Let $X$ be a continuous random variable defined on $[0, 1]$ such that its probability density function $f(x) = 1$ for $0 \leq x \leq 1$ and $0$ otherwise. Let $Y = \log_e (X + 1)$. Then the expected value of $Y$ is _____ . (rounded off to 2 decimal places)
A ram in the form of a rectangular body of size $l = 9 \text{ m}$ and $b = 2 \text{ m}$ is suspended by two parallel ropes of lengths $7 \text{ m}$. Assume the center-of-mass of the body is at its geometric center and $g = 9.81 \text{ m/s}^2$. For striking the object P with a horizontal velocity of $5 \text{ m/s}$, what is the angle $\theta$ with the vertical from which the ram should be released from rest?
![GATE ME 2024 Engineering Mechanics - Engineering Mechanics Static and Dynamics Question 17 English](https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lx6r8vnx/c6e2d554-95c7-458f-8617-31882ab74a98/02c776e0-25f0-11ef-b9f4-0b0f0dea04e0/file-6y3zli1lx6r8vny.png?format=png)
A rigid massless tetrahedron is placed such that vertex O is at the origin and the other three vertices A, B, and C lie on the coordinate axes as shown in the figure. The body is acted on by three point loads, of which one is acting at A along x-axis and another at point B along y-axis. For the body to be in equilibrium, the third point load acting at point O must be
![GATE ME 2024 Engineering Mechanics - Engineering Mechanics Static and Dynamics Question 16 English](https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lx6rd87d/d89238f7-f720-4d4a-96fb-8fd1cee97274/7bb3bb90-25f0-11ef-b9f4-0b0f0dea04e0/file-6y3zli1lx6rd87e.png?format=png)
A three-hinge arch ABC in the form of a semi-circle is shown in the figure. The arch is in static equilibrium under vertical loads of $P = 100$ kN and $Q = 50$ kN. Neglect friction at all the hinges. The magnitude of the horizontal reaction at B is _________ kN (rounded off to 1 decimal place).
![GATE ME 2024 Engineering Mechanics - Engineering Mechanics Static and Dynamics Question 15 English](https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lx72ywot/5741a425-839d-479a-8d97-3e4186c4ccf1/db074ad0-261d-11ef-a279-4f9760cfcb83/file-6y3zli1lx72ywou.png?format=png)