Advertisements
Advertisements
प्रश्न
The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .
विकल्प
20
40
400
445
Advertisements
उत्तर
445
Given: a * b = a2 + b2 + ab + 1
\[2 * 3 = 2^2 + 3^2 + 2 \times 3 + 1\]
\[ = 4 + 9 + 6 + 1\]
\[ = 20\]
\[\left( 2 * 3 \right) * 2 = 20 * 2\]
\[ = {20}^2 + 2^2 + 20 \times 2 + 1\]
\[ = 400 + 4 + 40 + 1\]
\[ = 445\]
APPEARS IN
संबंधित प्रश्न
Determine whether or not of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
On Z+, define * by a * b = ab
Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
On Z+, define * by a * b = |a − b|
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
For each binary operation * defined below, determine whether * is commutative or associative.
On R − {−1}, define `a*b = a/(b+1)`
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find
(i) 5 * 7, 20 * 16
(ii) Is * commutative?
(iii) Is * associative?
(iv) Find the identity of * in N
(v) Which elements of N are invertible for the operation *?
Let * be a binary operation on the set Q of rational numbers as follows:
(i) a * b = a − b
(ii) a * b = a2 + b2
(iii) a * b = a + ab
(iv) a * b = (a − b)2
(v) a * b = ab/4
(vi) a * b = ab2
Find which of the binary operations are commutative and which are associative.
Let A = N × 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.
Consider the binary operations*: R ×R → and o: R × R → R defined as a * b = |a - b| and ao b = a, &mnForE;a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;a, b, c ∈ R, a*(b o c) = (a* b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
Determine whether the following operation define a binary operation on the given set or not :
\[' +_6 ' \text{on S} = \left\{ 0, 1, 2, 3, 4, 5 \right\} \text{defined by}\]
\[a +_6 b = \begin{cases}a + b & ,\text{ if a} + b < 6 \\ a + b - 6 & , \text{if a} + b \geq 6\end{cases}\]
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On R, define * by a * b = a + 4b2
Here, Z+ denotes the set of all non-negative integers.
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b \[-\] ab, for all a, b \[\in\] S:
Prove that * is commutative as well as associative ?
Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?
Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :
Find the invertible elements in A ?
Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {1} is _______________ .
Determine whether * is a binary operation on the sets-given below.
a * b – a.|b| on R
Determine whether * is a binary operation on the sets-given below.
a * b = min (a, b) on A = {1, 2, 3, 4, 5}
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
Let * be defined on R by (a * b) = a + b + ab – 7. Is * binary on R? If so, find 3 * `((-7)/15)`
Fill in the following table so that the binary operation * on A = {a, b, c} is commutative.
| * | a | b | c |
| a | b | ||
| b | c | b | a |
| c | a | c |
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the existence of identity, existence of inverse properties for the operation * on M
Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.
Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.
A binary operation on a set has always the identity element.
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
Which of the following is not a binary operation on the indicated set?
