \) x \(\le \) 6} = {4,5,6}
Cartesian product of A and B = A x B = {1,2} x {3 ,4}
= {(1,3),(1,4),(2,3),(2,4)}
Let D = A x B = {(1,3),(1,4),(2,3),(2,4)}
Now D x C = A x B x C = {(1,3),(1,4),(2,3),(2,4)} x {4,5,6}
= {(1,3,4),(1,3,5),(1,3,6),(1,4,4),(1,4,5),(1,4,6),(2,3,4),(2,3,5),(2,3,6),(2,4,4),(2,4,5),(2,4,6)}
It is the required cartesian product of three sets.

15.

\(\left\{ (2,0),(4,8),(2,1),(3,6) \right\} \)
It is not a function because first elements of (2,0) and (2,1) are same.

16.

\( \left\{ (3,3),(4,2),(5,1),(6,0),(7,7) \right\} \)
It is a function because the first element of each ordered pair is different.

17.

We know that, two ordered pairs are equal, if their corresponding elements are equal.
(a-3,b+7)=(3,7)\(\Rightarrow \) a-3=3 and b+7=7
[equating corresponding elements]
\(\Rightarrow \) a=3+3 and b=7-7
\(\Rightarrow \) a=6 and b=0

Generic 3

2 x 3 = 6

18.

Domain = {7, 9, 13, 17} Range = {8, 27, 125, 343}

19.

A x B = {(1, 4), (1, 6), (2, 4), (2, 6), (3, 4)(3, 6)}
B x A = {(4, 1),(4, 2),(4, 3),(6, 1),(6, 2),(6, 3)}

Generic 5

2 x 5 = 10

20.

Domain of R = {1, 2, 3, 4}
Range of R = {4, 7, 10, 13}

21.

Here (-1,0) \(\in\) A x A \(\Rightarrow\)-1, 0 \(\in\) A and (0, 1) \(\in\) A x A \(\Rightarrow\)0, 1\(\in\) A
\(\therefore -1,0,1\)\(\in\) A.
It is given that n(A x A) = 9 which implies that n(A) = 3
\(\therefore\)A = {-1,0,1}
\(\therefore\)A x A = {(-1, -1), (-1, 0), (-1, 1),(0, -1), (0, 0), (0, 1), (1, -1),(1, 0), (1, 1)}
So the remaining elements of A x A are
(-1, 1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0) and (1,1).