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)=\(\frac { \left( { x }^{ 2 }-cosx \right) \left( 1+{ x }^{ 2 } \right) }{ \left( x-sinx \right) \left( 2x-{ x }^{ 3 } \right) } +{ e }^{ -\left| x \right| }\) 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.

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