KCET · Maths · Basic of Mathematics
If \(\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}\), then the value of \(x^{b+c} \cdot y^{c+a} \cdot z^{a+b}\) is
- A 1
- B 2
- C 0
- D \(-1\)
Answer & Solution
Correct Answer
(A) 1
Step-by-step Solution
Detailed explanation
Given, \(\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}=k\) (say)
\(\ldots(i)\) \(\Rightarrow \quad x=e^{k(b-c)}, y=e^{k(c-a)}, z=e^{k(a-b)}\)
Now, \(\quad x^{b+c} \cdot y^{c+a} \cdot z^{a+b}\) \(=e^{k(b-c)(b+c)} \cdot e^{k(c+a)(c-a)} \cdot e^{k(a+b)(a-b)}\)
\[
\begin{aligned}
&=e^{k\left(b^{2}-c^{2}\right)} \cdot e^{k\left(c^{2}-a^{2}\right)} \cdot e^{k\left(a^{2}-b^{2}\right)} \\
&=e^{k\left(b^{2}-c^{2}+c^{2}-a^{2}+a^{2}-b^{2}\right)}=e^{k \cdot 0}=1
\end{aligned}
\]
\(\ldots(i)\) \(\Rightarrow \quad x=e^{k(b-c)}, y=e^{k(c-a)}, z=e^{k(a-b)}\)
Now, \(\quad x^{b+c} \cdot y^{c+a} \cdot z^{a+b}\) \(=e^{k(b-c)(b+c)} \cdot e^{k(c+a)(c-a)} \cdot e^{k(a+b)(a-b)}\)
\[
\begin{aligned}
&=e^{k\left(b^{2}-c^{2}\right)} \cdot e^{k\left(c^{2}-a^{2}\right)} \cdot e^{k\left(a^{2}-b^{2}\right)} \\
&=e^{k\left(b^{2}-c^{2}+c^{2}-a^{2}+a^{2}-b^{2}\right)}=e^{k \cdot 0}=1
\end{aligned}
\]
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
- \( \int_{0}^{1} \sqrt{\frac{1+x}{1-x}} d x= \)KCET 2019 Medium
- If \( f(x)=|\cos x-\sin x| \), then \( f^{\prime}\left(\frac{\Pi}{6}\right) \) is equal toKCET 2018 Medium
- If \( y=f\left(x^{2}+2\right) \) and \( f^{\prime}(3)=5 . \) then \( \frac{d y}{d x} \) at \( x=1 \) isKCET 2015 Medium
- If \(f(x) = \begin{cases} x^2 - 1 & \text{if } x \geq 2 \\ x + 1 & \text{if } x < 2 \end{cases}\), then \(\lim\limits_{x \to 1} f(x) + \lim\limits_{x \to 2} f(x) = \)KCET 2026 Medium
- \(\lim _{n \rightarrow \infty} \frac{3 \cdot 2^{n+1}-4 \cdot 5^{n+1}}{5 \cdot 2^{n}+7 \cdot 5^{n}}\) is equal toKCET 2009 Easy
- If \(A_n=\left[\begin{array}{cc}1-n & n \\ n & 1-n\end{array}\right]\), then \(\left|A_1\right|+\left|A_2\right|+\ldots\left|A_{2021}\right|=\)KCET 2022 Medium
More PYQs from KCET
- Which cleansing agent gets precipitated in hard water?KCET 2019 Hard
- An organic compound \(X\) on treatment with PCC in dichloromethane gives the compound \(Y\). Compound \(Y\) reacts with \(\mathrm{I}_{2}\) and alkali to form yellow precipitate of triiodomethane. The compound \(X\) isKCET 2021 Easy
- Suppose \( \vec{a}+\vec{b}+\vec{c}=0,|\vec{a}|=3,|\vec{b}|=5,|\vec{c}|=7 \), then the angle between \( \vec{a} \& \vec{b} \) isKCET 2016 Hard
- The monomer used in Novolac, a polymer usedin paintsKCET 2017 Medium
- Let \(f: R \rightarrow R\) be defined by \(f(x)=x^2+1\). Then, the pre images of 17 and -3 . respectively areKCET 2024 Easy
- Let \(A=\{x, y, z, u\}\) and \(B=\{a, b\}\). A function \(f: A \rightarrow B\) is selected randomly. The probability that the function is an onto function isKCET 2023 Easy