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
- If \( A = \begin{bmatrix} 2 & 1
3 & 4 \end{bmatrix} \), then the determinant of matrix \( A \) is:- MHT CET - 2025
- Mathematics
- Determinants
If \(\begin{vmatrix} 2x & 3 \\ x & -8 \\ \end{vmatrix} = 0\), then the value of \(x\) is:
- If \(A\) is an identity matrix of order \(n\), then \(A (\text{Adj } A)\) is a/an:
- The value of the determinant \[ \begin{vmatrix} \cos 75^\circ & \sin 75^\circ \\ \sin 15^\circ & \cos 15^\circ \end{vmatrix} \] is:
- The area of a triangle with vertices \((3, 0), (0, k), (-3, 0)\) is \(9\) sq units. The value of \(k\) is:
Questions Asked in JEE Advanced exam
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 ________.
- JEE Advanced - 2025
- binomial expansion formula
- 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 $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
- JEE Advanced - 2025
- Differential equations
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____
- JEE Advanced - 2025
- Thermodynamics
- The total number of real solutions of the equation $$ \theta = \tan^{-1}(2 \tan \theta) - \frac{1}{2} \sin^{-1} \left( \frac{6 \tan \theta}{9 + \tan^2 \theta} \right) $$ is
(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
- Inverse Trigonometric Functions
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.