For the following questions answer them individually
The sum of all the real solutions of the equation
$$\log_{(x+3)}{(6x^{2}+28x+30)}=5-2\log_{(6x+10)}{(x^{2}+6x+9)}$$ is equal to
Let $$\overrightarrow{a}=\widehat{i}-2\widehat{j}+3\widehat{k}, \overrightarrow{b}=2\widehat{i}+\widehat{j}-\widehat{k}, \overrightarrow{c}=\lambda \widehat{i}+\widehat{j}+\widehat{k}$$ and $$\overrightarrow{v}= \overrightarrow{a} \times \overrightarrow{b}$$. If $$\overrightarrow{v}\cdot\overrightarrow{c}=11$$ and the length of the projection of $$\overrightarrow{b}$$ on $$\overrightarrow{c}$$ is p, then $$9p^{2}$$ is equal to
If $$f(x)= \begin{cases}\frac{a|x|+x^{2}-2(\sin|x|)(\cos|x|)}{x} & ,x \neq 0\\b & ,x = 0\end{cases}$$
is continuous at x = 0, then a + b is equal to
Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is $$\dfrac{p}{q},gcd(p,q)=1,$$ then $$p+q$$ is equal to
If $$z = \frac{\sqrt{3}}{2}+\frac{i}{2},i=\sqrt{-1},\text{ then }(z^{201}-i)^{8}\text{ is equal to }$$
The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange , is
Let A = {0 ,1,2,...,9}. Let R be a relation on A defined by (x,y) $$\in$$ R if and only if $$\mid x - y \mid $$ is a multiple of 3.
Given below are two statements:
Statement I: $$n (R) = 36.$$
Statement II: R is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below
Consider two sets $$A=\left\{x\in Z:|(|x-3|-3)\leq1\right\}$$ and $$B=\left\{x \in \mathbb R-\left\{1,2\right\}:\frac{(x-2)(x-4)}{x-1}\log_{e}(|x-2|)=0 \right\}$$. Then the number of onto functions $$f:A\rightarrow B$$ is equal to
If the points of intersection of the ellipses $$x^{2}+2y^{2}-6x-12y+23=0$$ and $$4x^{2}+2y^{2}-20x-12y+35=0$$ lie on a circle of radius r and centre (a, b), then the value of $$ab+18r^{2}$$ is
An equilateral triangle OAB is inscribed in the parabola $$y^{2} = 4x$$ with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is
