Chapter Questions
$A, B$ and $C$ are three non-empty sets. If $A \subset B$ and $B \subset C$, then which of the following is true?(1) $\mathrm{B}-\mathrm{A}=\mathrm{C}-\mathrm{B}$(2) $A \cap B \cap C=B$(3) $\mathrm{A} \cup \mathrm{B}=\mathrm{B} \cap \mathrm{C}$(4) $A \cup B \cup C=A$
If $S$ is the set of squares and $R$ is the set of rectangles, then $(S \cup R)-(S \cap R)$ is(1) S.(2) $\mathrm{R}$.(3) set of squares but not rectangles.(4) set of rectangles but not squares.
If $\mathrm{A}=\{1,2,3,4,5,6\}$, then how many subsets of $\mathrm{A}$ contain the elements 2,3 and 5 ?(1) 4(2) 8(3) 16(4) 32
If $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}, \mathrm{B}=\{\mathrm{a}, \mathrm{c}, \mathrm{e}, \mathrm{g}\}$ and $\mathrm{C}=\{\mathrm{b}, \mathrm{d}, \mathrm{e}, \mathrm{g}\}$, then which of the following is true?(1) $\mathrm{C} \subset(\mathrm{A} \cup \mathrm{B})$(2) $\mathrm{C} \subset(\mathrm{A} \cap \mathrm{B})$(3) $\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cup \mathrm{C}$(4) Both (1) and (3)
If $\mathrm{A}_{1} \subset \mathrm{A}_{2} \subset \mathrm{A}_{3} \subset \ldots \subset \mathrm{A}_{50}$ and $\mathrm{n}\left(\mathrm{A}_{\mathrm{x}}\right)=\mathrm{x}-1$, then find $\mathrm{n}\left[\bigcap_{\mathrm{x}=11}^{50} \mathrm{~A}_{\mathrm{x}}\right]$.(1) 49(2) 50(3) 11(4) 10
A group of 30 people take either tea or coffee. If 12 people do not take tea and 15 people take coffee, then how many people take tea?(1) 18(2) 16(3) 15(4) 12
7. If $\mathrm{P}$ is the set of parallelograms, and $\mathrm{T}$ is the set of trapeziums, then $\mathrm{P} \cap \mathrm{T}$ is(1) P.(2) $\mathrm{T}$.(3) \phi.(4) None of these
If $\mathrm{X}, \mathrm{Y}$ and $\mathrm{Z}$ are three sets such that $\mathrm{X} \supset \mathrm{Y} \supset \mathrm{Z}$, then $(\mathrm{X} \cup \mathrm{Y} \cup \mathrm{Z})-(\mathrm{X} \cap \mathrm{Y} \cap \mathrm{Z})=$(1) $\mathrm{X}-\mathrm{Y}$(2) $\mathrm{Y}-\mathrm{Z}$(3) $\mathrm{X}-\mathrm{Z}$(4) None of these
If $\mathrm{n}\left(\mathrm{A}_{x}\right)=\mathrm{x}+1$ and $\mathrm{A}_{1} \subset \mathrm{A}_{2} \subset \mathrm{A}_{3} \subset \ldots \ldots \ldots \subset \mathrm{A}_{99}$, then $\mathrm{n}\left[\bigcup_{\mathrm{x}=1}^{99} \mathrm{~A}_{\mathrm{x}}\right]=$(1) 99(2) 98(3) 100(4) 101
In a class every student can speak either English or Telugu. The number of students who can speak only English, the number of students who can speak only telugu and the number of students who can speak both English and Telugu are equal. Then which of the following can represent the number of students of the class?(1) 20(2) 25(3) 45(4) 50
If $(2 x-y, x+y)=(1,11)$, then the values of $x$ and $y$ respectively are(1) 6,5(2) 7,4(3) 4,7(4) 7,3
A relation between two persons is defined as follows:a $R$ b 'if a and $b$ born in different months', $R$ is(1) reflexive(2) symmetric(3) transitive(4) equivalence
If A is a non-empty set, then which of the following is false? p: There is atleast one reflexive relation on $A$ q: There is atleast one symmetric relation on $\mathrm{A}$(1) $\mathrm{p}$ alone(2) $\mathrm{q}$ alone(3) Both $\mathrm{p}$ and $\mathrm{q}$(4) Neither p nor q
