ISRO Scientist or Engineer Civil 2020

Consider a $$3 \times 3$$ symmetric matrix A such that two of its Eigen values are a $$\neq$$ 0 and b $$\neq$$ 0with respective Eigen vectors $$\begin{bmatrix}x_{1} \\x_{2}\\ x_{3} \end{bmatrix} $$ , $$\begin{bmatrix}y_{1} \\y_{2}\\ y_{3} \end{bmatrix} $$.  If a $$\neq$$ b, then $$x_{1} y_{1}+ x_{2} y_{2}+ x_{3} y_{3 }$$ equals

The area enclosed between the parabola $$y=x^{2}$$ and the straight line y=x is

The right circular cone of largest voume that can be enclosed by a sphere of 1 m radius has a height of

Consider the function f(x) =$$2x^{3}-3x^{2}$$ in the domain [-1,2]. The global minimum of f(x) is

The solution of $$x\frac{dy}{dx} + y = x^{4}$$ with the condition $$y(1) = \frac{6}{5}$$ is:

The inverse Laplace transform of the function $$F(s) = \frac{1}{S(S+1)}$$ is

Laplace transform of $$e^{-2t} \cos (4t)$$

A box contains 2 washers,3 nuts and 4bolts . Items are drawn from the box at random ,one at a time with out replacement .The probabilty of drawing 2 washers first followed by 3 nuts and subsequently the 4 bolts is

Consider an unbiased cubic dice with opposite faces coloured identically and each face coloured red, blue and green such that each color appears only two times on the dice .If the dice thrown thrice, the probabilty of obtaining red colour on the top face of dice at least twice is

The argument of the complex number $$\frac{1+i}{1-i}$$, where $$i = \sqrt{-1}$$, is