The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by {{{x^2}} \over {xy - {x^2}{y^2} - 1}}. If the curve passes through the point (1, 1), then e . y(e) is equal to

Let $$\lambda$$$${}^{\ast }$$ be the largest value of $$\lambda$$ for which the function $$f_{\lambda }(x)=4\lambda x^3-36\lambda x^2+36x+48$$ is increasing for all x $$\in$$ R. Then $$f_{{\lambda }^{\ast }}(1)+f_{{\lambda }^{\ast }}(-1)$$ is equal to :

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

For the function $$f(x)=4{\log }_e(x-1)-2x^2+4x+5,x>1$$, which one of the following is NOT correct?

If the tangent at the point (x1, y1) on the curve $$y=x^3+3x^2+5$$ passes through the origin, then (x1, y1) does NOT lie on the curve :

The sum of absolute maximum and absolute minimum values of the function $$f(x)=|2x^2+3x-2|+\sin x\cos x$$ in the interval [0, 1] is :

Let $$\lambda x-2y=\mu$$ be a tangent to the hyperbola $$a^2{x}^2-y^2=b^2$$. Then {\left( {{\lambda \over a}} \right)^2} - {\left( {{\mu \over b}} \right)^2} is equal to :

If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + $$\alpha$$, 0) and (0, 50 + $$\alpha$$), $$\alpha$$ > 0, then (x, y) also lies on the line :

Let $$f(x)=4x^3-11x^2+8x-5,x\in R$$. Then f :

If the absolute maximum value of the function $$f(x)=\left(x^2-2x+7\right){\mathrm{e}}^{\left(4x^3-12x^2-180x+31\right)}$$ in the interval $$[-3,0]$$ is $$f(\alpha )$$, then :

The curve $$y(x)=a{x}^3+b{x}^2+cx+5$$ touches the $$x$$-axis at the point $$\mathrm{P}(-2,0)$$ and cuts the $$y$$-axis at the point $$Q$$, where $$y^{\prime }$$ is equal to 3 . Then the local maximum value of $$y(x)$$ is:

If the maximum value of $$a$$, for which the function $$f_a(x)={\tan }^{-1}2x-3ax+7$$ is non-decreasing in $$\left(-\frac{\pi }{6},\frac{\pi }{6}\right)$$, is $$\bar{a}$$, then $$f_{\bar{a}}\left(\frac{\pi }{8}\right)$$ is equal to :

If the minimum value of $$f(x)=\frac{5x^2}{2}+\frac{\alpha }{x^5},x>0$$, is 14 , then the value of $$\alpha$$ is equal to :

The function $$f(x)=x{\mathrm{e}}^{x(1-x)},x\in \mathbb{R}$$, is :

Let $$f(x)=3^{{\left(x^2-2\right)}^3+4},x\in \mathrm{R}$$. Then which of the following statements are true? $$\mathrm{P}:x=0$$ is a point of local minima of $$f$$ $$\mathrm{Q}:x=\sqrt{2}$$ is a point of inflection of $$f$$ $$R:f^{\prime }$$ is increasing for $$x>\sqrt{2}$$

The area enclosed by y2 = 8x and y = $$\sqrt{2}$$ x that lies outside the triangle formed by y = $$\sqrt{2}$$ x, x = 1, y = 2$$\sqrt{2}$$, is equal to:

The area of the bounded region enclosed by the curve y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1| and the x-axis is :

The area of the region S = {(x, y) : y2 $$\le$$ 8x, y $$\ge$$ $$\sqrt{2}$$x, x $$\ge$$ 1} is

The area bounded by the curve y = |x2 $$-$$ 9| and the line y = 3 is :

The area of the region bounded by y2 = 8x and y2 = 16(3 $$-$$ x) is equal to: