For the following questions answer them individually
Let S and S' be the foci of the ellipse $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$ and $$P(\alpha , \beta)$$ be a point on the ellipse in the first quadrant. If $$(SP)^{2}+(S'P)^{2}-SP\cdot S'P=37$$, then $$\alpha^{2}+\beta^{2}$$ is equal to :
Let $$f(x)= [x]^{2}-[x+3]-3, x\in \mathbb R$$, where $$[\cdot]$$ is the greatest integer funtion. Then
The area of the region $$A= \left\{(x,y): 4x^{2}+y^{2}\leq 8 \text{and } y^{2} \leq 4x \right\}$$ is :
If y=y(x) satisfies the differential equation
$$16(\sqrt{x+9\sqrt{x}})(4+\sqrt{9+\sqrt{x}}) \cos{y}dy=(1+2 \sin y)dx, x>0 \text{and} y(256) = \frac{\pi}{2}, y(49)=\alpha$$, then $$2\sin \alpha$$ is equal to :
Let $$\left[\cdot\right]$$ denote the greatest integer function, and let f (x) = $$\min \left\{\sqrt{2x},x^{2}\right\}$$. Let S = $$\left\{x \in (-2,2): \text{the function,} g(x)= |x|\left[x^{2}\right]\text{is discontinuous at x} \right\}.$$ Then $$\sum_{x\in S}f(x)$$ equals
The number of elements in the relation $$R= \left\{(x,y): 4x^{2}+y^{2}<52,x,y\in Z\right\}$$ is
If $$X=\begin{bmatrix}x \\y \\z \end{bmatrix}$$ is a solution of the system of equations AX= B, where adj $$A= \begin{bmatrix}4 & 2 & 2 \\-5 & 0 & 5 \\1 & -2 & 3 \end{bmatrix}$$ and $$B=\begin{bmatrix}4 \\0 \\2 \end{bmatrix}$$, then |x+y+z| is equal to :
Let L be the line $$\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$$ and let S be the set of all points (a, b, c) on L, whose distance from the line $$\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0}$$a long the line L is 7. Then $$\sum_{(a,b,c)\in S} (a+b+c) $$ is equal to :
Let $$C_{r}$$ denote the coefficient of $$x^{r}$$ in the binomial expansion of $$(1+x)^{n}, n\in N, 0\leq r\leq n$$. If $$P_{n}= C_{0}-C_{1}+\frac{2^{2}}{3}C_{2}-\frac{2^{3}}{4}C_{3}+.....+\frac{(-2)^{n}}{n+1}C_{n}, \text{then the value of} \sum_{n=1}^{25} \frac{1}{P_{2n}} $$ equals.
Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}+\widehat{k}$$ and $$\overrightarrow{b}= \lambda \widehat{j}+2\widehat{k}, \lambda\in Z$$ be two vectors. Let $$\overrightarrow{c}= \overrightarrow{a} \times \overrightarrow{b} \text{and } \overrightarrow{d}$$ be a vector of magnitude 2 in yz-plane. If $$|\overrightarrow{c}|=\sqrt{53}$$, then the maximum possible value of $$\left(\overrightarrow{c}\cdot\overrightarrow{d}\right)^{2}$$ is equal to :
