The directional derivative of the function \( f \) given below at the point \( (1, 0) \) in the direction of \( \frac{1}{2} (\hat{i} + \sqrt{3} \hat{j}) \) is (rounded off to 1 decimal place). \[ f(x, y) = x^2 + xy^2 \]
If \( C \) is the unit circle in the complex plane with its center at the origin, then the value of \( n \) in the equation given below is (rounded off to 1 decimal place). \[ \int_C \frac{z^3}{(z^2 + 4)(z^2 - 4)} \, dz = 2 \pi i n \]
The system shown in the figure below consists of a cantilever beam (with flexural rigidity \( EI \) and negligible mass), a spring (with spring constant \( K \) and negligible mass) and a block of mass \( m \). Assuming a lumped parameter model for the system, the fundamental natural frequency (\( \omega_n \)) of the system is
In relation to additive manufacturing, match the following:
Match the mold elements in the casting process with the most suitable function:
Considering the actual demand and the forecast for a product given in the table below, the mean forecast error and the mean absolute deviation, respectively, are:
During a welding operation, thermal power of 2500 W is incident normally on a metallic surface. As shown in the figure below (figure is NOT to scale), the heated area is circular. Out of the incident power, 85% of the power is absorbed within a circle of radius 5 mm while 65% is absorbed within an inner concentric circle of radius 3 mm. The power density in the shaded area is _________ W mm^-2 (rounded off to 2 decimal places).
The values of a function \( f \) obtained for different values of \( x \) are shown in the table below.
Using Simpson’s one-third rule, approximate the integral \[ \int_0^1 f(x) \, dx \quad \text{(rounded off to 2 decimal places)}. \]
A truss structure is loaded as shown in the figure below. Among the options given, which member in the truss is a zero-force member?
\[ {Given: } F = 1000\,{N} \]
A rigid circular disc of radius \(r\) (in m) is rolling without slipping on a flat surface as shown in the figure below. The angular velocity of the disc is \(\omega\) (in rad/ssuperscript{-1}). The velocities (in m/ssuperscript{-1}) at points 0 and A, respectively, are:
Consider two identical tanks with a bottom hole of diameter \( d \). One tank is filled with water and the other tank is filled with engine oil. The height of the fluid column \( h \) is the same in both cases. The fluid exit velocity in the two tanks are \( V_1 \) and \( V_2 \). Neglecting all losses, which one of the following options is correct?
