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Recent questions tagged ch1
Questions
If $f : R \rightarrow R$ is defined by $f(x)=x^2-3x+2$, write $f(f(x))$.
cbse
class12
ch1
q6
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let A={a,b,c,d} and the function f={(a,b),(b,d),(c,a),(d,c)},Write $f^{-1}$.
cbse
class12
ch1
q5
p11
short-answer
exemplar
sec-a
easy
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let f : $R \to R$ be the function defined by $f(x)=2x-3$ $\forall x \in R$.Write $f^{-1}.$
cbse
class12
ch1
q4
p11
short-answer
exemplar
sec-b
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $f,g:R \rightarrow $R be defined by $f(x)=2x+1$ and $g(x)=x^2-2,\forall x \in R,$respectively. Then, find $g\;of(x)$.
cbse
class12
ch1
q3
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $D$ be the domain of the real valued function $f$ defined by $f(x)=\sqrt{25-x^2}.$Then, write $D$
cbse
class12
ch1
q2
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $A={a,b,c}$ and the relation $R$ be defined on $A$ as follows: $R= {(a,a)(b,c)(a,b)}$. Write the minimum number of ordered pairs to be ordered pairs to be added in $R$ to make $R$ reflexive and transitive.
cbse
class12
ch1
q1
p11
short-answer
exemplar
sec-a
math
asked
Dec 18, 2012
by
sreemathi.v
1
answer
Let $f:\;N\;\to N\;be\;defined\;by\;f(n)= \left\{ \begin{array}{1 1} \frac{n+1}{2}, & if\;n\;is\;odd\\ \frac{n}{2}, & if\;is\;even\end{array} \right. \qquad for\;all\;n \in N$ \[\text{state whether the function f is bijective.Justify your answer.} \]
cbse
class12
bookproblem
ch1
sec2
q9
p11
medium
sec-b
math
asked
Dec 10, 2012
by
sreemathi.v
1
answer
Number of binary operations on the set $\{a, b\}$ are
cbse
class12
bookproblem
ch1
misc
q19
p31
sec-a
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let \(f : R \to R\) be the Signum Function defined as \[ f(x) = \left \{ \begin {array} {1 1} 1, & \quad \text { x $>$ 0} \\ 0, & \quad \text { x $=$0} \\-1, & \quad \text { x $<$0} \\ \end {array} \right. \] and \(g:R \to R\) be the greatest Integer Function given by \(g(x)=[x]\) where \([x]\) is a greatest integer less thar or equal to \(x\) Then, does \(fog\) and \(gof\) coincide in \((0,1]\)?.
cbse
class12
bookproblem
ch1
misc
q18
p31
sec-b
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let $A = \{1, 2, 3\}$. Then number of equivalence relations containing $(1, 2)$ is
cbse
class12
bookproblem
ch1
misc
q17
p30
sec-a
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let $ A = \{1, 2, 3\}$. Then number of relations containing $(1, 2)\;$ and $\;(1, 3)$ which are reflexive and symmetric but not transitive is
cbse
class12
bookproblem
ch1
misc
q16
p30
sec-a
math
asked
Nov 27, 2012
by
vaishali.a
1
answer
Let $A=\{\text{-1,0,1,2}\}$ and $B=\{\text{-4,-2,0,2}\} and $f,g: A $\rightarrow B$ be functions defined by $f(x)=x^2-x, \;x \in A$ and $g(x)=2 |x- \frac {1} {2} | -1,\; x \in A$. Are $f$ and $g$ equal?
cbse
class12
bookproblem
ch1
misc
q15
p30
sec-b
math
asked
Nov 26, 2012
by
vaishali.a
1
answer
Define a binary operation \(\ast\) on the set \(\{0, 1, 2, 3, 4, 5\}\) as \[ a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a$+$b $<$ 6} \\ a+b-6, & \quad \text{ if a+b $\geq$ 6} \\ \end{array} \right. \] Show that zero is the identity for this operation and each element $a\neq0$ of the set is invertible with $6-a$ being the inverse of $a$.
cbse
class12
bookproblem
ch1
misc
q14
p30
sec-b
medium
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Given a non-empty set \( X,\) let \(\ast :\; P(X)\; \times\; P(X) \to P(X) \) be defined as \(A \ast B = \; ( A-B)\; \cup \; (B-A),\; \forall A, B \in \; P(X).\). Show that the empty set \(\emptyset \) is the identity for the operation $\ast$ and all the elemnets \(A\) of \( P(X) \) are invertible with \( A^{-1} \;= A\).
