For the following questions answer them individually
A rectangle is formed by the lines x= O, y = O, x=3 and y = 4. Let the line L be perpendicular to 3x +y + 6 = 0 and divide the area of the rectangle into two equal parts. Then the distance of the point $$\left(\frac{1}{2},-5\right)$$ from the line L is equal to :
Let $$\alpha$$ and $$\beta$$ respectively be the maximum and the minimum values of the function $$f(0)=4\left(\sin^{4}\left(\frac{7\pi}{2}-\theta\right)+\sin^{4}(11\pi+\theta)\right)-2\left(\sin^{6}\left(\frac{3\pi}{2}-\theta\right)+\sin^{6}(9\pi-\theta)\right),\theta\epsilon R$$. Then $$\alpha+2\beta$$ is equal to:
Let the direction cosines of two lines satisfy the equations : 4l + m - n =0 and 2mn + l0nl +3lm= 0. Then the cosine of the acute angle between these lines is :
If $$\alpha$$ and $$\beta$$ ($$\alpha < \beta$$) are the roots of the equation $$(-2+\sqrt{3})(|\sqrt{x}-3|)+(x-6\sqrt{x})+(9-2\sqrt{3})=0,x\geq0\text{ then }\sqrt{\frac{\beta}{\alpha}}+\sqrt{\alpha\beta}$$ is equal to:
Let the mean and variance of 8 numbers - 10, - 7, - 1, x, y, 9, 2, 16 be $$\frac{7}{2}\text{ and }\frac{293}{4}$$ respectively.
Then the mean of 4 numbers x, y, x + y + 1, |x-y| is:
Among the statements :
I: If $$ \begin{vmatrix}\mathbf{1} & \mathbf{\cos\alpha} & \mathbf{\cos\beta} \\\mathbf{\cos\alpha} & \mathbf{1} & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & \mathbf{1}\end{vmatrix}=\begin{vmatrix}\mathbf{0} & \mathbf{\cos\alpha}&\mathbf{\cos\beta} \\\mathbf{\cos\alpha} & \mathbf{0} & \mathbf{\cos\gamma} \\\mathbf{\cos\beta} & \mathbf{\cos\gamma} & \mathbf{0}\end{vmatrix}$$, then $$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\frac{3}{2},and$$
II: $$\begin{vmatrix}\mathbf{x^{2}-x} & \mathbf{x+1} & \mathbf{x-2} \\\mathbf{2x^{2}+3-1} & \mathbf{3x} & \mathbf {3x-3} \\\mathbf{x^{2}+2x+3} & \mathbf{2x-1} & \mathbf{2x-1}\end{vmatrix}$$ = px + q, then $$p^{2}=196q^{2}$$
Let the line y - x = l intersect the ellipse $$\frac{x^{2}}{2}+\frac{y^{2}}{1}=$$ at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:
Let $$f(x)=\int\frac{(2-x^{2}).e^{x}}{(\sqrt{1+x})(1-x)^{\frac{3}{2}}}dx$$. If f(0)=0, then $$f\left(\frac{1}{2}\right)$$ is equal to:
Let A= {- 2, - 1, 0, 1, 2, 3, 4}. Let R be a relation on A defined by xRy if and only if $$$$ Let l be the number of elements in R. Let m and n be the minimun number of elements required to be added in R to make it reflexive and symmetric relations respectively. Then l+ m + n is equal to:
Let y = y(x) be the solution of the differential equation $$x^{4}dy+(4x^{3}y+2\sin x)dx=0,x > 0,y\left(\frac{\pi}{2}\right)=0$$. Then $$\pi^{4}y\left(\frac{\pi}{3}\right)$$ is equal to :
