Choose the correct answer out of the given four options. Let * be… - Vidyadip

Question

Choose the correct answer out of the given four options.

Let * be binary operation defined on R by a * b = 1 + ab ∀ a, b ∈ R. Then the operation * is:

Commutative but not associative.
Associative but not commutative.
Neither commutative nor associative.
Both commutative and associative.

✓

Answer

Commutative but not associative.

Solution:

We are given that, a * b = 1 + ab ∀ a, b ∈ R

Consider, a * b = ab + 1

= ba + 1

= b * a

Hence, * is a communicative binary operation.

Also, a * (b * c) = a * (bc + 1) $[\because$ b * c = bc + 1$]$

= a(bc + 1) + 1

= a + abc + 1

Now, (a * b) * c = (ab + 1) * c $[\because$ a * b = ab + 1$]$

= (1 + ab)c + 1

= c + abc +1

Now, $\text{a}+\text{abc}+1\neq\text{c}+\text{abc}+1$

$\Rightarrow\ \text{a}\ ^*\ (\text{b}\ ^* \ \text{c})\neq(\text{a}\ ^* \ \text{b})\ ^* \ \text{c}$

Therefore, * is not associative.

Hence, * is communicative 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 $a=i+j$ and $b=2i-k$ are two vectors, then the point of intersection of two lines $r \times a = b \times a$ and $r \times a = a \times b$ is

The direction ratios of the diagonal of the cube joining the origin to the opposite corner are (when the 3 concurrent edges of the cube are coordinate axes).

$\frac{2}{\sqrt{3}},\frac{2}{3},\frac{2}{3}$
1, 1, 1
2, −2, 1
1, 2, 3

The order of the differential equation
$2\text{x}^2\frac{\text{d}^2\text{y}}{\text{dx}^2}-3\frac{\text{dy}}{\text{dx}}+\text{y}=0 \ \text{is}$

2
1
0
not defined.

The area bounded by the curves x + 2y² = 0 and x + 3y² = 1 is:

$1\text{ sq.}\text{units}$
$\frac{1}{3}\text{ sq.}\text{units}$
$\frac{2}{3}\text{ sq.}\text{units}$
$\frac{4}{3}\text{ sq.}\text{units}$

The eqution of the plane which cute equal intercepts of unit length on the coordinate axes is:

x + y + z = 1
x + y + z = 0
x + y - z = 1
x + y + z = 2

For non-singular square matrix A, B and C of the same order (AB^-1 C) =

A^-1 BC^-1
C^-1 B^-1 A^-1
CBA^-1
C^-1 BA^-1

Let $O$ be the origin and $\overline{ OA }=2 \hat{ i }+2 \hat{ j }+\hat{ k }, \overline{ OB }=\hat{ i }-2 \hat{ j }+2 \hat{ k }$ and $\overline{ OC }=\frac{1}{2}(\overline{ OB }-\lambda \overline{ OA })$ for some $\lambda>0$. If $|\overline{ OB } \times \overline{ OC }|=\frac{9}{2}$, then which of the following statements is (are) TRUE?

$(A)$ Projection of $\overline{ OC }$ on $\overline{ OA }$ is $-\frac{3}{2}$

$(B)$ Area of the triangle $OAB$ is $\frac{9}{2}$

$(C)$ Area of the triangle $ABC$ is $\frac{9}{2}$

$(D)$ The acute angle between the diagonals of the parallelogram with adjacent sides $\overline{ OA }$ and $\overline{ OC }$ is $\frac{\pi}{3}$

If one point on the vector 2i − 4j − k is (2, 1, 3), the other point is?

(−4, 3, 2)
(4, −3, −2)
(3, 2, 1)
(4, −3, 2)

$f(x) = \left\{ \begin{array}{l}
2 - \left| {{x^2} + 5x + 6} \right|,\,\,\,x \ne - 2\\
{a^2} + 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = - 2
\end{array} \right.$ . Then the range of $a$ , so that $f(x)$ has maximum at $x = -2$, is

The distance between the planes 2x + 2y - z +2 = 0 and 4x + 4y - 2z + 5 = 0 is:

$\frac{1}{2}$
$\frac{1}{4}$
$\frac{1}{6}$
$\text{None of these}$