cbse
class12
bookproblem
ch1
misc
q13
p30
medium
sec-b
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Consider the binary operation $\ast :\; R \times R \rightarrow R$ and $o :\; R \times R \rightarrow R$ defined as $a \ast b = | a \text{-b}|$ and \(\;a\;o\;b=a, \forall a,\;b \in R.\) Show that \(\ast\) is commutative but not associative, \(o\) is associative but not commutative. Further, show that \(\forall\; a,\; b,\; c \in R,\; a\; \ast\; (b\; o\; c) = (a \ast b) \;o\; (a \ast c)\). [If it is so, we say that the operation $\ast$ distributes over $o$]. Does $o$ distribute over? Justify your answer.
cbse
class12
bookproblem
ch1
misc
q12
p30
sec-b
medium
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Find the number of all onto functions from the set $\{1, 2, 3, ... , n\}$ to itself.
cbse
class12
bookproblem
ch1
misc
q10
p30
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Given a non-empty set \(X\), consider the binary operation \(\ast : P(X) × P(X) \to P(X)\) given by \(A \ast B=A \cap B\; \forall A, \) \( B \;in\; P(X),\) where \(P(X)\) is the power set of \(X\). Show that \(X\) is the identity element for this operation and \(X\) is the only invertible element in \(P(X)\) with respect to the operation \(\ast\).
cbse
class12
bookproblem
ch1
misc
q9
p30
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Let $S=\{a,b,c\}\;$ and$ \;T = \{1,2,3\}$. Find the inverse of the following function \(F\) from \(S\) to \(T\), if it exists - \[\;\; F=\{(a,3), (b,2), (c,1)\}\]
cbse
class12
bookproblem
ch1
misc
q11
q11-1
p30
easy
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Given a non empty set $X$, consider $P(X)$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows: For subsets $A,\; B$ in $ P(X),\; ARB$ if and only if $ A \subset B $ Is $R$ an equivalence relation on $P(X)$?
cbse
class12
bookproblem
ch1
misc
q8
p29
sec-a
math
asked
Nov 22, 2012
by
vaishali.a
1
answer
Give examples of two functions \(f: N \to N\) and \(g: N \to N\) such that \(g\;o\;f \) is onto but \(f\) is not onto.
cbse
class12
bookproblem
ch1
misc
q7
p29
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Give examples of two functions \(f : N \to Z \) and \(g: Z \to Z\) such that \(g\;o\;f\) is injective but \(g\) is not injective.
cbse
class12
bookproblem
ch1
misc
q6
p29
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Show that the function \(f : R \to R\) given by \(f (x) = x^3\) is injective.
cbse
class12
bookproblem
ch1
misc
q5
p29
sec-a
easy
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Show that the function f:R $\rightarrow \{ x \in$ R:-1$<$x$<$1 $\}$ defined by $f(x) = \frac {x} { 1+|\;x\;|}, x \in R$ is one-one and onto function.
cbse
class12
bookproblem
ch1
misc
q4
p29
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
If $(f:R \to R)$ is defined by $f(x) = x^2$ - $3x+2$. Find $f(f(x))$:
cbse
class12
bookproblem
ch1
misc
q3
p29
sec-a
easy
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let \(f:W \to W\) be defined as $f(n)=n$ - $1$, if \(n\;is\;odd\;and\; f(n)=n+1,\;if\;n\;is\; even.\) Show that \(f\) is invertible. Find the inverse of \(f\). Here, \(W\) is the set of all whole numbers.
cbse
class12
bookproblem
ch1
misc
q2
p29
medium
sec-b
math
modelpaper-2014
q15
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let \(f:R \to R\) be defined as \(f(x)=10x+7.\)Find the function \(g:R \to R\) such that \(g\;o\;f = f\;o\;g = I_R.\)
cbse
class12
bookproblem
ch1
misc
q1
p29
sec-b
easy
modelpaper
2012
q11
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Consider a binary operation $\ast$ on $N$ defined as $a \ast b = a^3 + b^3$. Choose the correct answer:
cbse
class12
bookproblem
ch1
sec4
q13
p26
easy
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let $(A=N \times N \,and\, * )$ be the binary operation on $(A)$ defined by $( (a, b) * (c, d) = (a + c, b + d))$. Show that * is commutative and associative. Find the identity element for * on $( A )$, if any.
