Let f(x)=2 x^{2}-x-1 and S ={n ∈ Z :|f(n)| ≤ 800} . Then value of Σ

English

Gujarati

Hindi

1.Relation and Function

normal

Let $f(x)=2 x^{2}-x-1$ and $S =\{n \in Z :|f(n)| \leq 800\}$ . Then value of $\sum_{n \in S} f(n)$ is . . . . .

$10620$

$10630$

$10640$

$10650$

(JEE MAIN-2022)

Solution

$f ( x )=2 x ^{2}- x -1$

$| f ( x )| \leq 800$

$2 n ^{2}- n -801 \leq 0$

$n^{2}-\frac{1}{2} n-\frac{801}{2} \leq 0$

$\left(n-\frac{1}{4}\right)^{2}-\frac{801}{2}-\frac{1}{16} \leq 0$

$\left(n-\frac{1}{4}\right)^{2}-\frac{6409}{16} \leq 0$

$\left(n-\frac{1}{4}-\frac{\sqrt{6409}}{4}\right)\left(n-\frac{1}{4}+\frac{\sqrt{6409}}{16}\right) \leq 0$

$\frac{1-\sqrt{6409}}{4} \leq n \leq \frac{1+\sqrt{6409}}{4}$

$n=\{-19,-18-17, \ldots \ldots 0,1,2, \ldots \ldots, 20\}$

$\sum_{n \in s} f(x)=\sum\left(2 x^{2}-x-1\right)$

$=2\left[19^{2}+18^{2}+\ldots . .+1^{2}+1^{2}+2^{2}+\ldots .+19^{2}+20^{2}\right]$

$=4\left[1^{2}+2^{2}+\ldots . .+19^{2}\right]+2\left[20^{2}\right]-20-40$

$=\frac{4 \times 19 \times 20 \times(2 \times 19+1)}{6}+2 \times 400-60$

$=\frac{4 \times 19 \times 20 \times 39}{6}+800-60-9880+800-60$

$=10620$

Standard 12

Mathematics

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,

normal

(KVPY-2019)

If $f(x) = \cos (\log x)$, then $f(x)f(y) – \frac{1}{2}[f(x/y) + f(xy)] = $

medium

(IIT-1983)

Let $2{\sin ^2}x + 3\sin x – 2 > 0$ and ${x^2} – x – 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

hard

(IIT-1994)

The graph of the function $f(x)=x+\frac{1}{8} \sin (2 \pi x), 0 \leq x \leq 1$ is shown below. Define $f_1(x)=f(x), f_{n+1}(x)=f\left(f_n(x)\right)$, for $n \geq 1$.

Which of the following statements are true?

$I.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=0$

$II.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=\frac{1}{2}$

$III.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=1$

$IV.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)$ does not exist.

normal

(KVPY-2016)

If a function $g(x)$ is defined in $[-1, 1]$ and two vertices of an equilateral triangle are $(0, 0)$ and $(x, g(x))$ and its area is $\frac{\sqrt 3}{4}$ , then $g(x)$ equals :-

normal

Let $f(x)=2 x^{2}-x-1$ and $S =\{n \in Z :|f(n)| \leq 800\}$ . Then value of $\sum_{n \in S} f(n)$ is . . . . .

Solution

Similar Questions

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements. $I.$ $f$ is an odd function. $II.$ $f$ is an even function. $III$. $f$ is differentiable everywhere. Then,

If $f(x) = \cos (\log x)$, then $f(x)f(y) – \frac{1}{2}[f(x/y) + f(xy)] = $

Let $2{\sin ^2}x + 3\sin x – 2 > 0$ and ${x^2} – x – 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

If a function $g(x)$ is defined in $[-1, 1]$ and two vertices of an equilateral triangle are $(0, 0)$ and $(x, g(x))$ and its area is $\frac{\sqrt 3}{4}$ , then $g(x)$ equals :-

Start a Free Trial Now

Let $f: R \rightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x \in R$. Consider the following statements.

$I.$ $f$ is an odd function.

$II.$ $f$ is an even function.

$III$. $f$ is differentiable everywhere. Then,