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JEE Advanced · Mathematics · 22. Functions

Let $f : R \to R$ be a differentiable function with $f (0) = 1$ and satisfying the equation $f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y)$ for all $x, y \in R$ . Then, the value of $\log_{e} (f (4))$ is

A 9
B 7
C 2
D 4

Verified Solution

Answer & Solution

Correct Answer

(C) 2

Step-by-step Solution

Detailed explanation

P (x, y) : f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y) \forall x, y \in R

P (0,0) : f (0) = f (0) f^{'} (0) + f^{'} (0) f (0)

\Rightarrow 1 = 2 f^{'} (0)

\Rightarrow f^{'} (0) = \frac{1}{2}

P (x, 0) : f (x) = f (x) . f^{'} (0) + f^{'} (x) . f (0)

\Rightarrow f (x) = \frac{1}{2} f (x) + f^{'} (x)

\Rightarrow f^{'} (x) = \frac{1}{2} f (x)

\Rightarrow f (x) = e^{\frac{1}{2} x}

\Rightarrow \ln (f (4)) = 2

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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From JEE Advanced

The wave function,

ψ_{n, l, m_{l}}

is a mathematical function whose value depends upon spherical polar coordinates

(r, θ, ϕ)

of the electron and characterized by the quantum numbers

n, l

m_{l}

r

is the distance from the nucleus,

θ

is colatitude and

ϕ

is azimuth. In the mathematical functions given in the Table,

Z

is the atomic number and

a_{o}

Column 1	Column 2	Column 3
(I) 1s orbital	\(\text {(i) } \psi_{ n , 1, m_1}\) \(\propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{3}{2}} e ^{-\left(\frac{ Zr }{ e _{ o }}\right)}\)	(P)
(II) 2s orbital	(ii) One radial node	(Q) Probability density at the nucleus \(\propto \frac{1}{ a _{ o }^3}\).
(III) \(2 p _{ z }\) orbital	\(\text {(iii) } \psi_{ n }, 1, m_{ l }\) \(\propto\left(\frac{ Z }{ a _{ o }}\right)^{\frac{5}{2}} re ^{-\left(\frac{ Zr }{2 a _{ o }}\right)} \cos \theta\)	(R) Probability density is maximum at the nucleus.
(IV) \(3 d_{ z }^2\) orbital	(iv) \(x y\)-plane is a nodal plane	(S) Energy needed to excite an electron from \(n =2\) state to \(n =4\) state is \(\frac{27}{32}\) times the energy needed to excite are electron from \(n =2\) state to \(n =6\) state.

For hydrogen atom, what is the only CORRECT combination?

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