Let \(m\) be the minimum possible value of \(\log _{3}\left(3^{y_{1}}+3^{y_{2}}+3^{y_{3}}\right)\), where \(y_{1}, y_{2}, y_{3}\) are real numbers for which \(y_{1}+y_{2}+y_{3}=9\) .Let \(M\) be the maximum possible value of \(\left(\log _{3} x_{1}+\log _{3} x_{2}+\log _{3} x_{3}\right)\), where \(x_{1}, x_{2}, x_{3}\) are positive real numbers for which \(x_{1}+x_{2}+x_{3}=9\) .Then the value of \(\log _{2}\left(m^{3}\right)+\log _{3}\left(M^{2}\right)\) is_______
Correct Answer: 8
Solution and Explanation
\(\text{The Value of}\) \(\log_2(m^3)+\log_3(M^2)\) \(is\;\underline{8}.\)
Top Questions on Determinants
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- Let the system of equations be: $ 2x + 3y + 5z = 9, $ $ 7x + 3y - 2z = 8, $ $ 12x + 3y - (4 + \lambda)z = 16 - \mu, $ which has infinitely many solutions. Then the radius of the circle centered at $ (\lambda, \mu) $ and touching the line $ 4x = 3y $ is:
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Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______
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Questions Asked in JEE Advanced exam
- Consider the vectors $$ \vec{x} = \hat{i} + 2\hat{j} + 3\hat{k},\quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},\quad \vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}. $$ For two distinct positive real numbers $ \alpha $ and $ \beta $, define $$ \vec{X} = \alpha \vec{x} + \beta \vec{y} - \vec{z},\quad \vec{Y} = \alpha \vec{y} + \beta \vec{z} - \vec{x},\quad \vec{Z} = \alpha \vec{z} + \beta \vec{x} - \vec{y}. $$ If the vectors $ \vec{X}, \vec{Y}, \vec{Z} $ lie in a plane, then the value of $ \alpha + \beta - 3 $ is ________.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
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(Here, the inverse trigonometric functions $ \sin^{-1} x $ and $ \tan^{-1} x $ assume values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ and $(-\frac{\pi}{2}, \frac{\pi}{2})$, respectively.)- JEE Advanced - 2025
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A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is:
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- Let $$ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \cdots + \frac{1}{\sin 118^\circ \sin 119^\circ}. $$ Then the value of $$ \left( \frac{\csc 1^\circ}{\alpha} \right)^2 $$ is \rule{1cm}{0.15mm}.
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Concepts Used:
Determiners
Determiners are words that are used to introduce and specify nouns in a sentence. They provide information about the number, ownership, quantity, and definiteness of the noun. In just 200 words, let's explore the key aspects and usage of determiners.
Determiners include articles (a, an, the), demonstratives (this, that, these, those), possessives (my, your, his, her, its, our, their), quantifiers (some, any, few, several, many, much), numbers (one, two, three), interrogatives (which, what), and distributives (each, every, either, neither).
Articles:
- Definite Article: "The" is used to refer to a specific noun, such as "the book" or "the car."
- Indefinite Articles: "A" and "an" are used to refer to a non-specific noun, such as "a book" or "an apple."
Demonstratives:
- These determiners point to specific objects or people. For example, "this car" or "those books."
Possessives:
- These determiners indicate ownership or possession. For instance, "my house" or "their dog."
Quantifiers:
- Quantifiers provide information about the quantity or amount of the noun. For example, "some water" or "many books."
Numbers:
- Numbers are used to indicate the exact quantity of a noun. For instance, "three cats" or "ten students."
Interrogatives:
- These determiners introduce questions and seek information. For example, "which color" or "what time."
Distributives:
- Distributives refer to individual members of a group. For instance, "each student" or "every child."
Determiners play a crucial role in specifying and clarifying nouns in a sentence. They help convey information about the noun's definiteness, ownership, quantity, or question being asked. Correct usage of determiners is essential for clear and precise communication.
It is important to note that determiners are used at the beginning of noun phrases and are followed by the noun they modify. They agree with the noun in terms of number and gender.
Overall, determiners provide important contextual information about nouns and contribute to the overall meaning and structure of a sentence. Their accurate use enhances clarity and specificity in communication.