Column I (A) In \(\mathbb{R}^{2}\), if the magnitude of the projection vector of the vector \(\alpha \hat{i}+\beta \hat{j}\) on \(\sqrt{3} \hat{i}+\hat{j}\) is \(\sqrt{3}\) and if \(\alpha=2+\sqrt{3} \beta\), then possible value \((\mathrm{s})\) of \(|\alpha|\) is (are) (B) Let \(a\) and \(b\) be real numbers such that the function $$ f(x)=\left\\{\begin{aligned} -3 a x^{2}-2, & x<1 \\ b x+a^{2}, & x \geq 1 \end{aligned}\right. $$ is differentiable for all \(x \in \mathbb{R}\). Then possible value(s) of \(a\) is (are) (C) Let \(\omega \neq 1\) be a complex cube root of unity. If \(\left(3-3 \omega+2 \omega^{2}\right)^{4 n+3}+\) \(\left(2+3 \omega-3 \omega^{2}\right)^{4 n+3}+\left(-3+2 \omega+3 \omega^{2}\right)^{4 n+3}=0\) then possible value(s) of \(n\) is (are) (D) Let the harmonic mean of two positive real numbers \(a\) and \(b\) be \(4 .\) If \(q\) is a positive real number such that \(a, 5, q, b\) is an arithmetic progression, then the value(s) of \(|q-a|\) is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 \((\mathrm{T}) \quad 5\)

One of the fundamental concepts in vector analysis is the vector projection. Vector projection can be thought of as the shadow of one vector onto another. Imagine you're shining a light at a vector and looking at the shadow it casts on another vector; this shadow corresponds to the vector projection. Formally, it is defined as a vector that is parallel to a given vector but has a length that is the dot product of the two vectors divided by the magnitude squared of the vector onto which you're projecting.

Mathematically, if we have two vectors \( \vec{u} \) and \( \vec{v} \) the projection of \( \vec{u} \) onto \( \vec{v} \) is given by \( \text{proj}_{\vec{v}} \vec{u} = \frac{\vec{u} \cdot \vec{v}}{||\vec{v}||^2} \vec{v} \) where \( \vec{u} \cdot \vec{v} \) is the dot product of \( \vec{u} \) and \( \vec{v} \) and \( ||\vec{v}|| \) is the magnitude of \( \vec{v} \). In simpler terms, you're scaling the vector \( \vec{v} \) to match the 'length' of the component of \( \vec{u} \) that lies along \( \vec{v} \). This concept is not just theoretical; it is widely used in physics, computer graphics, and engineering to analyze forces, project images onto screens, and more.

Differentiability is a cornerstone of calculus, referring to the ability of a function to have a derivative at every point in its domain. A function \( f(x) \) is said to be differentiable at a point \( x_0 \) if the derivative \( f'(x_0) \) exists, meaning that there's a specific slope to the tangent at that point on the graph of \( f(x) \).

For a piecewise function, like the one in the exercise, differentiability becomes a bit more complex. It requires that both pieces of the function not only are smooth on their own but also connect smoothly at the boundary point. This involves ensuring the limit of \( f'(x) \) as \( x \) approaches the boundary point is the same from both sides. In practical terms, a differentiable function is smooth and continuous; it has no sharp points or breaks, an essential property for modelling real-world phenomena where abrupt changes are rare.

The complex cube roots of unity are solutions to the equation \( z^3 = 1 \), where \( z \) is a complex number. These roots hold particular importance in complex number theory because they represent the concept of 'unity' extended into two dimensions. Besides the obvious solution \( z = 1 \), there are two additional complex roots which can be represented as \( \omega \) and \( \omega^2 \) where \( \omega \) is generally used to denote the primary complex cube root. It's interesting to note that these roots are evenly spaced in the complex plane and form the vertices of an equilateral triangle centered at the origin.

One intriguing property of these cube roots of unity is that the sum \( 1 + \omega + \omega^2 \) equals zero. This property often simplifies complex expressions, such as the one in the exercise, where the exponents of \( \omega \) can be managed through this identity. This simplification is essential for breaking down more complex equations into solvable parts in areas like signal processing, quantum physics, and number theory.

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is formed by adding a constant to the previous term. This constant is known as the common difference, denoted by \( d \). For instance, in the sequence \( 2, 5, 8, 11, ... \), the common difference is \( 3 \) because each term increases by that amount.

Formulas associated with AP are highly useful for solving various types of problems. The nth term of an AP is given by \( a_n = a_1 + (n-1)d \) where \( a_1 \) is the first term. The sum of the first \( n \) terms can be calculated with \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \). Understanding AP is not just for academic exercises; it’s crucial in financial calculations like mortgage payments, engineering problems, and in everyday situations such as planning a schedule. The harmonic mean, mentioned in the question, is a type of average commonly used in situations where rates are involved, such as speeds or densities. In an AP, the harmonic mean between two terms is related to the other terms in the progression, providing a linkage that unwraps the structure of the sequence.