List of practice Questions

In a healthy adult, which one of the following regions of the brain contains primarily white matter?
Evaluation of the integral \[ \int \frac{dx}{\sqrt{2x - x^2}} \] results in
If \(\vec{V} = a\hat{i} + b\hat{j} + c\hat{k}\), identify the INVALID operation on \(\vec{V}\).
$x(t)$ is a real continuous-time signal whose magnitude frequency response ($|X(j\omega)|$) is shown below. After sampling $x(t)$ at $100\ \text{rad s}^{-1}$, the spectral point P is down-converted to _________ rad·s$^{-1}$ in the spectrum of the sampled signal.
Discrete signals \( x[n] \) and \( y[n] \) are shown below. The cross-correlation \( r_{xy}[0] \) is _________
A sample of air analyzed at 25\(^\circ\)C and 1 atm pressure is reported to contain 0.04 ppm of SO\(_2\). Atomic mass of S = 32, O = 16. The equivalent SO\(_2\) concentration (in µg/m\(^3\)) will be _________. (round off to the nearest integer)
A parabolic vertical crest curve connects two road segments with grades +1.0% and -2.0%. If a 200 m stopping sight distance is needed for a driver at a height of 1.2 m to avoid an obstacle of height 0.15 m, then the minimum curve length should be _________ \text{m}. (round off to the nearest integer)
Assuming that traffic on a highway obeys the Greenshields model, the speed of a shockwave between two traffic streams (P) and (Q) as shown in the schematic is _________ \text{km/h}. (in integer)
It is given that an aggregate mix has 260 grams of coarse aggregates and 240 grams of fine aggregates. The specific gravities of the coarse and fine aggregates are 2.6 and 2.4, respectively. The bulk specific gravity of the mix is 2.3. The percentage air voids in the mix is _________. (round off to the nearest integer)
The lane configuration with lane volumes in vehicles per hour of a four-arm signalized intersection is shown in the figure. There are only two phases: the first phase is for the East-West and the West-East through movements, and the second phase is for the North-South and the South-North through movements. There are no turning movements. Assume that the saturation flow is 1800 vehicles per hour per lane for each lane and the total lost time for the first and the second phases together is 9 seconds.
The optimum cycle length (in seconds), as per the Webster’s method, is _________. (round off to the nearest integer)
A uniform rod \( KJ \) of weight \( w \) shown in the figure rests against a frictionless vertical wall at the point \( K \) and a rough horizontal surface at point \( J \). It is given that \( w = 10 \, \text{kN} \), \( a = 4 \, \text{m} \), and \( b = 3 \, \text{m} \). The minimum coefficient of static friction that is required at the point \( J \) to hold the rod in equilibrium is _________ (round off to three decimal places).
The activities of a project are given in the following table along with their durations and dependency.
The total float of the activity \( E \) (in days) is _________ (in integer).
A group of total 16 piles are arranged in a square grid format. The center-to-center spacing (s) between adjacent piles is 3 m. The diameter (d) and length of embedment of each pile are 1 m and 20 m, respectively. The design capacity of each pile is 1000 kN in the vertical downward direction. The pile group efficiency \( \eta_g \) is given by: \[ \eta_g = 1 - \frac{\theta}{90} \left[ (n-1)m + (m-1)n \right] \frac{1}{mn} \] Where m and n are the number of rows and columns in the plan grid of pile arrangement, and \( \theta = \tan^{-1} \left( \frac{d}{s} \right) \). The design value of the pile group capacity (in kN) in the vertical downward direction is _________ \text{. (round off to the nearest integer)}
A saturated compressible clay layer of thickness h is sandwiched between two sand layers, as shown in the figure. Initially, the total vertical stress and pore water pressure at point P, which is located at the mid-depth of the clay layer, were 150 kPa and 25 kPa, respectively. Construction of a building caused an additional total vertical stress of 100 kPa at P. When the vertical effective stress at P is 175 kPa, the percentage of consolidation in the clay layer at P is _________ \text{. (in integer)}
A hydraulic jump takes place in a 6 m wide rectangular channel at a point where the upstream depth is 0.5 m (just before the jump). If the discharge in the channel is 30 m\(^3\)/s and the energy loss in the jump is 1.6 m, then the Froude number computed at the end of the jump is _________ (round off to two decimal places).
A pump with an efficiency of 80% is used to draw groundwater from a well for irrigating a flat field of area 108 hectares. The base period and delta for paddy crop on this field are 120 days and 144 cm, respectively. Water application efficiency in the field is 80%. The lowest level of water in the well is 10 m below the ground. The minimum required horse power (h.p.) of the pump is _________ (round off to two decimal places).
Two discrete spherical particles (P and Q) of equal mass density are independently released in water. Particle P and particle Q have diameters of 0.5 mm and 1.0 mm, respectively. Assume Stokes' law is valid. The drag force on particle Q will be _________ \text{ times the drag force on particle P.} (round off to the nearest integer)
At a municipal waste handling facility, 30 metric ton mixture of food waste, yard waste, and paper waste was available. The moisture content of this mixture was found to be 10%. The ideal moisture content for composting this mixture is 50%. The amount of water to be added to this mixture to bring its moisture content to the ideal condition is _________ \text{ metric ton.} (in integer)
A sewage treatment plant receives sewage at a flow rate of 5000 m\(^3\)/day. The total suspended solids (TSS) concentration in the sewage at the inlet of primary clarifier is 200 mg/L. After the primary treatment, the TSS concentration in sewage is reduced by 60 %. The sludge from the primary clarifier contains 2 % solids concentration. Subsequently, the sludge is subjected to gravity thickening process to achieve a solids concentration of 6 %. Assume that the density of sludge, before and after thickening, is 1000 kg/m\(^3\). The daily volume of the thickened sludge (in m\(^3\)/day) will be _________. (round off to the nearest integer)
Let \(y\) be a non-zero vector of size \(2022 \times 1\). Which of the following statement(s) is/are TRUE?
Which of the following statement(s) is/are correct?
Water is flowing in a horizontal, frictionless, rectangular channel. A smooth hump is built on the channel floor at a section and its height is gradually increased to reach choked condition in the channel. The depth of water at this section is \(y_2\) and that at its upstream section is \(y_1\). The correct statement(s) for the choked and unchoked conditions in the channel is/are
A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is _________ (round off to three decimal places).
Consider two linearly elastic rods HI and IJ, each of length b, as shown in the figure. The rods are co-linear, and confined between two fixed supports at H and J. Both the rods are initially stress free. The coefficient of linear thermal expansion is \( \alpha = 10^{-6} \, \degree C^{-1} \), and the temperature of rod IJ is raised by \( \Delta T = 50 \degree C \, \text{, whereas the temperature of rod HI remains unchanged. An external horizontal force P is now applied at node I. It is given that the axial rigidities of the rods HI and IJ are 2AE and AE, respectively.
The linearly elastic planar structure shown in the figure is acted upon by two vertical concentrated forces. The horizontal beams UV and WX are connected with the help of the vertical linear spring with spring constant k = 20 kN/m. The fixed supports are provided at U and X. It is given that flexural rigidity \( EI = 10^5 \) kN-m², P = 100 kN, and a = 5 m. Force \( Q \) is applied at the center of beam WX such that the force in the spring VW becomes zero. The magnitude of force \( Q \) (in kN) is _________ \text{ kN}. (round off to the nearest integer)