# Limits and Derivatives - Class 11 - Mathematics

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## Extra Questions - Limits and Derivatives | NCERT | Mathematics | Class 11

Let $[k]$ denotes the greatest integer less than or equal to $k$. Then the number of positive integral solutions of the equation $\left[\frac{x}{\left[\pi^{2}\right]}\right]=\left[\frac{x}{\left[11 \frac{1}{2}\right]}\right]$

A) 29

B) 24

C) 21

D) 34

The correct option is **B) 24**

To solve the equation

$$\left[\frac{x}{\left[\pi^{2}\right]}\right]=\left[\frac{x}{\left[11 \frac{1}{2}\right]}\right]$$

it is necessary to determine the floor values of $\pi^2$ and $11 \frac{1}{2}$.

The calculations show that:

$\left[\pi^{2}\right] = \left[9.8696\right] = 9$

$\left[11 \frac{1}{2}\right] = [11.5] = 11$

The equation hence simplifies to: $$ \left[\frac{x}{9}\right] = \left[\frac{x}{11}\right] $$

**Case I: $0 \leq \frac{x}{9} < 1$ and $0 \leq \frac{x}{11} < 1$**

$\Rightarrow 0 \leq x < 9 \text{ and } 0 \leq x < 11$

Common values of $x$ are ${1, 2, 3, \ldots, 8}$ (8 values)

**Case II: $1 \leq \frac{x}{9} < 2$ and $1 \leq \frac{x}{11} < 2$**

$\Rightarrow 9 \leq x < 18 \text{ and } 11 \leq x < 22$

$\Rightarrow x \in {11, 12, \ldots, 17}$ (7 values)

**Case III: $2 \leq \frac{x}{9} < 3$ and $2 \leq \frac{x}{11} < 3$**

$\Rightarrow 18 \leq x < 27 \text{ and } 22 \leq x < 33$

$\Rightarrow x \in {22, 23, \ldots, 26}$ (5 values)

**Case IV: $3 \leq \frac{x}{9} < 4$ and $3 \leq \frac{x}{11} < 4$**

$\Rightarrow 27 \leq x < 36 \text{ and } 33 \leq x < 44$

$\Rightarrow x \in {33, 34, 35}$ (3 values)

**Case V: $4 \leq \frac{x}{9} < 5$ and $4 \leq \frac{x}{11} < 5$**

$\Rightarrow 36 \leq x < 45 \text{ and } 44 \leq x < 55$

$\Rightarrow x \in {44}$ (1 value)

The **total number** of positive integers satisfying the conditions across the cases is:
$$
8 + 7 + 5 + 3 + 1 = 24
$$

Therefore, the number of positive integral solutions to the given equation is **24**.

The value of $\lim_{x\rightarrow\infty}\left(\frac{x^{2}-1}{x^{2}+1}\right)^{x^{2}}$ is

A) 1

B) $\mathrm{e}^{-1}$

C) $\mathrm{e}^{-2}$

D) $\mathrm{e}^{-3}$

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