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where a, b, n and m are natural numbers. What is
the value of
n x a = a^{2}b
n = ab ⟹ n^{2 }= a^{2}b^{2}
m x b = ab
m = ab ⟹ m^{2} = a^{2}b^{2}
= (m×n)(n×m) = (m×n)(n×m)
= (m × n × n × m)
= m^{2} x n^{2}
= a^{2}b^{2} x a^{2}b^{2}
= a^{4}b^{4}
For nonnegative integers, a, b, c, what would be the value of a + b+ c if log a + log b + log c = 0 ?
As a, b, c are nonnegative integers and given log a + log b + log c = 0
log (a*b*c) = log 1
⇒ a = b = c = 1
Which can be possible for simple values.
a = b = c = 1
Hence a + b + c = 1+1+1 = 3
Alternate Method:
Given, a = b = c = 1
log a + log b + log c = 0
As we know log 1 = 0, so each one of them can be zero if a = b = c = 1
log 1 + log 1 + log 1 = 0
By putting a =b= 1 equation satisfies
a + b + c = 1+1+1 = 3
A threemember committee has to be formed from a group of 9 people. How many such distinct committees can be formed?
“Although it does contain some pioneering ideas, one would hardly characterize the work as ____________.” The word that best fills the blank in the above sentence is
Innovative is similar to pioneer.
“His face _________ with joy when the solution of the puzzle was ________ to him.” The words that best fill the blanks in the above sentence are
Shone – It is past – participle and past form of shine.
Shown – To show means to reveal and point out something.
A faulty wall clock is known to gain 15 minutes every 24 hours. It is synchronized to the correct time at 9 AM on 11^{th} July. What will be the correct time to the nearest minute when the clock show 2 PM on 15^{th} July of the same year?
9 AM of 11 July of 2 PM on 15th July = 101 hours
hours of incorrect clock = 24 hours of correct clock
hours of IC = 24 hours of correct clock
1 hour of IC = 96/97 hours of correct clock
101 hour of IC hours of correct clock
= 99.958 hours of correct clock
= 99 hours + 0.95876 * 60 minutes of correct clock
= 99 hours + 57.525 minutes
= 99 hours and approx. 58 minutes
So, correct time will be
2 PM, 11^{th} July + (99 hours and 58 minutes) = 12 : 58 PM on 15^{th} July
Given that 10 for x ≠ y ≠ z, what is the value of the product PQR ?
In manufacturing industries, loss is usually taken to be proportional to the square of the deviation from a target. If the loss is Rs. 4900 for a deviation of 7 units, what would be the loss in Rupees for a deviation of 4 units from the target?
The annual average rainfall in a tropical city is 1000 mm. On a particular rainy day (24 hour period), the cumulative rainfall experienced by the city is shown in the graph. Over the 24hour period, 50% of the rainfall falling on a rooftop, which had an obstructionfree area of 50 m^{2}, was harvested into a tank. What is the total volume of water collected in the tank in liters?
Cumulative rainfall = 300 mm
50% of rainfall
Area = 50 m^{2}
⇒ Volume stored in tank = 150 * 10^{3} * 50 m^{3}
or = 7500 L
Each of the letters in the figure below represents a unique integer from 1 to 9. The letters are positioned in the figure such that each of ( A + B + C) , (C + D + E) , (E + F+ G) and (G + H+ K ) is equal to 13. Which integer does E represent?
A + B + C = C + D + E + E + F + G
= G + H + K = 13
If we add all, we will get = 4 * 13 = 52
But sum of all natural number 1 to 9
A + B + C + C + D + E + E + F + G + G + H + K = 50
A + B + C + D + E + F + G + H + K = 45
Substraction eq. (ii) from (i)
Hence, C + E + G = 7
Also, C + D + E = 13
Substraction eq. (iii) from (iv)
D  G = 6
E = 4
Alternative Method
By checking other equations
The clay mineral, whose structural units are held together by potassium bond is
Kaolinite → Hydrogen bond
Illite → lonic Bond/Potassium Bond
Montmorillonite → Water Bond
A structural member subjected to compression, has both translation and rotation restrained at one end, while only translation is restrained at the other end. As per IS 456 : 2000, the effective length factor recommended for design is
One end is fixed
Other end is pin jointed
Effective length of column (as per IS : 4562000)
= 0.80 L
For a given discharge in an open channel, there are two depths which have the same specific energy. These two depths are known as
Depth with same specific energy are called Alternate depths of flow. It represents a subcritical
depth of flow (Y_{2}) and a supercritical depth of flow (Y_{1} ).
