From the chapter Sets, Relations and Functions, some of the important questions are covered in this question paper. The questions are covers from the book back and PTA question.

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Questions:

Multiple Choice Question

15 x 1 = 15

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

(a)

(b)

(c)

(d)

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)

-\(1\over f(x)\)

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)

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

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

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

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)

\((-∞,-1]\cup [\frac { 1 }{ 3 } ,∞)\)

If n((A \(\times\) B) ∩(A \(\times\) C)) = 8 and n(B ∩ C) = 2, then n(A) is

(a)

(b)

(c)

(d)

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

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.

10.

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

(a)

\(f(x)=\left\{\begin{array}{lll} -2 x & \text { if } & x \in(-\infty,-2] \\ 4 & \text { if } & x \in(-2,2] \\ 2 x & \text { if } & x \in(2, \infty) \end{array}\right.\)

(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)

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

11.

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

12.

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

(d)

aRb ⇔ a=b

13.

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

14.

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.

15.

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

(a)

[-9,9]

(b)

(c)

[-3,3]

(d)

[0,9]

Generic 5

3 x 5 = 15

16.

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.

17.

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.

18.

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

Generic 2

5 x 2 = 10

19.

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.

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.

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.

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.

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)

Generic 3

3 x 3 = 9

24.

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

25.

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.

26.

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 \((A\cup B)'=A'\cap B'\).

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Answers Key:

Multiple Choice Question

15 x 1 = 15

(b)
4

(a)
f(x)

(b)
constant function

(d)
2¹²

\(-1 \leq \sin x \leq 1 \)
\(-2 \leq 2 \sin x \leq 2 \)
\(2 \geq-2 \sin x \geq-2 \)
\(-2 \leq-2 \sin x \leq 2 \)
Adding 1,
\(1-2 \leq 1-2 \sin x \leq 1+2 \)
\(-1 \leq 1-2 \sin x \leq 3 \)
\(-1 \geq \frac{1}{1-2 \sin x} \geq \frac{1}{3} \)
\(\frac{1}{3} \leq \frac{1}{1-2 \sin x} \leq-1 \)
\(\therefore \text { Range is }(-\infty,-1] \cup\left[\frac{1}{3}, \infty\right)\)

\((A \times B) \cap(A \times C) =A \times(B \cap C) \)
\(n[(A \times B) \cap(A \times C)] =n(A) \times n(B \cap C) \)
\(\Rightarrow 8 =n(A) \times 2 \Rightarrow n(A)=4 \)

\(M \cap C=70 \text { which is } 10 \% \text { of } M \text { and } 14 \% \text { of } C\)
\(M =700, C=500\)
\(M \cup C=700+500-70=1130\)

\(\mathrm{T}: x-y \text { is an integer } \Rightarrow x \mathrm{R} y\)
\(\text { i) } x-x=0 \text { is an integer }\)
\(\therefore \text { T is reflexive. }\)
\(\text { ii) }(x-y) \text { is an integer } \Rightarrow y-x \text { is also an integer }\)
\(\therefore \mathrm{T} \text { is symmetry }\)
iii) If (x - y)is an integer and y- z is also an integer, by adding
x - z is also an integer.
\(\therefore \mathrm{T} \text { is transitive }\)
Thus, T is an equivalence relation.
\(\mathrm{S}: \mathrm{y}=x+1 \Rightarrow x \mathrm{~S} y\)
\(\text { i) } x=x+1 \Rightarrow x S x \text { is not true. }\)
\(\therefore S \text { is not reflexive. }\)
Hence T is an equivalence relation but S is not an equivalence relation

10.

