Download App

Relations and Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions1 Mark
MCQ
Consider a binary operation $∗$ on $N$ defined as $a^ ∗ b=a^3+b^3$
  • A
    $∗$ is both associative and commutative.
  • ✓
    $∗$ is commutative but not associative.
  • C
    $∗$ is neither commutative nor associative.
  • D
    $∗$ is associative but not commutative.
✓

Answer

Correct option: B.
$∗$ is commutative but not associative.
Given that the binary operation $∗$ on $N$ is defined as $a^∗b=a^3+b^3$
Apply the given binary operation on $b^∗a.$
$b^ * a=b^3+a^3=a^3+b^3$
It shows that the value of $a^ * b$ is equal to that of $b^ * a$.
So, the operation is commutative.
Consider different values of the variable as $a =1, b=2$ and $c =3$.
Apply the given binary operation on $(a^ * b)^ * c$.
$(a^ * b)^ * c=(1^ * 2)^ * 3=\left(1^3+2^3\right)^ * 3=9^3+3^3=729+27=756$
Apply the given binary operation on $a^ *(b^ * c)$.
$(a^ * b)^ * c=1^ *(2^ * 3)=1^ *\left(2^3+3^3\right)=1^3+35^3=42876$
$(a^ * b)^ * c \neq a^ *(b^ * c)$
So the operation is not associative.
Therefore, the given operation is commutative but not associative.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Relations and Functions→Relations and Functions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec a = 2\hat i + \hat j + \hat k,\vec b = \hat i + 2\vec j + 2\vec k,\vec c = \vec i + \vec j + 2\hat k$ and $\left( {1 + \alpha } \right)\hat i + \beta \left( {1 + \alpha } \right)\hat j + \gamma \left( {1 + \alpha } \right)\left( {1 + \beta } \right)\hat k = \hat a \times \left( {\vec b \times \vec c} \right)$ ,  then $\alpha ,\beta ,\gamma $ are
→
The position vectors of two points $ A$ and $B$  are $i + j - k$ and $2i - j + k$ respectively. Then $|\overrightarrow {AB} |\,\, = $
→
The inverse of the matrix $\left[ {\begin{array}{*{20}{c}}5&{ - 2}\\3&1\end{array}} \right]$ is
→
The corner points of the bounded feasible region are $(0,0),(2,0),(4,2),(2,4)$ and $\left(0, \frac{10}{3}\right)$

Then for the objective function $z=-x+2 y$

$(i)$ Maximum value of $z$ has at $\ldots \ldots \ldots . . .$

$(ii)$ Minimum value of $z$ has at $\ldots \ldots \ldots . . .$

$(iii)$ The maximum value of $z$ is $\ldots \ldots \ldots . . .$

$(iv)$ The minimum value of $z$ is $\ldots \ldots \ldots . . .$

→
The area bounded by the curve $y=\cos x$, the line joining $(-\pi / 4, \cos (-\pi / 4))$ and $(0,2)$ and the line joining $(\pi / 4, \cos (\pi / 4))$ and $(0,2)$ is
→
Statement $- 1:$ The function $x^2 (e^x + e^{-x})$ is increasing for all $x > 0.$

Statement $-2:$ The functions $x^2e^x$ and $x^2e^{-x}$ are increasing for all $x > 0$ and the sum of two increasing functions in any interval $(a, b)$ is an increasing function in $(a, b).$

→
If the angle between $a $ and $ b $ be ${30^o}$, then the angle between  $3 a$  and -$ 4 b$  will be ............ $^o$
→
Let $A = \{1, 2, 3\}. $Then, the number of relations containing $(1, 2)$ and $(1, 3)$ which are reflexive and symmetric but not transitive is:
→
The adjacent sides of a rectangle with given perimeter as  $100\, cm $ and enclosing maximum area are
→
Let $\left| {\vec a} \right| = \left| {\vec b} \right| = \left| {\vec a - \vec b} \right| = \,1$ then angle between $\vec a$ and $\vec b$ is
→