KCET · Maths · Binomial Theorem
Binary operation * on \( R-\{-1\} \) defined by
\( a^{*} b=\frac{a}{b+1} \) is
- A \( { }^{*} \) is associative and commutative
- B \( { }^{*} \) is associative but not commutative
- C \( { }^{*} \) is neither associative nor commutative
- D \( { }^{*} \) is commutative but not associative
Answer & Solution
Correct Answer
(C) \( { }^{*} \) is neither associative nor commutative
Step-by-step Solution
Detailed explanation
Given that
\[
\begin{array}{l}
a * b=\frac{a}{b+1} \\
\text { So, } b * a=\frac{b}{a+1}
\end{array}
\]
Since, \( a * b \neq b^{*} a \)
It is not commutative.
Now, \( \left(a^{*} b\right)^{*} c=\left(\frac{a}{b+1}\right)^{*} c \) \( =\frac{\left(\frac{a}{b+1}\right)}{c+1}=\frac{a}{(b+1)(c+1)} \) \( a *(b * c)=a^{*} \frac{b}{c+1} \) \( =\frac{a}{\left(\frac{b}{c+1}+1\right)}=\frac{a(c+1)}{(b+c+1)} \) Since, \( (a * b)^{*} C \neq a *\left(b^{*} c\right) \) It is not associative. Therefore, \( { }^{*} \) is neither commutative nor associative.
Therefore, \( { }^{*} \) is neither commutative nor associative.
\[
\begin{array}{l}
a * b=\frac{a}{b+1} \\
\text { So, } b * a=\frac{b}{a+1}
\end{array}
\]
Since, \( a * b \neq b^{*} a \)
It is not commutative.
Now, \( \left(a^{*} b\right)^{*} c=\left(\frac{a}{b+1}\right)^{*} c \) \( =\frac{\left(\frac{a}{b+1}\right)}{c+1}=\frac{a}{(b+1)(c+1)} \) \( a *(b * c)=a^{*} \frac{b}{c+1} \) \( =\frac{a}{\left(\frac{b}{c+1}+1\right)}=\frac{a(c+1)}{(b+c+1)} \) Since, \( (a * b)^{*} C \neq a *\left(b^{*} c\right) \) It is not associative. Therefore, \( { }^{*} \) is neither commutative nor associative.
Therefore, \( { }^{*} \) is neither commutative nor associative.
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Suppose \( f(x)=(x+1)^{2} \) for \( x \geq-1 \). If \( g(x) \) is a function whose graph is the reflection of the
graph of \( f(x) \) in the line \( y=x \), then \( g(x)= \)KCET 2014 Medium - The plane containing the point \((3,2,0)\) and the line \(\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}\) isKCET 2024 Easy
- If the vectors \(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) are coplanar, then the value of \(\lambda\) isKCET 2020 Easy
- \( \int \frac{1}{\sqrt{x}+x \sqrt{x}} d x= \)KCET 2019 Medium
- If \(P(A)=0.59, P(B)=0.30\) and \(P(A \cap B)=0.21\) then \(P\left(A^{\prime} \cap B^{\prime}\right)\) is equal toKCET 2021 Easy
- Area lying between the curves \( y^{2}=2 x \) and \( y=x \)KCET 2016 Easy
More PYQs from KCET
- If \(A\) and \(B\) are invertible matrices then which of the following is not correct ?KCET 2021 Easy
- The conductivity of semiconductor increases with increase in temperature becauseKCET 2019 Easy
- The surface temperature of the sun which has maximum energy emission at \(500 \mathrm{~nm}\) is \(6000 \mathrm{~K}\). The temperature of a star which has maximum energy emission at \(400 \mathrm{~nm}\) will beKCET 2007 Easy
- A tiny spherical oil drop carrying a net charge \(q\) is balanced in still air, with a vertical uniform electric field of strèngth \(\frac{81}{7} \pi \times 10^5 \mathrm{~V} / \mathrm{m}\). When the field is switched OFF, the drop is observed to fall with terminal velocity \(2 \times 10^{-3} \mathrm{~ms}^{-1}\). Here \(g=9.8 \mathrm{~m} / \mathrm{s}^2\), viscosity of air is \(1.8 \times 10^{-5} \mathrm{Ns} / \mathrm{m}^2\) and density of oil is \(900 \mathrm{~kg}\) \(\mathrm{m}^{-3}\). The magnitude of \(q\) isKCET 2022 Hard
- Which of the following statements are correct with respect to Frogs?
A. Bidder's canals are present in male Frogs
B. Copulatory pads are present in female Frogs
C. Sound producing vocal sacs are present in male Frogs
D. Cloaca is present in male Frog only
Choose the most appropriate answer from the options given below:KCET 2025 Medium - Read the following statements:
Statement I: All vertebrates develop a row of vestigial gill slits during embryonic stage.
Statement II: Embryos always pass through the adult stages of other animals.
Which of the following options is correct with reference to these statements?KCET 2024 Hard