In a set of teachers of a school, two teachers are said to be related if 'they teach the same subject', then the relation is(1) reflexive and symmetric(2) symmetric and transitive(3) reflexive and transitive(4) equivalence
If $A=\{x, y, z\}$, then the relation $R=\{(x, x),(y, y),(z, z),(z, x),(z, y)\}$ is(1) symmetric(2) anti symmetric(3) transitive(4) both (2) and (3)
In the set of animals, a relation $R$ is defined in each question.a $\mathrm{R} \mathrm{b}$ if ${ }^{\prime \prime} \mathrm{a}$ and $\mathrm{b}$ are in different zoological parks," then $\mathrm{R}$ is(1) only reflexive(2) only symmetric(3) only transitive(4) equivalence
In the set of animals, a relation $\mathrm{R}$ is defined in each question.On the set of human beings a relation $\mathrm{R}$ is defined as follows:$a \mathrm{R} \mathrm{b}$ if $^{\prime \prime} \mathrm{a}$ and $\mathrm{b}$ have the same brother", then $\mathrm{R}$ is(1) only reflexive(2) only symmetric(3) only transitive(4) equivalence
Consider the following statements:p: Every reflexive relation is a symmetric relation. q: Every anti-symmetric relation is reflexive.Which of the following is/are true?(1) $\mathrm{p}$ alone(2) $\mathrm{q}$ alone(3) Both $\mathrm{p}$ and $\mathrm{q}$(4) Neither $\mathrm{p}$ nor $\mathrm{q}$
In a set of ants in a locality, two ants are said to be related if they walk on a same straight line, then the relation is(1) reflexive and symmetric(2) symmetric and transitive(3) reflexive and transitive(4) equivalence
If $\mathrm{n}(\mathrm{A})=4$ and $\mathrm{n}(\mathrm{B})=4$, then find the number of subsets of $\mathrm{A} \times \mathrm{B}$.(1) 65636(2) 65536(3) 65532(4) None of these
A function $\mathrm{f}$ is constant from set $\mathrm{A}=\{1,2,3\}$ onto set $\mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$ such that $\mathrm{f}(1)=\mathrm{a}$, then the range of $\mathrm{f}$ is(1) $\{\mathrm{a}, \mathrm{c}\}$(2) $\{\mathrm{a}\}$(3) $\{\mathrm{a}, \mathrm{b}\}$(4) $\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$
Which of the following is an odd function?(1) $\mathrm{x}+\mathrm{x}^{3}$(2) $x^{3}-x^{2}-5$(3) $\mathrm{x}^{2}+\mathrm{x}^{4}$(4) $\frac{3 x^{2}}{x^{2}+1}$
Which of the following relations is a function?
Which of the following relation is not a function?
If $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is an onto function defined by $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-4$ and $\mathrm{A}=\{0,1,2,3\}$, then the co-domain of fis(1) $\{-4,0,2,5\}$.(2) $\{-1,2,5,6\}$.(3) $\{-4,-1,2,5\}$.(4) None of these
Which of the following graphs represents a function?
If $\mathrm{f}(\mathrm{x})=2 \mathrm{x}-3 \mathrm{x}^{2}-5$ and $\mathrm{g}(\mathrm{x})=\frac{\mathrm{f}(\mathrm{x})+\mathrm{f}(-\mathrm{x})}{2}$, then $\mathrm{g}(\mathrm{x})$ is(1) odd(2) even(3) even as well as odd(4) neither even nor odd
Domain of the function $\mathrm{f}(\mathrm{x})=\frac{5-\mathrm{x}}{|3-\mathrm{x}|}$ is(1) $\mathrm{x} \in \mathrm{R}$(2) $\mathrm{x} \in \mathrm{Z}$(3) $\mathrm{R}-\{3\}$(4) $\mathrm{R}-\{5\}$
\mathrm{A}=\{-1,0,1,2\}, \mathrm{B}=\{0,1,2\}$ and$\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}$, then $\mathrm{f}$ is(1) only one-one function(2) only onto function(3) bijective(4) not a function
