#### Sets, Relations and Functions - Important Question Paper

11th Standard

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Mathematics

Time : 01:00:00 Hrs
Total Marks : 50

Part A

15 x 1 = 15
1. If the function f:[-3,3]➝S defined by f(x)=x2 is onto, then S is

(a)

[-9,9]

(b)

R

(c)

[-3,3]

(d)

[0,9]

2. The function f:R➝R is defined by f(x)=$\frac { \left( { x }^{ 2 }+cosx \right) \left( 1+{ x }^{ 4 } \right) }{ \left( x-sinx \right) \left( 2x-{ x }^{ 3 } \right) } +{ e }^{ -\left| x \right| }$ is

(a)

an odd function

(b)

neither an odd function nor an even function

(c)

an even function

(d)

both odd function and even function.

3. If A={1,2,3}, B={1,4,6,9} and R is a relation from A to B defined by "x is greater than y". The range of R is

(a)

{1,4,6,9}

(b)

{4,6,9}

(c)

{1}

(d)

None of these

4. Which of the following is not an equivalence relation on z?

(a)

aRb ⇔ a+b is an even integer

(b)

aRb ⇔ a-b is an even integer

(c)

aRb ⇔ a<b

(d)

aRb ⇔ a=b

5. If A = {(x,y) : y = sin x, x ∈ R} and B = {(x,y) : y = cos x, x ∈ R} then A∩B contains

(a)

no element

(b)

infinitely many elements

(c)

only one element

(d)

cannot be determined

6. If f(x) = |x - 2| + |x + 2|, x ∈ R, then

(a)

$f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\4\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}$

(b)

$f(x)=\begin{cases}2x\ if\ x∈(-∞,-2] \\4x\ if \ x∈(-2,2]\\ - 2x\ if\ x∈(2,∞)\end{cases}$

(c)

$f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\-4x\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}$

(d)

$f(x)=\begin{cases}-2x\ if\ x∈(-∞,-2] \\2x\ if \ x∈(-2,2]\\ 2x\ if\ x∈(2,∞)\end{cases}$

7. Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

(a)

T is an equivalence relation but S is not an equivalence relation

(b)

Neither S nor T is an equivalence relation

(c)

Both S and T are equivalence relation

(d)

S is an equivalence relation but T is not an equivalence relation.

8. The number of students who take both the subjects Mathematics and Chemistry is 70. This represents 10% of the enrollment in Mathematics and 14% of the enrollment in Chemistry. The number of students take at least one of these two subjects, is

(a)

1120

(b)

1130

(c)

1100

(d)

insufficient data

9. If n((A x B) ∩(A x C)) = 8 and n(B ∩ C) = 2, then n(A) is

(a)

6

(b)

4

(c)

8

(d)

16

10. The range of the function ${1\over 1-2sinx}$ is

(a)

$(-∞,-1)\cup\left( {1\over 3},\infty\right)$

(b)

$\left( -1,{1\over 3}\right)$

(c)

$\left[ -1,{1\over 3}\right]$

(d)

$(-∞,-1]\cup [\frac { 1 }{ 3 } ,∞)$

11. The number of relations from a set containing 4 elements to a set containing 3 elements is:

(a)

216

(b)

25

(c)

27

(d)

212

12. Let X = {a, b,c},y = (1,2,3) then $f:x\rightarrow y$ given by (a, 1) (b, 1) (c, 1) is called:

(a)

onto

(b)

constant function

(c)

one one

(d)

bijective

13. Which of the following functions is an even function?

(a)

$f(x)={2^x+2^{-n}\over 2^x-2^{-x}}$

(b)

$f(x)={3^x+1\over 3^x-1}$

(c)

$f(x)={x.3^x-1\over 3^x+1}$

(d)

$f(x) = log (x +\sqrt{x^2 + 1})$

14. If $f(x)={1-x\over 1+x},(x\neq0)$ then f-1(x) =

(a)

f(x)

(b)

$1\over f(x)$

(c)

-f(x)

(d)

-$1\over f(x)$

15. If A = {1,2}, B = {1,3} then n(A x B) =

(a)

2

(b)

4

(c)

8

(d)

0

16. Part B

5 x 2 = 10
17. Graph the function f(x)=x3 and $g(x)\sqrt[3]x$ on the same co-ordinate plane. Find fog and graph it on the plane as well. Explain your results.

18. By taking suitable sets A, B, C, verify the following results:
(A$\times$ B)$\cap$(B$\times$A) = (A$\cap$B) $\times$ (B$\cap$A)

19. Show that the relation R on the set A={1,2,3} given by R={(1,1)(2,2)(3,3)(1,2)(2,3)} is reflexive but neither symmetric nor transitive.

20. If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find $n((A\cup B)\times(A\cap B)\times(A \triangle B))$

21. Part C

5 x 3 = 15
22. Discuss the following relations for reflexivity, symmetricity and transitivity:
Let P denote the set of all straight lines in a plane. The relation R defined by "lRm if l is perpendicular to m".

23. On the set of natural number let R be the relation defined by aRb if 2a + 3b = 30. Write down the relation by listing all the pairs. Check whether it is equivalence.

24. Let U = {-2, -1, 0, 1, 2, 3, ... 10} A = {-2, 2, 3, 4, 5} and B = {1, 3, 5, 8, 9}. Verify the De Morgans' law $(A\cup B)'=A'\cap B'$.

25. Prove that $((A\cup B'\cup C)\cap(A\cap B'\cap C'))\cup((A\cup B\cup C')\cap (B'\cap C'))=B'\cap C'.$

26. If $f:R-\{ -1,1\}\rightarrow R$ is defined by $f(x)={x \over x^2-1},$ verify whether f is one-to-one or not.

27. Part D

2 x 5 = 10
28. Show that the relation R defined on the set A of all polygons as R = {(P1 P2): P1 and P2 have same number of sides} is an equivalence relation.

29. The weight of the muscles of a man is a function of his body weight x and can be expressed as W(x)=0.35x. Determine the domain of this function.