Dupuit’s assumptions are valid for
unconfined aquifer
A vertical load of 10 kN acts on a hinge located at a distance of L /4 from the roller support Q of a beam of length L (see figure).
The vertical reaction at support Q is
Bending moment about hinge point A = 0 (Consider the right hand side of A)
Peak Hour Factor (PHF) is used to represent the proportion of peak subhourly traffic flow within the peak hour. If 15minute subhours are considered, the theoretically possible range of PHF will be
0.25 ≤ PHF_{15} ≤1
A fillet weld is simultaneously subjected to factored normal and shear stresses of 120 MPa and 50 MPa, respectively. As per IS 800 : 2007, the equivalent stress (in MPa, up to two decimal places) is ___________
Factored normal stress, f_{a} = 120 MPa
Factored shear stress, q = 50 MPa
According to IS800 : 2007, clause 10.5.10.1.1
The equivalent stress,
Note : The above check for combination of stresses need not be done for fillet welds where the sum of normal and shear stresses does not exceed [clause 10.5.10.1.2(b)] Hence, sum of normal
and shear stresses = 120 + 50 = 170 MPa ≥ fwd ( = 184.75 MPa)
So, the above check need not be done.
So, weld is designed for a resultant shear stress of (f_{r} )
So, the design stress is however equivalent stress 147.99 MPa.
As per IS 456 : 2000, the minimum percentage of tension reinforcement (up to two decimal places) required in reinforcedconcrete beams of rectangular crosssection (considering effective depth in the calculation of area) using Fe500 grade steel is __________________.
Minimum percentage of steel (for Fe500)
A reinforcedconcrete slab with effective depth of 80 mm is simply supported at two opposite ends on 230 mm thick masonry walls. The centretocentre distance between the walls is 3.3 m. As per IS 456 : 2000, the effective span of the slab (in m, up to two decimal places) is _________.
Effective depth
d = 80 mm
Width of support = 230 mm
c/c distance between walls = 3.30 m
Clear span of slab = 3.30− 0.23= 3.07 m
Effective span
So,
L_{eff }= 3.15 m
Probability (up to one decimal place) of consecutively picking 3 red balls without replacement from a box containing 5 red balls and 1 white ball is _________
Probability,
The contact pressure and settlement distribution for a footing are shown in the figure.
Q. The figure corresponds to a
Rigid footing on granular soil.
A culvert is designed for a flood frequency of 100 years and a useful life of 20 years. The risk involved in the design of the culvert (in percentage, up to two decimal places) is ___________
Risk = The probability of a flood to occur at least once in nsuccessive years.
Which one of the following statements is NOT correct?
Boussinesq’s assumed soil as isotropic hence not applicable for stratified soil.
The initial concavity in the loadpenetration curve of a CBR test is NOT due to
Initial concavity in CBR test due to :
• Improper compaction
• Soft top layer
• Inclined plunger
As per IS 10500:2012, for drinking water in the absence of alternate source of water, the permissible limits for chloride and sulphate, in mg/L, respectively are
As per IS1500: 2012
Permissible limit in absence of alternate source.
The quadratic equation 2x^{2} − 3x + 3 = 0 is to be solved numerically starting with an initial guess x_{0}= 2.The new estimate of x after the first iteration using NewtonRaphson method is _____
Given
The setting time of cement is determined using
Vicat apparatus is used to determine the normal consistency, IST, FST of cement.
A flownet below a dam consists of 24 equipotential drops and 7 flow channels. The difference between the upstream and downstream water levels is 6 m. The length of the flow line adjacent to the toe of the dam at exit is 1 m. The specific gravity and void ratio of the soil below the dam are 2.70 and 0.70, respectively. The factor of safety against piping is
N_{f} = 7 N_{d} = 24 H = 6 m
Critical Hydraulic Gradient
The solution of the equation passing through the point (1,1) is
When
The graph of a function f (x ) is shown in the figure.
For f (x ) to be a valid probability density function, the value of h is
A probability distribution with right skew is shown in the figure.
The correct statement for the probability distribution is
T_{L} <t_{mean}= Curve is skew to right.
mode < mean
i.e., mean > median and mode
As per IRC:372012, in order to control subgrade rutting in flexible pavements, the parameter to be considered is
As per IRC : 372012
Where
N = Number of cumulative standard axle
∈_{v}Vertical strain in the subgrade
All the members of the planar truss (see figure), have the same properties in terms of area of crosssection ( A) and modulus of clasticity ( E ) .
Since member BD neither elongate nor contract.