If two sets $\mathrm{A}$ and $\mathrm{B}$ have $\mathrm{p}$ and $\mathrm{q}$ no. of elements respectively and $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is one-one, then the relation between $\mathrm{p}$ and $\mathrm{q}$ is(1) $\mathrm{p} \geq \mathrm{q}$(2) $\mathrm{p}>\mathrm{q}$(3) $\mathrm{p} \leq \mathrm{q}$(4) $\mathrm{p}=\mathrm{q}$
If, $X=\left\{\begin{array}{c}p: \text { where } p=\frac{(n+2)\left(2 n^{5}+3 n^{4}+4 n^{3}+5 n^{2}+6\right)}{n^{2}+2 n} \\ \text { and } n, p \in Z^{+}\end{array}\right\}$, then find the number of elementsin the set $\mathrm{X}$.(1) 2(2) 3(3) 4(4) 6
If $n(A \cap B)=10, n(B \cap C)=20$ and $n(A \cap C)=30$, then find the greatest possible value of $\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})$.(1) 15(2) 20(3) 10(4) 4
If $X, Y$ and $Z$ are any three non empty sets such that any two of them are disjoint, then $(X \cup Y \cup Z)$ $\cap(\mathrm{X} \cap \mathrm{Y} \cap \mathrm{Z})$ is(1) X(2) Y(3) Z(4) $\phi$
$\mathrm{A}$ and $\mathrm{B}$ are any two non empty sets and $\mathrm{A}$ is proper subset of $\mathrm{B}$. If $\mathrm{n}(\mathrm{A})=5$, then find the minimum possible value of $\mathrm{n}(\mathrm{A} \Delta \mathrm{B})$.(1) 1(2) 5(3) Cannot be determined(4) None of these
If $\mathrm{n}(\mathrm{A} \cap \mathrm{B})=5, \mathrm{n}(\mathrm{A} \cap \mathrm{C})=7$ and $\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=3$, then the minimum possible value ofn $(B \cap C)$ is(1) 0(2) 1(3) 3(4) 2
If a set contains $\mathrm{n}$ elements, then which of the following cannot be the number of reflexive relations in the set?(1) $2^{\text {n }}$(2) $2^{\mathrm{n}-1}$(3) $2^{n^{2}-1}$(4) $2^{\mathrm{n}+1}$
If $\mathrm{A}=\{4,6,10,12\}$ and $\mathrm{R}$ is a relation defined on $\mathrm{A}$ as "two elements are related if they have exactly one common factor other than $1 "$ ". Then the relation $\mathrm{R}$ is(1) anti symmetric.(2) only transitive.(3) only symmetric.(4) equivalence.
$\mathrm{X}$ is the set of all members in a colony and $\mathrm{R}$ is a relation defined on $\mathrm{X}$ as "two persons are related if they speak same language". The relation $\mathrm{R}$ is(1) only symmetric.(2) only reflexive.(3) both symmetric and reflexive but not transitive.(4) equivalence.
If $\mathrm{A}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{x}, \mathrm{y}, \mathrm{z}\}$, then the maximum number of elements in any relation on $\mathrm{A}$ is(1) 12(2) 16(3) 32(4) 36
The relation 'is a factor of' on the set of natural numbers is not(1) reflexive(2) symmetric(3) anti symmetric(4) transitive
If $f(x)=\log x$, then $\frac{f(x y)+f(x / y)}{f(x) f(y)}=$(1) $\frac{2}{\log x}$(2) $2 \log \mathrm{y}$(3) $2 \log \mathrm{x}$(4) $\frac{2}{\log y}$
If $f(x)=\frac{x-1}{x+1}, x \neq-1 ;$ then find $f\left(\frac{x-1}{x+1}\right)$.(1) $x$(2) $-\frac{1}{\mathrm{x}}$(3) $\mathrm{f}(\mathrm{x})$(4) $\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)$
If $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ defined by $\mathrm{f}(\mathrm{x})=3 \mathrm{x}-5$, then $\mathrm{f}^{-1}(\{-1,-2,1,2\})=$(1) $\left\{1, \frac{4}{3}, \frac{7}{3}\right\}$(2) $\left\{-1,2, \frac{-4}{3}\right\}$(3) $\left\{1,2, \frac{4}{3}, \frac{7}{3}\right\}$(4) $\{1,2,-1,-2\}$
