Download App

Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Check the commutativity and associativity of the following binary operations:
'⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q.
✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}$ then,

a ⊙ b = a2 + b2

= b2 + a2

= b ⊙ a

Therefore,

a ⊙ b = b ⊙ a, $\forall\ \text{a, b}\in\text{Q}$

Thus, ⊙ is commuatative on Q.

Associativity: Let $\text{a, b, c}\in\text{Q.}$

a ⊙ (b ⊙ c) = a ⊙ (b2 + c2)

= ab2 + (b2 + c2)2

= ab2 + b4 + c4 + 2b2c2

(a ⊙ b) ⊙ c = (a2 + b2) ⊙ c

= (a2 + b2)2 + c2

= a4 + b4 + 2a2b2 + c2

Therefore,

$\text{a}\odot\text{b}\odot\text{c}\neq\text{a}\odot\text{b}\odot\text{c}$

Thus, ⊙ is not associative on Q.

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 "3 Marks" from Binary Operations→Binary Operations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
→
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate.
→
If $\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix},$ verify that ATA = I2.
→
Evaluate the following integrals:
$\int\frac{\text{e}^{2\text{x}}}{1+\text{e}^\text{x}}\text{ dx}$
→
Evaluate:
$\int\frac{\text{e}^{\log\sqrt{\text{x}}}}{\text{x}}\text{dx}$
→
ABCD is a quadrilateral. Find the sum of the vectors $\overrightarrow{\text{BA}},\overrightarrow{\text{BC}},\overrightarrow{\text{CD}}\text{ and }\overrightarrow{\text{DA}}$.
→
Show that the vectores $\vec{\text{a}}=3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-3\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=2\hat{\text{i}}+\hat{\text{j}}-4\hat{\text{k}}$from a right-angled triangle.
→
For each of the differential equations given in find the general solution:
$\frac{\text{dy}}{\text{dx}}+\sec\text{xy}=\tan\text{x}\Big(0\leq\text{x}<\frac{\pi}{2}\Big)$
→
Find the shortest distance between the lines $I_1$ and $l_2$ whose vector equation are
$\begin{array}{l}
\vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k}) \text { and } \\
\vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})
\end{array}$
→
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R.
→