The area of the region enclosed between the parabolas y2 = 2x $$-$$ 1 and y2 = 4x $$-$$ 3 is

The area of the region given by $$A=\left\{(x,y):x^2\le y\le \min \{x+2,4-3x\}\right\}$$ is :

Let the locus of the centre $$(\alpha ,\beta ),\beta >0$$, of the circle which touches the circle $$x^2+(y-1{)}^2=1$$ externally and also touches the $$x$$-axis be $$\mathrm{L}$$. Then the area bounded by $$\mathrm{L}$$ and the line $$y=4$$ is:

The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is {{364} \over 3}, is equal to :

The area bounded by the curves $$y=\left|x^2-1\right|$$ and $$y=1$$ is

The area of the smaller region enclosed by the curves $$y^2=8x+4$$ and $$x^2+y^2+4\sqrt{3}x-4=0$$ is equal to

The area of the region enclosed by $$y\le 4x^2,x^2\le 9y$$ and $$y\le 4$$, is equal to :

Consider a curve $$y=y(x)$$ in the first quadrant as shown in the figure. Let the area $${\mathrm{A}}_1$$ is twice the area $${\mathrm{A}}_2$$. Then the normal to the curve perpendicular to the line $$2x-12y=15$$ does NOT pass through the point.

The area enclosed by the curves $$y={\log }_e\left(x+{\mathrm{e}}^2\right),x={\log }_e\left(\frac{2}{y}\right)$$ and $$x={\log }_{\mathrm{e}}2$$, above the line $$y=1$$ is:

The area of the region $$\left\{(x,y):|x-1|\le y\le \sqrt{5-x^2}\right\}$$ is equal to :

Let $$\Delta$$ $$\in$$ {$$\wedge$$, $$\vee$$, $$\Rightarrow$$, $$\iff$$} be such that (p $$\wedge$$ q) $$\Delta$$ ((p $$\vee$$ q) $$\Rightarrow$$ q) is a tautology. Then $$\Delta$$ is equal to :

Negation of the Boolean statement (p $$\vee$$ q) $$\Rightarrow$$ (($$\sim$$ r) $$\vee$$ p) is equivalent to :

Let p, q, r be three logical statements. Consider the compound statements $$S_1:((\sim p)\vee q)\vee ((\sim p)\vee r)$$ and $$S_2:p\to (q\vee r)$$ Then, which of the following is NOT true?

Which of the following statement is a tautology?

The boolean expression $$(\sim (p\wedge q))\vee q$$ is equivalent to :

Let $$\Delta$$, $$\nabla$$ $$\in$$ {$$\wedge$$, $$\vee$$} be such that p $$\nabla$$ q $$\Rightarrow$$ ((p $$\Delta$$ q) $$\nabla$$ r) is a tautology. Then (p $$\nabla$$ q) $$\Delta$$ r is logically equivalent to :

Let r $$\in$$ {p, q, $$\sim$$p, $$\sim$$q} be such that the logical statement r $$\vee$$ ($$\sim$$p) $$\Rightarrow$$ (p $$\wedge$$ q) $$\vee$$ r is a tautology. Then r is equal to :

The negation of the Boolean expression (($$\sim$$ q) $$\wedge$$ p) $$\Rightarrow$$ (($$\sim$$ p) $$\vee$$ q) is logically equivalent to :

Consider the following two propositions: $$P1:\sim (p\to \sim q)$$ $$P2:(p\wedge \sim q)\wedge ((\sim p)\vee q)$$ If the proposition $$p\to ((\sim p)\vee q)$$ is evaluated as FALSE, then :

Consider the following statements: A : Rishi is a judge. B : Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is