A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is :

A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

The number of real roots of the equation $$e^{4x}+2e^{3x}-e^x-6=0$$ is :

The function $$f(x)=x^3-6x^2+ax+b$$ is such that $$f(2)=f(4)=0$$. Consider two statements : Statement 1 : there exists x1, x2 $$\in$$(2, 4), x1 < x2, such that f'(x1) = $$-$$1 and f'(x2) = 0. Statement 2 : there exists x3, x4 $$\in$$ (2, 4), x3 < x4, such that f is decreasing in (2, x4), increasing in (x4, 4) and $$2f^{\prime }(x_3)=\sqrt{3}f(x_4)$$. Then

The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola y2 = 9x, is :

The area of the region : $$R=\{(x,y):5x^2\le y\le 2x^2+9\}$$ is :

Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = {\pi \over 2} in the first quadrant. Then,

The area bounded by the curve 4y2 = x2(4 $$-$$ x)(x $$-$$ 2) is equal to :

The area (in sq. units) of the region, given by the set $$\{(x,y)\in R\times R|x\ge 0,2x^2\le y\le 4-2x\}$$ is :

The area of the region bounded by y $$-$$ x = 2 and x2 = y is equal to :

If the area of the bounded region R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\} is , $$\alpha ({\log }_{e}2{)}^{-1}+\beta ({\log }_{e}2)+\gamma$$, then the value of $$(\alpha +\beta -2\lambda {)}^2$$ is equal to :

The area of the region bounded by the parabola (y $$-$$ 2)2 = (x $$-$$ 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :

The area, enclosed by the curves $$y=\sin x+\cos x$$ and $$y=\left|\cos x-\sin x\right|$$ and the lines x = 0,x = {\pi \over 2}, is :

The statement among the following that is a tautology is :

The negation of the statement $$\sim p\wedge (p\vee q)$$ is :

For the statements p and q, consider the following compound statements : (a) $$(\sim q\wedge (p\to q))\to \sim p$$ (b) $$((p\vee q)\wedge \sim p)\to q$$ Then which of the following statements is correct?

The statement A $$\to$$ (B $$\to$$ A) is equivalent to :

The contrapositive of the statement "If you will work, you will earn money" is :

Let F1(A, B, C) = (A $$\wedge$$ $$\sim$$ B) $$\vee$$ [$$\sim$$C $$\wedge$$ (A $$\vee$$ B)] $$\vee$$ $$\sim$$ A and F2(A, B) = (A $$\vee$$ B) $$\vee$$ (B $$\to$$ $$\sim$$A) be two logical expressions. Then :

Which of the following Boolean expression is a tautology?