Let \(y=y(x), x>1\), be the solution of the differential equation \((x-1) \frac{d y}{d x}+2 x y=\frac{1}{x-1}\), with \(y(2)=\frac{1+e^{4}}{2 e^{4}}\). If \(y(3)=\frac{e^{\alpha}+1}{\beta e^{\alpha}}\). then the value of \(\alpha+\beta\) is equal to

The term independent of \(x\) in the binomial expansion of \(\left( {1 - \frac{1}{x} + 3{x^5}} \right){\left( {2{x^2} - \frac{1}{x}} \right)^8}\) is

Let \(P Q\) be a chord of the parabola \(y^2=12 x\) and the midpoint of \(PQ\) be at \((4,1)\). Then, which of the following point lies on the line passing through the points \(\mathrm{P}\) and \(\mathrm{Q}\) ?

\(\lim _{n \rightarrow \infty}\left(\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots+\frac{1}{2 n}\right)\) is equal to :-

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

Let \(e_1\) and \(e_2\) be two distinct roots of the equation \(x^2 - ax + 2 = 0\). Let the sets \(\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of hyperbolas}\} = (\alpha, \beta)\), and \(\{a \in \mathbb{R} : e_1 \text{ and } e_2 \text{ are the eccentricities of an ellipse and a hyperbola, respectively}\} = (\gamma, \infty)\). Then \(\alpha^2 + \beta^2 + \gamma^2\) is equal to:

The number of distinct real solutions of the equation \(x|x+4|+3|x+2|+10=0\) is

Let the point A be the foot of perpendicular drawn from the point \(P(a, b, 0)\) on the line \(\dfrac{x-1}{2} = \dfrac{y-2}{1} = \dfrac{z-\alpha}{3}\). If the midpoint of the line segment PA is \(\left(0, \dfrac{3}{4}, \dfrac{-1}{4}\right)\), then the value of \(a^2 + b^2 + \alpha^2\) is equal to:

Let \(P\) and \(Q\) be two distinct points on a circle which has center at \(C(2,3)\) and which passes through origin \(O\) , If \(O C\) is perpendicular to both the line segments \(C P\) and \(C Q\), then the set \(\{\mathrm{P}, \mathrm{Q}\}\) is equal to:

The area (in sq. units) of the region \(\mathrm{S}=\{\mathrm{z} \in \mathbb{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{lm}(\mathrm{z}) \geq 0\}\) is

If \(0 \le x < 2\pi \) , then the number of real values of \(x,\) which satisfy the equation  \(\cos x + \cos 2x + \cos 3x + \cos 4x = 0\) is  . .  .

The number of real roots of the equation \(\sqrt{x^2-4 x+3}+\sqrt{x^2-9}=\sqrt{4 x^2-14 x+6}\), is:

If \(A=\left[\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right]\) and \(M=A+A^{2}+A^{3}+\ldots .+A^{20}\), then the sum of all the elements of the matrix \(\mathrm{M}\) is equal to \(.....\)