Let \(\Delta, \nabla \in\{\Lambda, V\}\) be such that \(( p \rightarrow q ) \Delta( p \nabla q )\) is a tautology. Then
Let \(\Delta, \nabla \in\{\Lambda, V\}\) be such that \(( p \rightarrow q ) \Delta( p \nabla q )\) is a tautology. Then
- $\Delta=\Lambda, \nabla=\Lambda$
- $\Delta=V, \nabla=V$
- $\Delta=V, \nabla=\wedge$
- $\Delta=\Lambda, \nabla=V$
The Correct Option is B
Approach Solution - 1
For the given expression to be a tautology, every possible valuation of \(p\) and \(q\) must make the expression true. Since \(p \to q\) is equivalent to \(\neg p \lor q\), the expression simplifies as:
\( (\neg p \lor q) \land (p \lor q) \)
Using distributive laws:
\( (\neg p \land p) \lor (\neg p \land q) \lor (p \land q) \lor (q \land q) \)
Simplifying further, knowing \(\neg p \land p\) is always false:
\( (\neg p \land q) \lor (p \land q) \lor q = q \)
Hence, for the expression to be a tautology, it must always evaluate to true, which is the case when \(\land\) and \(\lor\) are defined such that the final result of any expression involving these operators is always true.
Approach Solution -2
The correct answer is (B) : \(\Delta=V, \nabla=V\)
Given (p→q)Δ(p∇q)
Option I Δ=∧,∇=∨
Hence, it is tautology.
Option 4Δ=∧,∇=∧
Top Questions on Vectors
যদি \( \vec{a} = 4\hat{i} - \hat{j} + \hat{k} \) এবং \( \vec{b} = 2\hat{i} - 2\hat{j} + \hat{k} \) হয়, তবে \( \vec{a} + \vec{b} \) ভেক্টরের সমান্তরাল একটি একক ভেক্টর নির্ণয় কর।
যদি ভেক্টর \( \vec{\alpha} = a\hat{i} + a\hat{j} + c\hat{k}, \quad \vec{\beta} = \hat{i} + \hat{k}, \quad \vec{\gamma} = c\hat{i} + c\hat{j} + b\hat{k} \) একই সমতলে অবস্থিত (coplanar) হয়, তবে প্রমাণ কর যে \( c^2 = ab \)।
- If the sum of two vectors is a unit vector, then the magnitude of their difference is:
- Let \( \vec{a} \) be a position vector whose tip is the point (2, -3). If \( \overrightarrow{AB} = \vec{a} \), where coordinates of A are (–4, 5), then the coordinates of B are:
The respective values of \( |\vec{a}| \) and} \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:
Questions Asked in JEE Main exam
- Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to
- JEE Main - 2026
- Vector Algebra
A substance 'X' (1.5 g) dissolved in 150 g of a solvent 'Y' (molar mass = 300 g mol$^{-1}$) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent 'Y' is $____________ \(\times 10^{-2}\). (nearest integer)
[Given : $K_{b}$ of the solvent = 5.0 K kg mol$^{-1}$]
Assume the solution to be dilute and no association or dissociation of X takes place in solution.- JEE Main - 2026
- Solutions
- The wavelength of photon 'A' is 400 nm. The frequency of photon 'B' is \(10^{16}\,\text{s}^{-1}\). The wave number of photon 'C' is \(10^{5}\,\text{cm}^{-1}\). The correct order of energy of these photons is:
- JEE Main - 2026
- General Chemistry
- Given below are two statements: Statement I: The increasing order of boiling point of hydrogen halides is HCl<HBr<HI<HF. Statement II: The increasing order of melting point of hydrogen halides is HCl<HBr<HF<HI. In the light of the above statements, choose the correct answer from the options given below:
- JEE Main - 2026
- General Chemistry
Inductance of a coil with \(10^4\) turns is \(10\,\text{mH}\) and it is connected to a DC source of \(10\,\text{V}\) with internal resistance \(10\,\Omega\). The energy density in the inductor when the current reaches \( \left(\frac{1}{e}\right) \) of its maximum value is \[ \alpha \pi \times \frac{1}{e^2}\ \text{J m}^{-3}. \] The value of \( \alpha \) is _________.
\[ (\mu_0 = 4\pi \times 10^{-7}\ \text{TmA}^{-1}) \]- JEE Main - 2026
- Ray optics and optical instruments
Concepts Used:
Types of Vectors
In general, vectors are used in Maths and Science and are categorized into 10 different types of vectors such as:-
- Unit Vector - When a vector has a magnitude of 1 unit length, called a Unit Vector.
- Co-Initial Vector - Two or more vectors that have the same initial point are known to be Co-Initial Vectors.
- Coplanar Vector - Vectors that lie either in the same plane or are parallel to the same plane are called Coplanar vectors.
- Equal Vector - When two vectors have equal direction as well as magnitude, they are Equal Vectors, even if the initial point is different for both vectors.
- Negative of a Vector - When two vectors have the same magnitude but have exactly different directions.
- Zero Vector - When a vector has the same starting and ending point and has zero magnitudes is called a zero vector. The starting point needs to coincide with the terminal point. It is denoted by 0. It is also known as the null vector.
- Position Vector - A vector that indicates the location or the position of a point in a plane (three-dimensional Cartesian system) w.r.t. its origin. If A is a reference origin and there’s an arbitrary point B in the plane then AB will be called the position vector of the point.
- Like and Unlike Vectors - Like vectors are the vectors that have the same direction. And unlike vectors are the vectors that have opposite directions.
- Collinear Vector - Vectors either lying in the same line or which are parallel to the same line are Collinear vectors.
- Displacement Vector - The vector AB will be known as a displacement vector if a point is displaced from position B to position A.