Advertisements
Advertisements
प्रश्न
If the binary operation * on Z is defined by a * b = a2 − b2 + ab + 4, then value of (2 * 3) * 4 is ____________ .
विकल्प
233
33
55
-55
Advertisements
उत्तर
33
Given:- a * b = a2 − b2 + ab + 4
\[2 * 3 = 2^2 - 3^2 + 2 \times 3 + 4\]
\[ = 4 - 9 + 6 + 4\]
\[ = 5\]
\[\left( 2 * 3 \right) * 4 = 5 * 4\]
\[ = 5^2 - 4^2 + 5 \times 4 + 4\]
\[ = 25 - 16 + 20 + 4\]
\[ = 33\]
APPEARS IN
संबंधित प्रश्न
Let * be a binary operation, on the set of all non-zero real numbers, given by `a** b = (ab)/5` for all a,b∈ R-{0} that 2*(x*5)=10
Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer.
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = ∀ a a * N.
Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b= a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.
If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5) ∘ (10), where * and ∘ are binary operations.
Determine whether the following operation define a binary operation on the given set or not : 'O' on Z defined by a O b = ab for all a, b ∈ Z.
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N
Determine whether the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= ab + ba for all a, b ∈ N
The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.
Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N
Check the commutativity and associativity of '*' on N.
Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
If the binary operation o is defined by aob = a + b − ab on the set Q − {−1} of all rational numbers other than 1, shown that o is commutative on Q − [1].
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 I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].
Show that 'o' is both commutative and associate ?
On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Show that '*' is both commutative and associative on A ?
Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?
Define an associative binary operation on a set.
Write the identity element for the binary operation * defined on the set R of all real numbers by the rule
\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?
Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.
Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
The law a + b = b + a is called _________________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
For the multiplication of matrices as a binary operation on the set of all matrices of the form \[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\] a, b ∈ R the inverse of \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\] is ___________________ .
The number of binary operation that can be defined on a set of 2 elements is _________ .
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * on Q.
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 |
Choose the correct alternative:
In the set Q define a ⨀ b = a + b + ab. For what value of y, 3 ⨀ (y ⨀ 5) = 7?
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = `"ab"/4` for a, b ∈ Q.
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.
If * is a binary operation on the set of integers I defined by a * b = 3a + 4b - 2, then find the value of 4 * 5.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