\(\text { If } x \in(-\infty,-2), \text { Let } x=-3\)
\(\text { then } f(x)=|-5|+|1|=6=-2 x\)
\(\text { If } x \in(-2,-2), \text { Let } x=0 \text { then }\)
\(f(x)=|0-2|+|0+2|=4\)
\(\text { If } x \in(2, \infty), \text { Let } x=4\)
\(\text { then } f^{}(x)=|2|+|6|=8=2 x\)
\(\therefore \mathrm{f}(x)=\left\{\begin{array}{rl} -2 x & x \in(-\infty,-2] \\ 4 & x \in(-2,2] \\ 2 x & x \in(2, \infty) \end{array}\right.\)

11.

12.

13.

14.

\(f(x) =\frac{\left(x^{2}+\cos x\right)\left(1+x^{4}\right)}{(x-\sin x)\left(2 x-x^{3}\right)}+e^{-|x|} \)
\(f(-x) =\frac{\left[(-x)^{2}+\cos (-x)\right]\left[1+(-x)^{4}\right]}{[-x-\sin (-x)]\left[-2 x-\left(-x^{3}\right)\right]}+e^{+x \mid} \)
\(=\frac{\left(x^{2}+\cos x\right)\left(1+x^{4}\right)}{(x-\sin x)\left(2 x-x^{3}\right)}+e^{-|x|} \)
\(=f(x) \)
Hence f(x) is an even function

15.

f(0) = 0, f(-3) = 9 and f(3) = 9
.'. S is [0,9]

Generic 5

3 x 5 = 15

16.

Given relation is 2a + 3b = 30 for all a, b \(\in \) N.
2a + 3b = 30 \(\Rightarrow\) 2a =30 - 3b
\(\Rightarrow a=\frac { 30-3b }{ 2 } \)

a	12	9	6	3
b	2	4	6	8

\(\therefore\) The list of ordered pairs are (12, 2) (9, 4) (6, 6) (3, 8)
Equivalence : R is not an equivalence relation.

17.

The relation R on the set of all polygons is defined as R = {(P₁, P₂): P₁ and P₂ have same number of sides}
Reflexivity : Let P be any polygon in A. Then P and P have same number of sides.
⇒ (P, P) ∈ R
⇒ R is reflexive on A.
Symmetry : Let P₁ and P₂ be two polygons in A such that (P_1, P₂) ∈ R
(P₁, P₂) ∈ R ⇒ P₁ and P₂ have same number of sides.
⇒ P₂ and P₁ have same number of sides.
⇒ (P₂, P₁)∈ R
∴ R is symmetric on A
Transitivity : Let P₁, P₂, P₃ be three polygons in A such that (P₁, P₂) E Rand (P₂, P₃) ∈ R.
⇒ P₁ and P₂ have same number of sides and P2 and P3 have same number of sides.
⇒ P₁ and P₃ have same number of sides.
⇒ (P₁, P₃) ∈ R
∴ R is transitive.
Hence, R is an equivalence relation.

18.

Given functions are f(x) = x³ and g(x) = \(x^{ ^{ \frac { 1 }{ 3 } } }\)
Now, f o g(x) = f(g(x)
= \(f\left( { x }^{ \frac { 1 }{ 3 } } \right) \)
= \(\left( { x }^{ 3 } \right) ^{ \frac { 1 }{ 3 } }=x\)

Since f o g(x) = x is symmetric about the line y = x, g(x) is the inverse of f(x)
∴ g(x) = f^-1(x).
where f o g(x) = g o f(x) = x so that
(i) f o g is bijective.
(ii) both f(x) and g(x) also bijective.
(iii) f and g are symmetrical about y = x

Generic 2

5 x 2 = 10

19.

We start with the assumption f(x) = f(y). Then
\({x \over x^2-1}={y \over y^2-1}\)
\(\Rightarrow x(y^2-1)=(x^2-1)\)
\(\Rightarrow xy^2-x-yx^2+y-0\Rightarrow (y-x)(xy+1)=0\)
This implies that x =y or xy = - 1. So if we select two numbers x and y so that xy = -1, then f(x) = f(y). \(f(x)=f(y) \cdot\left(2,-\frac{1}{2}\right),\left(7,-\frac{1}{7}\right),\left(-2, \frac{1}{2}\right)\)are some among the infinitely many possible pairs. Thus \(f(2)=f\left(\frac{-1}{2}\right)=\frac{2}{3}\). That is, f(x) = f(y) does not imply x = y. Hence it is not one-to-one.

