If \(f(x) = (x - 2)^2 (x - 3)^3\) and \(x ∈ [1, 4]\) and If \(M\) and \(m\) denotes maximum and minimum values respectively, then \(M - m\) is
Solution and Explanation
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Relations and functions Questions
- If the domain of the function \( f(x) = \log_e \left( \frac{2x + 3}{4x^2 + x - 3} \right) + \cos^{-1} \left( \frac{2x - 1}{x + 2} \right) \) is \( (\alpha, \beta] \), then the value of \( 5\beta - 4\alpha \) is equal to
- If \(f(x) =ln(\frac {1-x^2}{1+x^2})\) then value of \(225(f'(x) – f''(x))\) at \(x=\frac 12\)
- Let \( A = \{ 1, 2, 3, \dots, 20 \} \). Let \( R_1 \) and \( R_2 \) be two relations on \( A \) such that \(R_1 = \{(a, b) : b \text{ is divisible by } a\}\)
and \(R_2 = \{(a, b) : a \text{ is an integral multiple of } b\}\).Then, the number of elements in \( R_1 - R_2 \) is equal to \(\_\_\_\_.\) - Let \( A = \{1, 2, 3, 4\} \) and \( R = \{(1, 2), (2, 3), (1, 4)\} \) be a relation on \( A \).Let \( S \) be the equivalence relation on \( A \) such that \( R \subseteq S \) and the number of elements in \( S \) is \( n \). Then, the minimum value of \( n \) is _____
- Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be a function defined by \[ f(x) = \frac{4^x}{4^x + 2} \] and \[ M = \int_{f(a)}^{f(1 - a)} x \sin^4 \left( x (1 - x) \right) \, dx, \] \[ N = \int_{f(a)}^{f(1 - a)} \sin^4 \left( x (1 - x) \right) \, dx; \quad a \neq \frac{1}{2}. \] If \( \alpha M = \beta N \), \( \alpha, \beta \in \mathbb{N} \), then the least value of \( \alpha^2 + \beta^2 \) is equal to _____
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
