The equations of the sides AB and AC of a triangle ABC are $$(\lambda +1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda )y+\lambda =0$$ respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the point C to the part of the parabola $$y^2=6x$$ in the first quadrant is :

Let a tangent to the curve $${\mathrm{y}}^2=24x$$ meet the curve $$xy=2$$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

Let $$\mathrm{PQ}$$ be a focal chord of the parabola $$y^2=36x$$ of length 100 , making an acute angle with the positive $$x$$-axis. Let the ordinate of $$\mathrm{P}$$ be positive and $$\mathrm{M}$$ be the point on the line segment PQ such that PM : MQ = 3 : 1. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $$\mathrm{PQ}$$?

Let $$\mathrm{A}(0,1),\mathrm{B}(1,1)$$ and $$\mathrm{C}(1,0)$$ be the mid-points of the sides of a triangle with incentre at the point $$\mathrm{D}$$. If the focus of the parabola $$y^2=4\mathrm{ax}$$ passing through $$\mathrm{D}$$ is $$(\alpha +\beta \sqrt{2},0)$$, where $$\alpha$$ and $$\beta$$ are rational numbers, then $$\frac{\alpha }{{\beta }^2}$$ is equal to :

Let $$R$$ be the focus of the parabola $$y^2=20x$$ and the line $$y=mx+c$$ intersect the parabola at two points $$P$$ and $$Q$$. Let the point $$G(10,10)$$ be the centroid of the triangle $$PQR$$. If $$c-m=6$$, then $$(PQ{)}^2$$ is :

The value of the integral \int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{x + {\pi \over 4}} \over {2 - \cos 2x}}dx} is :

Let $\alpha >0$. If $\underset{0}{\overset{\alpha }{\int }}\frac{x}{\sqrt{x+\alpha }-\sqrt{x}}\mathrm{d}x=\frac{16+20\sqrt{2}}{15}$, then $\alpha$ is equal to :

If $\varphi (x)=\frac{1}{\sqrt{x}}\underset{\frac{\pi }{4}}{\overset{x}{\int }}\left(4\sqrt{2}\sin t-3{\varphi }^{\prime }(t)\right)dt,x>0$, then ${\varnothing }^{\prime }\left(\frac{\pi }{4}\right)$ is equal to :

\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {1 + n}} + {1 \over {2 + n}} + {1 \over {3 + n}}\, + \,...\, + \,{1 \over {2n}}} \right] is equal to

Let $$\alpha \in (0,1)$$ and $$\beta ={\log }_e(1-\alpha )$$. Let {P_n}(x) = x + {{{x^2}} \over 2} + {{{x^3}} \over 3}\, + \,...\, + \,{{{x^n}} \over n},x \in (0,1). Then the integral \int\limits_0^\alpha {{{{t^{50}}} \over {1 - t}}dt} is equal to

The value of \int_\limits{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{(2+3 \sin x)}{\sin x(1+\cos x)} d x is equal to :

$\underset{n\to \infty }{\lim }\frac{3}{n}\left\{4+{\left(2+\frac{1}{n}\right)}^2+{\left(2+\frac{2}{n}\right)}^2+\dots +{\left(3-\frac{1}{n}\right)}^2\right\}$ is equal to :

If [t] denotes the greatest integer $$\le \mathrm{t}$$, then the value of {{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx} is :

The value of the integral \int_1^2 {\left( {{{{t^4} + 1} \over {{t^6} + 1}}} \right)dt} is

The value of the integral \int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx} is equal to :

Let f(x) = x + {a \over {{\pi ^2} - 4}}\sin x + {b \over {{\pi ^2} - 4}}\cos x,x \in R be a function which satisfies $$f(x)=x+\underset{0}{\overset{\pi /2}{\int }}\sin (x+y)f(y)dy$$. then $$(a+b)$$ is equal to

The integral 16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}} is equal to

The minimum value of the function $$f(x)=\underset{0}{\overset{2}{\int }}{e}^{|x-t|}dt$$ is :

\int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx} is equal to :

If $\underset{0}{\overset{1}{\int }}\frac{1}{\left(5+2x-2x^2\right)\left(1+e^{(2-4x)}\right)}dx=\frac{1}{\alpha }{\log }_e\left(\frac{\alpha +1}{\beta }\right),\alpha ,\beta >0$, then ${\alpha }^4-{\beta }^4$ is equal to :