If $\mathrm{f} \mathrm{R} \rightarrow \mathrm{R}$ is a function defined as $\mathrm{f}(\alpha-\mathrm{f}(\alpha))=5 \mathrm{f}(\alpha)$ and $\mathrm{f}(1)=7$, then find $\mathrm{f}(-6)$.(1) 37(2) 35(3) 7(4) 21
The domain of the function $f(x)=\frac{1}{x}+\frac{1}{\log (2-x)}$ is(1) $\mathrm{x}>2$(2) $x \in R-\{2\}$(3) $\mathrm{x}<2, \mathrm{x} \neq 0, \mathrm{x} \neq 1$(4) $\mathrm{x}<2, \mathrm{x} \neq 0$
All the students of a class like Horlicks, Maltova or Viva. Number of students who like only Horlicks and Moltova, only Maltova and Viva and only Horlicks and Viva are all equal to twice the number of students who like all the three foods. Number students who like only Horlicks, only Maltova and only Viva are all equal to thrice the number of students who like all the three foods. If four students like all the three, then find the number of students in the class.(1) 64(2) 48(3) 68(4) 52
If $f(x)=2 x+3$ and $g(x)=3 x-1$ then find $f^{-1} \operatorname{og}^{-1}$(1) $\frac{x+8}{6}$(2) $\frac{x-8}{6}$(3) $\frac{8-x}{6}$(4) $\frac{x-8}{2}$
The inverse of the function $\mathrm{f}(\mathrm{x})=\left(\mathrm{x}^{3}-1\right)^{4 / 2}-12$ is(1) $\left[1+(\mathrm{x}+12)^{3}\right]^{\mathrm{s}}$(2) $\left[1-(x+12)^{4}\right]^{3}$(3) $\left[(x+12)^{3}-1\right]^{3}$(4) $\left[1+(\mathrm{x}+12)^{4}\right]^{3}$
The relation, $R=\{(1,3),(3,5)\}$ is defined on the set with minimum number of elements of natural numbers. The minimum number of elements to be included in $\mathrm{R}$ so that $\mathrm{R}$ is equivalence is(1) 5(2) 6(3) 7(4) 8
If $\mathrm{f}(2 \mathrm{x}+3)=4 \mathrm{x}^{2}+12 \mathrm{x}+15$, then the value of $\mathrm{f}(3 \mathrm{x}+2)$ is(1) $9 \mathrm{x}^{2}-12 \mathrm{x}+36$(2) $9 \mathrm{x}^{2}+12 \mathrm{x}+10$(3) $9 \mathrm{x}^{2}-12 \mathrm{x}+24$(4) $9 \mathrm{x}^{2}-12 \mathrm{x}-5$
If a relation $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is defined by $\mathrm{f}(\mathrm{x})=\mathrm{x}+2$, where $\mathrm{A}=\{-1,0,1\}$ and $\mathrm{B}=\{1,2,3\}$,then $\mathrm{f}$ is(1) only one-one function.(2) only onto function.(3) bijective.(4) None of these
The domain of the function, $f(x)=\frac{|x|-2}{|x|-3}$ is(1) $\mathrm{R}$(2) $\mathrm{R}-\{2,3\}$(3) $\mathrm{R}-\{2,-2\}$(4) $\mathrm{R}-\{-3,3\}$
If $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{xy})$ and $\mathrm{f}(1)=5$, then find the value of $\sum_{\mathrm{k}=0}^{6} \mathrm{f}(\mathrm{k})$.(1) 25(2) 35(3) 36(4) 24
If $f(x)=(x+1)$ and $g(x)=(x-1)$, then find (fo g) ((g of) (2))(1) 1(2) 2(3) 3(4) 4
If $\mathrm{f}(\mathrm{x})=\mathrm{x}, \mathrm{g}(\mathrm{x})=\mathrm{x}^{2}$ and $\mathrm{h}(\mathrm{x})=\mathrm{x}^{3}$, then find $[(\mathrm{h} \circ \mathrm{g}) \circ \mathrm{f}](\mathrm{x})$(1) $\mathrm{x}$(2) $x^{2}$(3) $\mathrm{x}^{3}$(4) $\mathrm{x}^{6}$
If $\mathrm{f}=\{(2,4),(3,6),(4,8)\}$ and $\mathrm{g}=\{(4,3),(6,4),(8,2)\}$, then find fo g.(1) $\{(2,3),(3,4),(4,2)\}$(2) $\{(4,6),(6,8),(8,4)\}$(3) $\{(3,2),(4,3),(2,4)\}$(4) $\{(6,4),(8,6),(4,8)\}$
Find the domain of the function $f(x)=\frac{1}{\sqrt{2 x^{2}+5 x+2}}$(1) $\mathrm{R}$(2) $\left(-2, \frac{-1}{2}\right)$(3) $(-\infty,-2] \cup\left[\frac{-1}{2}, \infty\right)$(4) $(-\infty,-2) \cup\left(\frac{-1}{2}, \infty\right)$
Find the domain of function, $\sum_{p=1}^{10} \frac{1}{|2 x-p|}$.(1) $\mathrm{R}$(2) $\mathrm{R}-\left\{\frac{1}{2}, 1, \frac{3}{2}, 2 \ldots . .10\right\}$(3) $\mathrm{R}-\left\{\frac{1}{2}, 1, \frac{3}{2}, 2 \ldots . .5\right\}$(4) $\mathrm{R}-\{1,2 \ldots \ldots 10\}$
Find the inverse function of $f(x)=2 x-3$(1) $3 \mathrm{x}+2$(2) $3 x-2$(3) $(x+3) / 2$(4) $(\mathrm{x}-3) / 2$
If $\mathrm{f}(\mathrm{x})=\mathrm{x}+1$ and $\mathrm{g}(\mathrm{x})=\mathrm{x}-2$, then find $\left(\mathrm{f}^{-1} \mathrm{o} \mathrm{g}^{-1}\right)(\mathrm{x})$.(1) $\mathrm{x}-1$(2) $\mathrm{x}+2$(3) $\mathrm{g}(\mathrm{x})$(4) $\mathrm{f}(\mathrm{x})$
If $\mathrm{f}(\mathrm{x})+\mathrm{f}(1-\mathrm{x})=10$, then the value of $\mathrm{f}\left(\frac{1}{10}\right)+\mathrm{f}\left(\frac{2}{10}\right)+\ldots \ldots \ldots+\mathrm{f}\left(\frac{9}{10}\right)$(1) is 45(2) is 50(3) is 90(4) Cannot be determined
There are 60 students in a class. The number of students who passed in Mathematics is 45 and the number of students who passed in Physics is 40 . The number of students who failed in both the subjects is 5 . Find the number of students who passed in exactly one of the subjects.(1) 35(2) 25(3) 15(4) Cannot be determined
If $X=\{2,3,5,7,11\}$ and $Y=\{4,6,8,9,10\}$, then find the number of one-one functions from $\mathrm{X}$ to $\mathrm{Y}$(1) 720(2) 120(3) 24(4) 12
There are 40 students in a class. Each student speaks at least one of the languages Tamil, English and Hindi. Ten students speak exactly one of the languages. Twenty five students speak atmost two languages. How many students speak atleast two languages?(1) 15(2) 25(3) 30(4) 5
For any two sets $\mathrm{A}$ and $\mathrm{B}, \mathrm{n}(\mathrm{A})=15, \mathrm{n}(\mathrm{B})=12, \mathrm{~A} \cap \mathrm{B} \neq \phi$ and $\mathrm{B} \not \subset \mathrm{A}$Find the maximum possible value of $\mathrm{n}(\mathrm{A} \Delta \mathrm{B})$.(1) 27(2) 26(3) 24(4) 25
For any two sets $\mathrm{A}$ and $\mathrm{B}, \mathrm{n}(\mathrm{A})=15, \mathrm{n}(\mathrm{B})=12, \mathrm{~A} \cap \mathrm{B} \neq \phi$ and $\mathrm{B} \not \subset \mathrm{A}$Find the minimum possible value of $\mathrm{n}(\mathrm{A} \Delta \mathrm{B})$.(1) 3(2) 4(3) 5(4) 6
$\mathrm{A}$ and $\mathrm{B}$ are two finite sets. The difference of the number of elements of the power sets is 96 . $($ Assume $\mathrm{n}(\mathrm{A})>\mathrm{n}(\mathrm{B})$Find $\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})$.(1) 11(2) 12(3) 13(4) 14
$\mathrm{A}$ and $\mathrm{B}$ are two finite sets. The difference of the number of elements of the power sets is 96 . $($ Assume $\mathrm{n}(\mathrm{A})>\mathrm{n}(\mathrm{B})$ Find $\mathrm{n}(\mathrm{A})-\mathrm{n}(\mathrm{B})$.(1) 2(2) 3(3) 4(4) 5
The relation $\mathrm{R}$ is defined on a set $\mathrm{P}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$ and $\mathrm{R}$ is a reflexive relation.Which of the following is true about the number of elements of $\mathrm{R}$ ?(1) $1 \leq \mathrm{n}(\mathrm{R}) \leq 5$(2) $1 \leq n(R) \leq 2^{5}$(3) $5 \leq \mathrm{n}(\mathrm{R})<2^{5}$(4) $5 \leq \mathrm{n}(\mathrm{R}) \leq 25$
The relation $\mathrm{R}$ is defined on a set $\mathrm{P}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$ and $\mathrm{R}$ is a reflexive relation.How many reflexive relations are possible on $\mathrm{P}$ ?(1) $2^{5}$(2) $2^{25}$(3) $2^{20}$(4) $2^{18}$