For the following questions answer them individually
Let the line $$x = - 1$$ divide the area of the region $$ \left\{(x,y): 1+x^{2}\leq y \leq 3 -x\right\} $$ in the ratio m : n, gcd (m, n) = 1. Then m + n is equal to
Two distinct numbers a and b are selected at random from 1, 2, 3, ... , 50. The probability, that their product ab is divisible by 3, is
If the image of the point $$P(1 , 2, a)$$ in the line $$ \frac{x-6}{3} = \frac{y - 7}{2} = \frac{7 -z}{2}$$ is $$Q(5, b, c)$$, then $$ a^{2}+b^{2}+c^{2}$$ is equal to
The number of distinct real solutions of the equation $$ x\mid x+4\mid +3\mid x +2 \mid + 10 +0 $$ is
If the domain of the function $$ \large f(x)=\sin ^{-1} \left( \frac{5-x}{3+2x} \right)+\frac{1}{\log_{e}{(10-x)}} $$ is $$ \large (-\infty,\propto] \cup [\beta,\gamma) - \left\{ \delta\right\} $$, then $$ \large 6(\alpha+ \beta+ \gamma+\delta) $$ is equal to
If the sum of the first four tenns of an A.P. is 6 and the sum of its first six tenns is 4, then the sum of its first twelve terms is
If the chord joining the points $$ P_{1}(x_{1}, y_{1}) $$ and $$P_{2}(x_{2},y_{2})$$ on the parabola $$y^{2}=12x$$ subtends a right angle at the vertex of the parabola, then $$ x_{1}x_{2}-y_{1}y_{2} $$ is equal to
Let $$ P(\alpha,\beta, \gamma)$$ be the point on the line $$\frac{x-1}{2}=\frac{y+1}{-3}=z$$ at a distance $$4\sqrt{14}$$ from the point (1, -1, 0) and nearer to the origin. Then the shortest di stance, between the Lines $$\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}$$ and $$\frac{x+5}{2}= \frac{y-10}{1}=\frac{z-3}{1}$$, is equal to
Let $$\overrightarrow{AB} = 2\widehat{i}+4\widehat{j}-5\widehat{k}$$ and $$ \overrightarrow{AD} = \widehat{i}+2\widehat{j}+\lambda\widehat{k}, \lambda\text{ }\epsilon \text{ } R$$. Let the projection of the vector $$ \overrightarrow{v}=\widehat{i}+\widehat{j}+\widehat{k}$$ on the disgonal $$\overrightarrow{AC}$$ of the parallelogram ABCD be of length one unit. If $$\alpha> \beta$$, be the roots of the equation $$\lambda^{2}x^{2}-6\lambda x+5=0$$, then $$2\alpha-\beta$$ is equal to
Let $$ f(x)=x^{2025}-x^{2000}, x \text{ }\epsilon \text{ }[0,1] $$ and the minimmu value of the function $$ f(x)$$ in the interval [0, 1] be $$(80)^{80}(n)^{-81}$$. Then n is equal to