Hence,
F_{BD} = 0
So there are 2 tension members (AB and DC) and 3 zero force members
The intensity of irrigation for the Kharif season is 50% for an irrigation project with culturable command area of 50,000 hectares. The duty for the Kharif season is 1000 hectare/cumec. Assuming transmission loss of 10%, the required discharge (in cumec, up to two decimal places) at the head of the canal is _______
Culturable command area = 50000 ha
Intensity of irrigation for kharif season = 50%
∴ Area under kharif = 25000 ha
Duty for kharif season = 1000 ha/cumec
∴ Discharge at the head of field
Transmission/conveyance loss = 10%
∴
Discharge at the head of canal
In the figures, Group I represents the atmospheric temperature profiles (P, Q, R and S) and Group II represents dispersion of pollutants from a smoke stack (1, 2, 3 and 4). In the figures of Group I, the dashed line represents the dry adiabatic lapse rate, whereas the horizontal axis represents temperature and the vertical axis represents the altitude.
Q. The correct match is
An 8 m long simply supported elastic beam of rectangular crosssection (100 mm x 200 mm) is subjected to a uniformly distributed load of 10 kN/m over its entire span. The maximum principal stress (in MPa, up to two decimal places) at a point located at the extreme compression of a crosssection and at 2 m from the support is ____________.
So principal stress = 90 N/mm^{2}= 90 MPa
A level instrument at a height of 1.320 m has been placed at a station having a Reduced Level (RL) of 112.565 m. The instrument reads – 2.835 m on a levelling staff held at the bottom of a bridge deck. The RL (in m) of the bottom of the bridge deck is
RL of bottom of bridge deck
= 112.565 + 1.320 + 2.835)
= 116.720 m
A cable PQ of length 25 m is supported at two ends at the same level as shown in the figure. The horizontal distance between the supports is 20 m. A point load of 150 kN is applied at point R which divides it into two equal parts
Neglecting the selfweight of the cable, the tension (in kN, in integer value) in the cable due to applied load will be ________
12.5^{2} = x^{2} + 10 ⇒ x = 7.5 m
Bending moment at S = 0 {Consider the left part}
A group of nine piles in a 3 x 3 square pattern is embedded in a soil strata comprising dense sand underlying recently filled clay layer, as shown in the figure. The perimeter of an individual pile is 126 cm. The size of pile group is 240 cm ´ 240 cm. The recently filled clay has undrained shear strength of 15 kPa and unit weight of 16 kN/m^{3}.
The negative frictional load (in kN, up to two decimal places) acting on the pile group is _____________
pile group
Negative skin friction for group.
weight of soil in negative zone.
= 472.3 kN/m^{2}
Negative skin friction in individual action.
Negative skin friction is maximum of above (two)
Q_{nf} = 472.32 kN/m^{2}
A coal containing 2% sulfur is burned completely to ash in a brick kiln at a rate of 30 kg/min. The sulfur content in the ash was found to be 6% of the initial amount of sulfur present in the coal fed to the brick kiln. The molecular weights of S, H and O are 32, 1 and 16 g/mole, respectively. The annual rate of sulfur dioxide (SO2) emission from the kiln (in tonnes/year, up to two decimal places) is __________
Coal burned in one year = 30 × 24 × 60 × 365
= 1.5768 × 10^{7} kg
Sulfur content 315.36 tonnes/year
Sulfur content in ash = 315.36 tonnes/year
Sulfur converted to SO_{2} = 315.36 – 18.92 = 296.44 tonnes/year
1 mole of S is present in 1 mole of SO_{2}
32 gm of S is present in 64 gm of SO_{2}
∴ Rate of SO_{2} emission = 592.88 tonnes
A 3 m high vertical earth retaining wall retains a dry granular backfill with angle of internal friction of 30^{o} and unit weight of 20 kN/m^{3}. If the wall is prevented from yielding (no movement), the total horizontal thrust (in kN per unit length) on the wall is
Soil is dry sand
K_{0} = 1 − sinØ = 1  sin 30° = 0.5
Total horizontal thrust
A singlyreinforced rectangular concrete beam of width 300 mm and effective depth 400 mm is to be designed using M25 grade concrete and Fe500 grade reinforcing steel. For the beam to be underreinforced, the maximum number of 16 mm diameter reinforcing bars that can be provided is
B = 300 mm
d = 400 mm (effective depth)
M25 and Fe500
= 1142.64 mm^{2}
Number of 16 mm
For maximum number of bars to be provided is 5.
