Let the function $f:[0,1] \rightarrow \mathbb{R}$ be defined by $\mathrm{f}(\mathrm{x})=\frac{4^{\mathrm{x}}}{4^{\mathrm{x}}+2}$ Then the value of $\mathrm{f}\left(\frac{1}{40}\right)+\mathrm{f}\left(\frac{2}{40}\right)+\mathrm{f}\left(\frac{3}{40}\right)+\ldots \ldots \ldots+\mathrm{f}\left(\frac{39}{40}\right)-\mathrm{f}\left(\frac{1}{2}\right)$ is

Two fair dice, each with faces numbered 1,2,3,4,5 and 6, are rolled together and the sum of the numbers on...

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a...

An engineer is required to visit a factory for exactly four days during the first 15 days of every month...

For non-negative integers s and r, let
$...

Let a and b be positive real numbers. Suppose $\overrightarrow{\mathrm{PQ}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{PS}}=a \hat{\mathrm{i}}-b \hat{\mathrm{j}}$ are adjacent sides of...

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^2+\beta^2+\gamma^2 \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is...

Let 𝒇:ℝ→ℝ and 𝒈:ℝ→ℝ be functions satisfying 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦)+𝑓(𝑥)𝑓(𝑦) and 𝑓(𝑥)=𝑥𝑔(𝑥) for all 𝑥,𝑦∈ℝ. If $\lim _{x \rightarrow 0} g(x)=1$, then...

Let b be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $(0)=1$. If...

The value of the limit
$...

Let the functions : (−1,1)→ℝ and 𝑔 : (−1,1)→(−1,1) be defined by ƒ(𝑥) = |2𝑥−1|+|2𝑥+1| and 𝑔(𝑥) = 𝑥−[𝑥], where...

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a...

Let O be the centre of the circle $\mathrm{x}^2+\mathrm{y}^2=\mathrm{r}^2$, where $\mathrm{r}>\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle...

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least...

For a complex number z, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all...

Let e denote the base of the natural logarithm. The value of the real number a for which the right...

For a polynomial $\mathrm{g}(\mathrm{x})$ with real coefficient, let $\mathrm{m}_{\mathrm{g}}$ denote the number of distinct real roots of $\mathrm{g}(\mathrm{x})$. Suppose S...

Let $\mathrm{x} \in \mathbb{R}$ and let $...