For the following questions answer them individually
Let f be a function such that $$3f(x)+2f \left(\frac{m}{19x}\right) = 5x, x\neq 0$$, where $$m= \sum_{i-1}^9(i)^{2}$$. Then f(5) - f(2) is equal to
Lety = y (x) be a differentiable function in the interval $$(0, \infty)$$ such that y(l) = 2, and $$\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$$ for each x > 0. Then 2){2) is equal to
Let f(a) denote the area of the region in the first quadrant bounded by x = 0, x = 1, $$y^{2}=x$$ and y = |ax - 5| - |1 - ax| + $$ax^{2}$$. Then (f(O) + f(1)) is equal
Let the image of parabola $$x^{2}=4y$$, in the line x - y = 1 be $$(y+a)^{2}$$ = b(x-c), $$a,b,c \in N.$$ Then a + b + c is equal to
Let $$\overrightarrow{a}= 2\widehat{i}-\widehat{j}-\widehat{k}, \overrightarrow{b}=\widehat{i}+ 3\widehat{j}-\widehat{k}$$ and $$\overrightarrow{c} = 2\widehat{i}+\widehat{j}+3\widehat{k}.$$ Let $$\overrightarrow{\nu}$$ be the vector in the plane of the vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, such that the length of its projection on the vector $$\overrightarrow{C}$$ is $$\frac{1}{\sqrt{14}}$$. Then $$\mid \overrightarrow{\nu} \mid$$ is euqal to
Let $$\overrightarrow{a}= 2\widehat{i}-5\widehat{j}+5\widehat{k}$$ and $$\overrightarrow{b}= \widehat{i}-\widehat{j}+3\widehat{k}$$. If $$\overrightarrow{C}$$ is a vector such that $$2(\overrightarrow{a}\times\overrightarrow{c})+3(\overrightarrow{b}\times\overrightarrow{c})= \overrightarrow{0}$$ and $$(\overrightarrow{a}-\overrightarrow{b})\cdot\overrightarrow{c}=-97,$$ then $$\mid \overrightarrow{c}\times\widehat{k} \mid^{2}$$ is equal to
The sum of all values of $$\alpha$$, for which the sho1test distance between the lines
$$\frac{x+1}{\alpha}=\frac{y-2}{-1}=\frac{z-4}{-\alpha}$$ and $$\frac{x}{\alpha}=\frac{y-1}{2}=\frac{z-1}{2\alpha}$$ is $$\sqrt{2}$$, is
Let $$a_{1},a_{2},a_{3},a_{4}$$ be an A.P. of four term such that each term of the A.P. and its common difference l are integers. If $$a_{1} +a_{2}+a_{3}+a_{4}= 48$$ and $$a_{1} a_{2}a_{3}a_{4}6 + l^{4} = 361,$$ then the largest tenn of the A.P. is equal to
The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionaiy. The rank of the word "UDAYPUR" is
Let the length of the latus rectum of an ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b)$$ be 30. If its eccentricity is the maximum value of the function $$f(t)=-\frac{3}{4}+2t-t^{2}$$ then $$(a^{2}+b^{2})$$ is equal to
