Let \(A (0,1), B (1,1)\) and \(C (1,0)\) be the mid - points of the sides of a triangle with incentre at the point D. If the focus of the parabola \(y^2=4 a x\) passing through \(D\) is \((\alpha+\beta \sqrt{2}, 0)\), where \(\alpha\) and \(\beta\) are rational numbers, then \(\frac{\alpha}{\beta^2}\) is equal to

For \(0<\mathrm{c}<\mathrm{b}<\mathrm{a}\), let \((\mathrm{a}+\mathrm{b}-2 \mathrm{c}) \mathrm{x}^2+(\mathrm{b}+\mathrm{c}-2 \mathrm{a}) \mathrm{x}\) \(+(c+a-2 b)=0\) and \(\alpha \neq 1\) be one of its root. Then, among the two statements \((I)\) If \(\alpha \in(-1,0)\), then \(\mathrm{b}\) cannot be the geometric mean of \(\mathrm{a}\) and \(\mathrm{c}\) \((II)\) If \(\alpha \in(0,1)\), then \(\mathrm{b}\) may be the geometric mean of \(a\) and \(c\)

Let \(\alpha\) and \(\beta\) be the roots of the equation \(\mathrm{px}^2+\mathrm{qx}-\) \(r=0\), where \(p \neq 0\). If \(p, q\) and \(r\) be the consecutive terms of a non-constant G.P and \(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}\), then the value of \((\alpha-\beta)^2\) is :

Line \(L_1\) passes through the point \((1,2,3)\) and is parallel to Z -axis. Line \(\mathrm{L}_2\) passes through the point \((\lambda, 5,6)\) and is parallel to \(y\)-axis. Let for \(\lambda=\lambda_1, \lambda_2, \lambda_2 \lt \lambda_1\), the shortest distance between the two lines be 3 . Then the square of the distance of the point \(\left(\lambda_1, \lambda_2, 7\right)\) from the line \(\mathrm{L}_1\) is

Let \(P\) be a plane \(l x+m y+n z=0\) containing the line, \(\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3} .\) If plane \(P\) divides the line segment \(AB\) joining points \(A (-3,-6,1)\) and \(B (2,4,-3)\) in ratio \(k : 1\) then the value of \(k\) is equal to

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______

Let the plane \(P\) pass through the intersection of the planes \(2 x+3 y-z=2\) and \(x+2 y+3 z=6\), and be perpendicular to the plane \(2 x+y-z+1=0\). If \(d\) is the distance of \(P\) from the point \((-7,1,1)\), then \(d ^2\) is equal to :

Equation of the line of the shortest distance between the lines \(\frac{x}{1} = \frac{y}{{ - 1}} = \frac{z}{1}\) and \(\frac{{x - 1}}{0} = \frac{{y + 1}}{{ - 2}} = \frac{z}{1}\) is

If \(\alpha+\beta+\gamma=2 \pi\), then the system of equations \(x+(\cos \gamma) y+(\cos \beta) z=0\) \((\cos \gamma) x+y+(\cos \alpha) z=0\) \((\cos \beta) x+(\cos \alpha) y+z=0\) has :

Let sets \(A\) and \(B\) have \(5\) elements each. Let the mean of the elements in sets \(A\) and \(B\) be \(5\) and \(8\) respectively and the variance of the elements in sets \(A\) and \(B\) be \(12\) and \(20\) respectively \(A\) new set \(C\) of \(10\) elements is formed by subtracting \(3\) from each element of \(A\) and adding 2 to each element of B. Then the sum of the mean and variance of the elements of \(C\) is \(.......\).

 Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}=\frac{y}{x}\left(1+x y^2\left(1+\log _e x\right)\right)\) \(x > 0, y(1)=3\). Then \(\frac{y^2(x)}{9}\) is equal to :

A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point \(A\) on the path, he observes that the angle of elevation of the top of the pillar is \(30^o .\) After walking for \(10\) minutes from \(A\) in the same direction, at a point \(B,\) he observes that the angle of elevation of the top of the pillar is \(60^o .\) Then the time taken (in minutes) by him, from \(B\) to reach the pillar, is:

If the tangents drawn at the point \(O (0,0)\) and \(P (1+\sqrt{5}, 2)\) on the circle \(x ^{2}+ y ^{2}-2 x -4 y =0\) intersect at the point \(Q\), then the area of the triangle \(OPQ\) is equal to