cbse
class12
bookproblem
ch1
sec4
q11
p25
sec-b
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Find if the given operation has identity: $\;\; a \ast b = a^2 + b^2$
cbse
class12
bookproblem
ch1
sec4
p25
q10
q10-2
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
2
answers
Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows: $\;\; a \ast b = a-b$. Find which of the binary operations are commutative and which are associative.
cbse
class12
bookproblem
ch1
sec4
q9
q9-1
p25
easy
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let $\ast$ be the binary operation on $N$ defined by $a \ast b=H.C.F. $ of a and b. Is $\ast$ commutative? Is $\ast$ associative? Does there exist identity for this binary operation on $N$?
cbse
class12
bookproblem
ch1
sec4
q8
p25
easy
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Is $\ast$ defined on the set $\{1,2,3,4,5\}$ by $a\ast b=L.C.M$.of $a$ and $b$ a binary operation?
cbse
class12
bookproblem
ch1
sec4
q7
p25
sec-a
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Let \(\ast\) be the binary operation on \(N\) given by \(a\ast b=L.C.M\,.of\,a\,and\,b\). Find $\begin{array}{1 1}(i)\;\; 5 \ast7,\; 20 \ast 16 & (ii)\;\; Is\; \ast \;commutative?\\(iii)\;\; Is\; \ast \;associative? & (iv)\;\; Find\, the\,identity \, of \ast \, in N\\(v)\;\; Which \, elements\, of \, N\, are\, invertible\, for\, the\, operaation\,\ast ? & \;\end{array}$
cbse
class12
bookproblem
ch1
sec4
q6
p25
sec-b
math
asked
Nov 21, 2012
by
vaishali.a
2
answers
Let $\ast '$ be the binary operation on the set $\{1, 2, 3, 4, 5\}$ defined by $a \ast ' b = H.C.F$ of a and b. Is the operation same as the operation $\ast$ defined in the table below? $\begin{matrix} *&1&2&3&4&5 \\ 1&1&1&1&1&1 \\ 2&1&2&1&2&1 \\ 3&1&1&3&1&1 \\ 4&1&2&1&4&1 \\ 5&1&1&1&1&5 \end{matrix}$
cbse
class12
bookproblem
ch1
sec4
q5
p25
sec-a
easy
math
asked
Nov 21, 2012
by
vaishali.a
1
answer
Consider the binary operation \( \wedge\) on the set \(\{1, 2, 3, 4, 5\}\) defined by \(a \wedge b = min \{a, b\}\). Write the multiplication table of the operation \( \wedge\) .
cbse
class12
bookproblem
ch1
sec4
q3
p24
sec-a
easy
modelpaper
2012
q11
math
asked
Nov 20, 2012
by
vaishali.a
1
answer
For each operation $\ast$ defined below, determine whether $\ast$ is binary, commutative or associative. $\begin{array}{1 1}(i) \;\;\; On\, Z,\, define \,a*b\, = a-b & \;\\(ii) \;\;\; On\, Q,\, define \,a*b\, = ab+1 & \;\\(iii) \;\;\; On\, Q,\, define \,a*b\, = \frac {ab} {2} & \;\\(iv) \;\;\; On\, Z^+, \, define\, a*b= 2^{ab} & \;\\(v) \;\;\; On\, Z^+,\, define \,a*b\, = a^b & \;\\(vi) \;\;\; On R - \{ -1\},\, define\, a*b= \frac {a} {b+1} & \;\end{array}$
cbse
class12
bookproblem
ch1
sec4
q2
p24
sec-b
math
asked
Nov 20, 2012
by
vaishali.a
5
answers
Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this - On$ \; Z^+,\,$ defined $*\,$ by$\; a*b= a-b $
cbse
class12
bookproblem
ch1
sec4
q1
q1-1
p24
easy
sec-a
math
asked
Nov 19, 2012
by
vaishali.a
1
answer
Let \(f : R - \{ - \frac {4} {3} \} \to R \) be a function defined as \(f(x)= \frac {4x} {3x+4} \) .The inverse of \(f\) is the map \(g\): Range \(f \to R - \{ - \frac {4} {3} \} \) given by
cbse
class12
bookproblem
ch1
sec3
q14
p19
medium
sec-b
math
asked
Nov 19, 2012
by
vaishali.a
1
answer
If $f: R\to R$ be given by $f(x)=(3-x^3)^\frac {1}{3} $, then $fof(x)$ is
cbse
class12
bookproblem
ch1
sec3
q13
p19
sec-a
math
asked
Nov 19, 2012
by
vaishali.a
1
answer
Let \(f : X \to Y\) be an invertible function. Show that the inverse of $f^{-1}$ is $f$, i.e., $(f^{-1})^{-1} = f$.
