GATE IN 2016 Question Paper (31-Jan-2016) (Shift 2)

Let f : [-1, 1] → ℝ, where f(x) = 2x³ – x⁴ – 10. The minimum value of f(x) is ______ .

An urn contains 5 red and 7green balls. A ball is drawn at random and its colour is noted. The ball is placed back into the urn along with another ball of the same colour. The probability of getting a red ball in the next draw is

Consider the matrix A = ≤ft( {*20c 2&1&1\\ 2&3&4\\ - 1& - 1& - 2 } ) whose eigenvalues are 1, -1 and 3. Then Trace of (A³ – 3A²) is _________.

The relationship between the force f(t) and the displacement x(t) of a spring-mass system (with a mass M, viscous damping D and spring constant K) is

$M\frac{d^{2}x\left(t\right)}{d{t}^2}+D\frac{dx\left(t\right)}{dt}+Kx\left(t\right)=f\left(t\right)$

X(s) and F(s) are the Laplace transforms of x(t) and f(t) respectively. With M = 0.1, D = 2, K = 10 in appropriate units, the transfer function $G\left(s\right)=\frac{X\left(s\right)}{F\left(s\right)}is$

The value of the integral 1/2π i _C^ {z^2 + 1}{z^2 - 1}dz where z is a complex number and C is a unit circle with centre at 1 + 0j in the complex plane is __________ .