At present, a firm is manufacturing \(2000\) items. It is estimated that the rate of change of production \(P\) w.r.t. additional number of workers \(x\) is given by \(\frac{{dp}}{{dx}} = 100 - 12\sqrt x \) . If the firm employs \(25 \) more workers, then the new level of production of itmes is 

If the set of all values of a, for which the equation \(5 x^3-15 x-a=0\) has three distinct real roots, is the interval \((\alpha, \beta)\), then \(\beta-2 \alpha\) is equal to ______

The absolute difference between the squares of the radii of the two circles passing through the point \((-9,4)\) and touching the lines \(x+y=3\) and \(x-y=3\), is equal to ______.

Let \(f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)\) be a polynomial of degree 2, satisfying \(f(x) f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)\). If \(f(K)=-2 K\), then the sum of squares of all possible values of \(K\) is :

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let \(a, b, c\) be the length of three sides of a triangle satisfying the condition \(\left(a^2+b^2\right) x^2-2 b(a+c)\). \(x+\left(b^2+c^2\right)=0\). If the set of all possible values of \(x\) is the interval \((\alpha, \beta)\), then \(12\left(\alpha^2+\beta^2\right)\) is equal to ...........

If \(A=\{x \in R:|x|<2\}\) and \(B=\{x \in R:|x-2| \geq 3\}\) then

Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{ dy }{ dx }+\frac{1}{ x ^{2}-1} y =\left(\frac{ x -1}{ x +1}\right)^{\frac{1}{2}}\), \(x>1\) passing through the point \(\left(2, \sqrt{\frac{1}{3}}\right)\). Then \(\sqrt{7} y (8)\) is equal to.

If \(\lim _{x \rightarrow 0} \frac{\alpha e^{x}+\beta e^{-x}+\gamma \sin x}{x \sin ^{2} x}=\frac{2}{3}\), where \(\alpha, \beta, \gamma \in R\), then which of the following is \(NOT\) correct ?

The area (in sq. units) of the region \(\left\{ {x \in R:x \ge } \right.0,\,y \ge 0,\,y \ge x - 2\,and\,y \le \sqrt x \} \,,\,\) is

Let \(y=y(x)\) be the solution of the differential equation \(x\frac{dy}{dx}-sin~2y=x^{3}(2-x^{3})cos^{2}y,\) \(x\ne0.\) If \(y(2)=x,\) then \(tan(y(1))\) is equal to

If \(0 < a , b < 1,\) and \(\tan ^{-1} a +\tan ^{-1} b =\frac{\pi}{4},\) then the value of \((a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots\) is ..... .

The equation of the plane which contains the \(y-\)axis and passes through the point \((1,2,3)\). is