New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Functions question

2023 · 29 Jan · Shift 2 · Q30

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Functions
2023 · 29 Jan · Shift 2 · Q30

Functions question

2023 · 29 Jan · Shift 2 · Q30

JEE MainMathematicsFunctionsMCQ+4 / −1

Consider a function

f:\mathbb{N}\to\mathbb{R}

, satisfying

f(1)+2f(2)+3f(3)+....+xf(x)=x(x+1)f(x);x\ge2

with

f(1)=1

. Then

\frac{1}{f(2022)}+\frac{1}{f(2028)}

is equal to

A
8000
B
8400
C
8100
D
8200

View written solutionFree

Correct answer: 4050

Let

\cdots +xf(x).$$ Given, $$S_x=x(x+1)f(x), \qquad x\ge 2,$$ and also $f(1)=1$. 2. Write the relation for $x=n$ and $x=n-1$: $$S_n=n(n+1)f(n),$$ $$S_{n-1}=(n-1)n f(n-1).$$ But $$S_n=S_{n-1}+n f(n).$$ So, $$n(n+1)f(n)=(n-1)n f(n-1)+n f(n).$$ 3. Simplify: $$n[(n+1)f(n)-f(n)]=(n-1)n f(n-1)$$ $$n^2 f(n)=n(n-1)f(n-1).$$ For $n\ge 2$, $$n f(n)=(n-1)f(n-1),$$ so $$f(n)=\frac{n-1}{n}f(n-1).$$ 4. Use $f(1)=1$: $$f(2)=\frac{1}{2}f(1)=\frac12,$$ $$f(3)=\frac{2}{3}f(2)=\frac13,$$ and in general this telescopes to $$f(n)=\frac{1}{n}.$$ 5. Therefore, $$\frac{1}{f(2022)}+\frac{1}{f(2028)}=2022+2028=4050.$$ 6. Compare with options: - A: $8000$ - B: $8400$ - C: $8100$ - D: $8200$ None matches $4050$. So the stored correct answer $C$ is inconsistent with the given question. The correct value should be $$4050.$$

More from Functions

Let $S = {1, 2, 3, 4, 5, 6}$ . Then the number of one-one functions $f : S \to P (S)$ , where $P (S)$ denote the power set of $S$ , such that $f (n) \subset f (m)$ where $n \lt…2023 · Numerical
The range of the function $f (x) = 3 - x + 2 + x$ is :2023 · MCQ
Let $A = {1, 2, 3, 5, 8, 9}$ . Then the number of possible functions $f : A \to A$ such that $f (m \cdot n) = f (m) \cdot f (n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to .2023 · Numerical
If the domain of the function $f (x) = \frac{[ x ]}{1 + x ^{2}}$ , where $[x]$ is greatest integer $[2, 6)$ , is $[2, 6)$ , then its range is2023 · MCQ
Let $f : R - {2, 6} \to R$ be real valued function defined as $f (x) = \frac{x ^{2} + 2 x + 1}{x ^{2} - 8 x + 12}$ . Then range of $f$ is2023 · MCQ
The absolute minimum value, of the function $f (x) = x^{2} - x + 1 + [x^{2} - x + 1]$ , where $[t]$ denotes the greatest integer function, in the interval $[- 1, 2]$ , is :2023 · MCQ
The number of one-one functions f : {a, b, c, d} $\to$ {0, 1, 2, ......, 10} such that 2f(a) $-$ f(b) + 3f(c) + f(d) = 0 is .2022 · Numerical
The total number of functions, $f : {1, 2, 3, 4} \to {1, 2, 3, 4, 5, 6}$ such that $f (1) + f (2) = f (3)$ , is equal to :2022 · MCQ