(D) Let $e$ be the identity element in $Q^{+}$ such that $a * e = a$ for all $a \in Q^{+}$.$\frac{a \times e}{2010} = a \implies e = 2010$.Thus,the id

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(D) Let $e$ be the identity element in $Q^{+}$ such that $a * e = a$ for all $a \in Q^{+}$.$\frac{a \times e}{2010} = a \implies e = 2010$.Thus,the identity element is $2010$.Let $x$ be the inverse of $2010$. By definition,$2010 * x = e$.$\frac{2010 \times x}{2010} = 2010$.$x = 2010$.Therefore,the inverse of $2010$ is $2010$.

Class 12 Mathematics Relation and Function Binary Operation

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The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $$ defined by $a b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

KCET 2010 All KCET

A
$2009$
B
$2011$
C
$1$
D
$2010$

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Relation and Function

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Binary Operation

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KCET 2010 Paper All KCET years →

Show that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.

Medium

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The set $\{-1, 0, 1\}$ is not a multiplicative group because of the failure of

EasyKCET 2008

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Let $$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

Medium

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Consider a binary operation $$ on the set $\{1,2,3,4,5\}$ given by the following multiplication table. Is $^$ commutative?
(Hint: use the following table)

$^*$ $1$ $2$ $3$ $4$ $5$

$1$ $1$ $1$ $1$ $1$ $1$

$2$ $1$ $2$ $2$ $2$ $2$

$3$ $1$ $2$ $3$ $3$ $3$

$4$ $1$ $2$ $3$ $4$ $4$

$5$ $1$ $2$ $3$ $4$ $5$

Medium

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Let $^$ be the binary operation on $N$ given by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

Medium

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$^*$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$1$	$1$	$1$	$1$
$2$	$1$	$2$	$2$	$2$	$2$
$3$	$1$	$2$	$3$	$3$	$3$
$4$	$1$	$2$	$3$	$4$	$4$
$5$	$1$	$2$	$3$	$4$	$5$

The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $*$ defined by $a * b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

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Show that $-a$ is the inverse of $a$ for the addition operation '$+$' on $R$ and $\frac{1}{a}$ is the inverse of $a \neq 0$ for the multiplication operation '$\times$' on $R$.

The set $\{-1, 0, 1\}$ is not a multiplicative group because of the failure of

Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.

Let $^*$ be the binary operation on $N$ given by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

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The inverse of $2010$ in the group $Q^{+}$ of all positive rational numbers under the binary operation $$ defined by $a b = \frac{ab}{2010}, \forall a, b \in Q^{+}$,is

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Let $^$ be the binary operation on $N$ given by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?