List of practice Questions

A 12-hour storm occurs over a catchment and results in a direct runoff depth of 100 mm. The time-distribution of the rainfall intensity is shown in the figure (not to scale). The $\varphi$-index of the storm is (in mm, rounded off to two decimal places): 

A smooth vertical retaining wall supporting layered soils is shown in the figure. According to Rankine's earth pressure theory, the lateral active earth pressure acting at the base of the wall is  kPa (rounded off to one decimal place).

A $5$ cm long metal rod ($x$ in cm from $A$ to $B$) is governed by $\dfrac{\partial T}{\partial t}=D\,\dfrac{\partial^2 T}{\partial x^2}$ with $D=1.0~\text{cm}^2/\text{s}$ and both ends held at $0^\circ$C. The temperature field is \[ T(x,t)=\sum_{n=1,3,5,\ldots} C_n \sin\!\left(\frac{n\pi x}{5}\right)e^{-\beta n^{2} t}. \] Find $\beta$ (in s$^{-1}$, rounded to three decimals). 

Consider the singly reinforced section of a cantilever concrete beam under bending, as shown in the figure (M25 grade concrete, Fe415 grade steel). The stress block parameters for the section at ultimate limit state, as per IS 456: 2000 notations, are given. The ultimate moment of resistance for the section by the Limit State Method is  kN.m (round off to one decimal place).

A 2D thin plate (plane stress) has $E=1.0~\text{N/m}^2$ and Poisson’s ratio $\mu=0.5$. The displacement field is $u=Cx^2y$, $v=0$ (in m). Distances $x,y$ are in m. The stresses are $\sigma_{xx}=40xy~\text{N/m}^2$ and $\tau_{xy}=\alpha x^2~\text{N/m}^2$. Find $\alpha$ (in $\text{N/m}^4$, integer). 

An idealised frame supports a load as shown in the figure. The horizontal component of the force transferred from the horizontal member PQ to the vertical member RS at P is  N (round off to one decimal place).} 

A square footing is to be designed to carry a column load of 500 kN which is resting on a soil stratum having the following average properties: bulk unit weight = 19 kN/m³; angle of internal friction = 0° and cohesion = 25 kPa. Considering the depth of the footing as 1 m and adopting Meyerhof's bearing capacity theory with a factor of safety of 3, the width of the footing (in m) is  (round off to one decimal place)} 
 

A soil having the average properties, bulk unit weight $=19\\{kN/m}^3$, angle of internal friction =25 degree and cohesion $=15\\{kPa}$, is being formed on a rock slope at an inclination of 35 degree with the horizontal. The critical height (in m) of the soil formation up to which it would be stable without failure is (round off to one decimal place). [Assume the soil is formed parallel to the rock bedding plane and there is no ground water effect.] 

The infinitesimal element shown in the figure (not to scale) represents the state of stress at a point in a body. What is the magnitude of the maximum principal stress (in N/mm², in integer) at the point?

The differential equation \(\dfrac{du}{dt} + 2tu^{2} = 1\) is solved by a backward difference scheme. At the \((n-1)\)-th time step, \(u_{n-1}=1.75\) and \(t_{n-1}=3.14\,\text{s}\). With \(\Delta t=0.01\,\text{s}\), find \(u_n-u_{n-1}\) (round off to three decimals).

For a horizontal curve, the radius of a circular curve is 300 m with the design speed 15 m/s. If the allowable jerk is 0.75 m/s$^3$, what is the minimum length (in m, integer) of the transition curve?

The ordinates of a one-hour unit hydrograph (1-hr UH) for a catchment are:

Using superposition, a $D$-hour unit hydrograph is derived. Its ordinates are found to be $3\ \text{m}^3\!/\text{s}$ at $t=1$ hour and $10\ \text{m}^3\!/\text{s}$ at $t=2$ hour. Find the value of $D$ (integer).

A canal supplies water to an area growing wheat over 100 hectares. The duration between the first and last watering is 120 days, and the total depth of water required by the crop is 35 cm. The most intense watering is required over a period of 30 days and requires a total depth of water equal to 12 cm. Assuming precipitation to be negligible and neglecting all losses, the minimum discharge (in m\(^3\)/s, rounded off to three decimal places) in the canal to satisfy the crop requirement is

A drained direct shear test was carried out on a sandy soil. Under a normal stress of 50 kPa, the test specimen failed at a shear stress of 35 kPa. The angle of internal friction of the sample is  degree (round off to the nearest integer).

Consider the fillet-welded lap joint shown in the figure (not to scale). The length of the weld shown is the effective length. The welded surfaces meet at right angle. The weld size is 8 mm, and the permissible stress in the weld is 120 MPa. What is the safe load $P$ (in kN, rounded off to one decimal place) that can be transmitted by this welded joint? 

In the differential equation $\dfrac{dy}{dx} + \alpha x y = 0$, $\alpha$ is a positive constant. If $y=1.0$ at $x=0.0$, and $y=0.8$ at $x=1.0$, the value of $\alpha$ is (rounded off to three decimal places).

The probabilities of occurrences of two independent events \( A \) and \( B \) are 0.5 and 0.8, respectively. What is the probability of occurrence of at least \( A \) or \( B \) (rounded off to one decimal place)?

Consider the Deterministic Finite-state Automaton (DFA) A shown below. The DFA runs on the alphabet \(\{0, 1\}\), and has the set of states \(\{s, p, q, r\}\), with \(s\) being the start state and \(p\) being the only final state.

Which one of the following regular expressions correctly describes the language accepted by A?
 

Consider the following definition of a lexical token id for an identifier in a programming language, using extended regular expressions: \[ \text{letter} \;\;\rightarrow\;\; [A\!-\!Za\!-\!z] \] \[ \text{digit} \;\;\rightarrow\;\; [0\!-\!9] \] \[ \text{id} \;\;\rightarrow\;\; \text{letter (letter | digit)}^* \] Which one of the following Non-deterministic Finite-state Automata with $\epsilon$-transitions accepts the set of valid identifiers? (A double-circle denotes a final state).