The maximum area of a triangle whose one vertex is at \((0,0)\) and the other two vertices lie on the curve \(y=-2 x^2+54\) at points \((x, y)\) and \((-x, y)\) where \(\mathrm{y}>0\) is :

The number of points on the curve \(y=54 x^5-\) \(135 x^4-70 x^3+180 x^2+210 x\) at which the normal lines are parallel to \(x+90 y+2=0\) is :

If \(f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _{\varepsilon}(123)}{x \log _{\varepsilon}(1234)-\left(\tan 1^{\circ}\right)}, x > 0\), then the least value of \(f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)\) 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,

The sum of squares of all possible values of \(k\), for which area of the region bounded by the parabolas \(2 \mathrm{y}^2=\mathrm{kx}\) and \(\mathrm{ky}^2=2(\mathrm{y}-\mathrm{x})\) is maximum, is equal to :

The values of \(a\) and \(b\), for which the system of equations    \(2 x+3 y+6 z=8\)   ;  \(x+2 y+a z=5\)     ;  \(3 x+5 y+9 z=b\)  has no solution, are:

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function such that \(f(0)=0, f(1)=1, f(2)=-1, f(3)=2\) and \(f(4)=-2\). Then, the minimum number of zeros of \(\left(3 f^{\prime} f^{\prime \prime}+f f^{\prime \prime \prime}\right)(x)\) is ...........

Let \(g:(0, \infty) \rightarrow R\) be a differentiable function such that  \(\int\left(\frac{x(\cos x-\sin x)}{e^{x}+1}+\frac{g(x)\left(e^{x}+1-x e^{x}\right)}{\left(e^{x}+1\right)^{2}}\right) d x=\frac{x g(x)}{e^{x}+1}+c\) for all \(x >0\), where \(c\) is an arbitrary constant. Then.

The least value of \(|z|\) where \(z\) is complex number which satisfies the inequality \(\exp \left(\frac{(|z|+3)(|z|-1)}{|| z|+1|} \log _{ e } 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i |\) \(i=\sqrt{-1},\) is equal to :

The mean and standard deviation of \(20\) observations were calculated as \(10\) and \(2.5\) respectively. It was found that by mistake one data value was taken as \(25\) instead of \(35 .\) If \(\alpha\) and \(\sqrt{\beta}\) are the mean and standard deviation respectively for correct data, then \((\alpha, \beta)\) is :

The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is

Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\) if the diagonal \(\overrightarrow{B D}=\hat{i}+2 \hat{j}+3 \hat{k}\) then the area of the parallelogram is equal to

Consider ellipses \(E _{ k }: kx ^2+ k ^2 y ^2=1, k =1,2, \ldots\),\(20\). Let \(C _{ k }\) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse \(E_k\), If \(r_k\) is the radius of the circle \(C _{ k }\), then the value of \(\sum \limits_{ k =1}^{20} \frac{1}{ I _{ k }^2}\) is \(.......\).