For the following questions answer them individually
Let the ellipse $$E:\frac{x^{2}}{144}+\frac{y^{2}}{169}=1$$ and the hyperbola $$H:\frac{x^{2}}{16}-\frac{y^{2}}{\lambda^{2}}=-1$$ have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H , then the value of 24(e+ L) is:
Given below are two statements :
Statement I : The function $$f:R\rightarrow R $$ defined by $$f(x)=\frac{x}{1+\mid x\mid}$$ is one-one.
Statement II : The function $$f:R\rightarrow R $$ defined by $$f(x)=\frac{x^{2}+4x-30}{x^{2}-8x+18}$$ is many-one.
In the light of the above statements, choose the correct answer from the options given below :
The sum of the coefficients of $$x^{499} \text{ and }x^{500} \text{ in }(1+x)x^{1000}+x(1+x)x^{999}+x^{2}(1+x)x^{998}+....+x^{1000} \text{ is: }$$
Let P be a point in the plane of the vectors $$ \overrightarrow{AB}=3\widehat{i} + \widehat{j}-\widehat{k} \text{ and }\overrightarrow{AC}=\widehat{i}-\widehat{j}+3\widehat{k}$$ such that P is equidistant from the Lines AB and AC. If $$ \mid \overrightarrow{AP} \mid=\frac{\sqrt{5}}{2} $$ then the area of the triangle ABP is:
The sum of all the elements in the range of $$f(x) =Sgn(\sin x) + Sgn(\cos x) +Sgn(\tan x) +Sg n(\cot x)$$, $$x \neq \frac{n\pi}{2}, n\epsilon Z, \text{ where } Sgn(t)=\begin{cases}1, & \text{ if } t>0\\-1 & \text{ if } t<0\end{cases} ,is:$$
Let $$P_1 : y=4x^2 \text{ and } P_2 : y=x^2 + 27$$ be two parabolas. If the area of the bounded region enclosed between P_1 and P_2 is six times the area of the bounded region enclosed between the line $$y = c\alpha x, \alpha > 0 \text{ and } P_1,$$ then $$\alpha$$ is equal to:
Let the circle $$x^{2}+y^{2}=4$$ interesect x-axis at the points A(a,0), a > 0 and B(b, 0). let $$P(2 \cos \alpha, 2 \sin \alpha),0 < \alpha < \frac{\pi}{2} \text{and } Q(2\cos \beta, 2\sin \beta)$$ be two points such that $$( \alpha - \beta) =\frac {\pi}{2}$$. Then the point of intersection of AQ and BP lies on:
Let $$f(x)=\int\frac{dx}{x^{\left(\frac{2}{3}\right)}+2x^{\left(\frac{1}{2}\right)}} $$ be such that $$f(0)=-26+24\log_{e}{(2)}. \text { If } f(1)=a+b \log_{e}{(3)}, \text{ where } a,b \in Z$$, then a+b is equal to:
$$\frac{6}{3^{26}}+\frac{10.1}{3^{25}}+\frac{10.2}{3^{24}}+\frac{10.2^{2}}{3^{23}}+...+\frac{10.2^{24}}{3}$$ is equal to :
Let $$[\cdot]$$ denote the greatest integer function. Then $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$$ is equal to:
