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If the Binary Operation * on the Set Z of Integers is Defined by a * B = a + 3b2, Find the Value of 2 * 4. - Mathematics

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प्रश्न

If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.

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उत्तर

Given: a * b = a + 3b2

Here,
         2 * 4 = 2 + 3 (4)2
                  = 2 + 3 (16)
                  = 2 + 48
                  = 50

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पाठ 3: Binary Operations - Exercise 3.6 [पृष्ठ ३६]

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आरडी शर्मा Mathematics [English] Class 12
पाठ 3 Binary Operations
Exercise 3.6 | Q 18 | पृष्ठ ३६

संबंधित प्रश्‍न

Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.


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 R, define * by ab2


For each binary operation * defined below, determine whether * is commutative or associative.

On Z, define − b


For each binary operation * defined below, determine whether * is commutative or associative.

On − {−1}, define `a*b = a/(b+1)`


Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following table)

* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 1 2 1
3 1 1 3 1 1
4 1 2 1 4 1
5 1 1 1 1 5

Find the total number of binary operations on {ab}.


Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all ab ∈ Z ?


Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = (a − b)2 for all ab ∈ Q ?


Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all ab ∈ Z ?


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 ?


If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.


For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.


For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.


For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.


Mark the correct alternative in the following question:-

For the binary operation * on Z defined by a * b = a + b + 1, the identity element is ________________ .


If G is the set of all matrices of the form

\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .


Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all ab ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .


Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .


The binary operation * defined on N by a * b = a + b + ab for all a, b N is ________________ .


Subtraction of integers is ___________________ .


A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .


For the binary operation * defined on R − {1} by the rule a * b = a + b + ab for all a, b ∈ R − {1}, the inverse of a is ________________ .


The number of binary operation that can be defined on a set of 2 elements is _________ .


The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .


Let A = ℝ × ℝ and let * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ ℝ × ℝ.
(i) Show that * is commutative on A.
(ii) Show that * is associative on A.
(iii) Find the identity element of * in A.


Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?


Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.


Determine whether * is a binary operation on the sets-given below.

(a * b) = `"a"sqrt("b")` is binary on R


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.


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 A be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A


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 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 ______.


The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.


The binary operation * defined on set R, given by a * b `= "a+b"/2` for all a, b ∈ R is ____________.


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.


a * b = `((a + b))/2` ∀a, b ∈ N is


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