Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x^2-px+\frac{5}{4}p=0$ are rational. Then the area of the region $\left\{(x,y):0\le y\le (x-q{)}^2,0\le x\le q\right\}$ is :

The area of the region A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\} is

Let $$\Delta$$ be the area of the region $$\left\{(x,y)\in R^2:x^2+y^2\le 21,y^2\le 4x,x\ge 1\right\}$$. Then {1 \over 2}\left( {\Delta - 21{{\sin }^{ - 1}}{2 \over {\sqrt 7 }}} \right) is equal to

Let $$[x]$$ denote the greatest integer $$\le x$$. Consider the function $$f(x)=\max \left\{x^2,1+[x]\right\}$$. Then the value of the integral $$\underset{0}{\overset{2}{\int }}f(x)dx$$ is

Let $$A=\left\{(x,y)\in {\mathbb{R}}^2:y\ge 0,2x\le y\le \sqrt{4-(x-1{)}^2}\right\}$$ and $$B=\left\{(x,y)\in \mathbb{R}\times \mathbb{R}:0\le y\le \min \left\{2x,\sqrt{4-(x-1{)}^2}\right\}\right\}\text{. }$$. Then the ratio of the area of A to the area of B is

The area enclosed by the curves $$y^2+4x=4$$ and $$y-2x=2$$ is :

The area of the region $$\left\{(x,y):x^2\le y\le \left|x^2-4\right|,y\ge 1\right\}$$ is

The area of the region enclosed by the curve $$f(x)=\max \{\sin x,\cos x\},-\pi \le x\le \pi$$ and the $$x$$-axis is

The area of the region enclosed by the curve $$y=x^3$$ and its tangent at the point $$(-1,-1)$$ is :

Area of the region $$\left\{(x,y):x^2+(y-2{)}^2\le 4,x^2\ge 2y\right\}$$ is

The area of the region $$\left\{(x,y):x^2\le y\le 8-x^2,y\le 7\right\}$$ is :

The area bounded by the curves $$y=|x-1|+|x-2|$$ and $$y=3$$ is equal to :

Which of the following statements is a tautology?

The number of values of $\mathrm{r}\in \{\mathrm{p},\mathrm{q},\sim \mathrm{p},\sim \mathrm{q}\}$ for which $((\mathrm{p}\wedge \mathrm{q})\Rightarrow (\mathrm{r}\vee \mathrm{q}))\wedge ((\mathrm{p}\wedge \mathrm{r})\Rightarrow \mathrm{q})$ is a tautology, is :

The negation of the expression $$q\vee \left((\sim q)\wedge p\right)$$ is equivalent to

$$(\mathrm{S}1)(p\Rightarrow q)\vee (p\wedge (\sim q))$$ is a tautology $$(\mathrm{S}2)((\sim p)\Rightarrow (\sim q))\wedge ((\sim p)\vee q)$$ is a contradiction. Then

Consider the following statements: P : I have fever Q: I will not take medicine $\mathrm{R}$ : I will take rest. The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to :

Among the statements : $$(\mathrm{S}1)((\mathrm{p}\vee \mathrm{q})\Rightarrow \mathrm{r})\iff (\mathrm{p}\Rightarrow \mathrm{r})$$ $$(\mathrm{S}2)((\mathrm{p}\vee \mathrm{q})\Rightarrow \mathrm{r})\iff ((\mathrm{p}\Rightarrow \mathrm{r})\vee (\mathrm{q}\Rightarrow \mathrm{r}))$$

The statement $$B\Rightarrow \left(\left(\sim A\right)\vee B\right)$$ is equivalent to :

If $$p,q$$ and $$r$$ are three propositions, then which of the following combination of truth values of $$p,q$$ and $$r$$ makes the logical expression $$\left\{(p\vee q)\wedge \left((\sim p)\vee r\right)\right\}\to \left((\sim q)\vee r\right)$$ false?