20.

We have \(n(A \cup B)=6,n(A\cap B)=2\) and \(n(A \triangle B)=4.\)
So, \(n((A\cup B)\times(A\cap B)\times(A \triangle B))=n(A \cup B)\times n(A\cap B)\times n(A\triangle B)= 6 \times 2 \times 4 = 48.\)

21.

Since A = {1, 2, 3}
(1,1) (2, 2) (3, 3) ∈ R ⇒ is reflexive.
Also (1, 2) ∈ R but (2, 1) ∉ R ⇒ R is not symmetric.
And (1, 2) ∈ R, (2, 3) ∈ R but (1, 3) ∉ R ⇒ R is not transitive.
R is reflexive but neither symmetric nor transitive.

22.

Let l, m, n ∈ p.
Reflexivity: We cannot say l is perpendicular to l itself.
∴ l R l \(\Rightarrow \) R is not reflexive.

Symmetry: lRm ≠ mRl
I is perpendicular to m ⇒ m is perpendicular to l
∴ R is symmetric
Transitive: lRm and mRn ≠ lRn.
l is perpendicular to m and m is perpendicular to n.
⇒ l is perpendicular to n.
∴ R is not transitive.
⇒ R is only symmetric.

23.

(A\(\times\) B) = {(1,4) (1,5) (1,6) (1,7) (2,4) (2,5) (2,6) (2,7) (3,4) (3,5) (3,6) (3,7)}
(B\(\times\)A) = {(4,1) (4,2) (4,3) (5,1) (5,2) (5,3) (6,1) (6,2) (6,3) (7,1) (7,2) (7,3)}
LHS = (A\(\times\)B)\(\cap \)(B\(\times\)A) = { }....(1)
(A\(\cap \)B) = { }, (B\(\cap \)A) = { }
\(\therefore\) RHS = (A\(\cap \)B) \(\times\) (B\(\cap \)A) = { }.....(2)
From (1) and (2), LHS = RHS

Generic 3

3 x 3 = 9

24.

We have \(A\cap B'\cap C' \subseteq A\subseteq A\cup B'\cup C\) and hence \((A\cup B'\cup C)\cap(A\cap B' \cap C')=A\cap B' \cap C'\)
Also, \(B'\cap C'\subseteq C'\subseteq A\cup B \cup C'\) and hence \((A \cup B \cup C')\cap B'\cap C'=B' \cap C'\)
Now as \(A \cap B'\cap C' \subseteq B'\cap C'\),
We have \(((A \cup B' \cup C)\cap(A \cap B'\cap C'))\cup((A \cup B\cup C')\cap (B'\cap C'))=B'\cap C'\)

25.

Given w(x) = 0.35x
Since x represents the number of men, it will take only positive integers.
\(\therefore\) W : W \(\rightarrow\) R⁺
Hence the domain is the set of whole numbers.

26.

Given A = {-2, 2,3,4, 5}, B = {1, 3, 5, 8, 9}
A U B = {-2, 1,2,3,4,5,8, 9}
Also U = {-2, -1,0, 1,2,3, ... 10}
∴ (A U B)' = U\ (A U B) = {-2, -1, 1,2,3, ... 10} {-2, 1,2,3,4,5,8, 9}
= {-1, 0, 6, 7, 10} ...(1)
A' = U\A = {-1, 0, 1,6, 7, 8,9, 10}
A' = U\B = {-2, -1,0,2,4,6, 7, 10}
∴ A' ⋂ B' = {-1, 0, 6, 7, 10} ...(2)
From (1) and (2), (A U B)' = A' ⋂ B'.