cbse
class12
bookproblem
ch1
sec3
q12
p19
easy
sec-a
math
asked
Nov 16, 2012
by
vaishali.a
1
answer
Consider\(f:\{1,2,3\} \to\{a,b,c\}\) given by \(f(1)=a, \,f(2)=b\) and \(f(3)=c\). Find \(f^{-1} \) and show that \((f^{-1})^{-1} = f\).
cbse
class12
bookproblem
ch1
sec3
q11
p19
easy
sec-a
math
asked
Nov 15, 2012
by
vaishali.a
1
answer
Let \(f : X \to Y\) be an invertible function. Show that \(f\) has unique inverse.
cbse
class12
bookproblem
ch1
sec3
q10
p19
easy
sec-a
math
asked
Nov 15, 2012
by
vaishali.a
1
answer
Consider \(f:R_+ \to [\;\text{–5}, \infty )\)given by \(f(x)=9x^2 +6x\)-\(5\).Show that \(f\) is invertible with \( f^{-1} (y) = \bigg(\frac{(\sqrt{y+6}) -1} { 3}\bigg) \)
cbse
class12
bookproblem
ch1
sec3
q9
p19
medium
sec-b
math
asked
Nov 15, 2012
by
vaishali.a
1
answer
Consider \( f:R_+ \to [4,\infty)\)given by \(f(x)=x^2 +4\). Show that \(f\) is invertible with the inverse \(f^{-1}\)of \(f\) given by \(f^{-1}(y) = \sqrt {y-4}\),where $R_+$ is the set of all non-negative real numbers.
cbse
class12
bookproblem
ch1
sec3
q8
p18
medium
sec-b
math
asked
Nov 14, 2012
by
vaishali.a
1
answer
Consider $f:R \to R$ given by $f(x)=4x+3$. Show that $f$ is invertible. Find the inverse of $ f$
cbse
class12
bookproblem
ch1
sec3
q7
p18
medium
sec-b
math
asked
Nov 14, 2012
by
vaishali.a
1
answer
Show that $f : [-1,1]$ $ \rightarrow R$, given by $f(x) =\frac {x } { (x+2)}$ is one-one. Find the inverse of of the $f : [-1,1] \rightarrow $ Range $ f$.
cbse
class12
bookproblem
ch1
sec3
q6
p18
medium
sec-b
math
asked
Nov 14, 2012
by
vaishali.a
1
answer
State with reason whether following functions have inverse: (iii) \(h: \{2,3,4,5,\} \to \{7,9,11,13\}\) with \(h=\{(2,7),(3,9),(4,11),(5,13) \} \)
cbse
class12
bookproblem
ch1
sec3
q5
p18
q5-3
sec-a
easy
math
asked
Nov 9, 2012
by
vaishali.a
1
answer
If $f(x) = \frac { (4x+3) } { (6x-4) }, x \neq \frac {2} {3}$, show that $ f(x) =x $, for all $ x \neq \frac {2} {3}$. What is the inverse of $f$
cbse
class12
bookproblem
ch1
sec3
q4
p18
medium
sec-b
math
asked
Nov 9, 2012
by
vaishali.a
1
answer
Find \( gof\) and \(fog\), if (i) \( f(x) = |\;x\;| \, and \, g(x) = |\;5x-2\;| \)
cbse
class12
bookproblem
ch1
sec3
q3
q3-1
p18
easy
sec-a
math
asked
Nov 9, 2012
by
vaishali.a
1
answer
Let \(f,\, g\, and\, h\) be functions from \(R\, to\, R.\) Show that \[ (f+g) oh = foh + goh\]
cbse
class12
bookproblem
ch1
sec3
q2
p18
easy
sec-a
q2-1
math
asked
Nov 9, 2012
by
vaishali.a
1
answer
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