A car follows a slow moving truck (travelling at a speed of 10 m/s) on a twolane twoway highway. The car reduces its speed to 10 m/s and follows the truck maintaining a distance of 16 m from the truck. On finding a clear gap in the opposing traffic stream, the car accelerates at an average rate of 4 m/s2, overtakes the truck and returns to its original lane. When it returns to its original lane, the distance between the car and the truck is 16 m. The total distance covered by the car during this period (from the time it leaves its lane and subsequently returns to its lane after overtaking) is
Overtaking time,
S = Space heady way = 16 m
a = Acceleration = 4 m/s2
Distance travelled by vehicle = S_{2
}
Three soil specimens (Soil 1, Soil 2 and Soil 3), each 150 mm long and 100 mm diameter, are placed in series in a constant head flow setup as shown in the figure. Suitable screens are provided at the boundaries of the specimens to keep them intact. The values of coefficient of permeability of Soil 1, Soil 2 and Soil 3 are 0.01, 0.003 and 0.03 cm/s, respectively.The value of h in the setup is
In normal to bedding plane flow (series arrangement), Discharge will be same and Head loss and Hydraulic gradient will be different.
Total head loss = H_{L} = 560 mm
h = 40 mm
A prismatic propped cantilever beam of span L and plastic moment capacity M_{P} is subjected to a concentrated load at its midspan. If the collapse load of the beam is the value of α is __________
A flocculation tank contains 1800 m^{3} of water, which is mixed using paddles at an average velocity gradient G of 100/s. The water temperature and the corresponding dynamic viscosity are 30^{o}C and 0.798 × 10^{–3} Ns/m^{2}, respectively. The theoretical power required to achieve the stated value of G (in kW, up to two decimal places) is ___________.
Power required
The value (up to two decimal places) of a line integral for along C which is a straight line joining (0,0) to (1,1) is _____________
At a small water treatment plant which has 4 filters, the rates of filtration and backwashing are 200 m^{3} /d/m^{2} and 1000 m^{3} /d/m^{2 }respectively. Backwashing is done for 15 min per day. The maturation, which occurs initially as the filter is put back into service after cleaning, takes 30 min. It is proposed to recover the water being wasted during backwashing and maturation. The percentage increase in the filtered water produced (up to two decimal places) would be ________
Let total area of filters be 1 m^{2}
Water used for backwashing
Water used for maturation
Total water wasted for backwashing and maturation = 10.4166 + 4.166 = 14.58 m^{3}
Water to be treated by filtered
% increase in filtered water produced
% increase in filtered water produced
A prismatic beam PQR of flexural rigidity is subjected to a moment of 180 kNm at Q as shown in the figure.
The rotation at Q (in rad, up to two decimal places) is _________
KQ = K_{QP} + K_{QR
}
Alternate solution
c is taken at a scale of 1:7500. The focal length of the camera is 15 cm. The altitude of the flight above mean sea level (in m, up to one decimal place) is ___________
h = 1400 m
Scale 1 : 7500
f = 15 cm
A rough pipe of 0.5 m diameter, 300 m length and roughness height of 0.25 mm, carries water (kinematic viscosity = 0.9 x 10^{6 }m^{2}s with velocity of 3 m/s. Friction factor ( f ) for laminar flow is given by f = 64 / Re, and for turbulent flow it is given by where, Re = Reynolds number, r = radius of pipe, k = roughness height and g = 9.81 m/s^{2} . The head loss (in m, up to three decimal places) in the pipe due to friction is ________.
Means turbulent flow.
So, = 4.594 m
The total horizontal and vertical stresses at a point X in a saturated sandy medium are 170 kPa and 300 kPa, respectively. The static porewater pressure is 30 kPa. At failure, the excess porewater pressure is measured to be 94.50 kPa, and the shear stresses on the vertical and horizontal planes passing through the point X are zero. Effective cohesion is 0 kPa and effective angle of internal friction is 36^{o}. The shear strength (in kPa, up to two decimal places) at point X is __________.
The matrix has
(complex eigen values)
∴ Complex Eigen values and complex Eigen vectors.
A 7.5 m wide twolane road on a plain terrain is to be laid along a horizontal curve of radius 510 m. For a design speed of 100 kmph, superelevation is provided as per IRC : 731980. Consider acceleration due to gravity as 9.81 m/s^{2}. The level difference between the inner and outer edges of the road (in m, up to three decimal places) is __________
The compression curve (void ratio, e vs. effective stress, for a certain clayey soil is a straight line in a semilogarithmic plot and it passes through the points and The compression index (up to two decimal places) of the soil is ___________
Compression Index
A threefluid system (immiscible) is connected to a vacuum pump. The specific gravity values of the fluids (S_{1},S_{2} ) are given in the figure.
The gauge pressure value (in kN/m^{2}, up to two decimal places) of p_{1} is _________.
Taking P_{1} is in gauge pressure.
The total rainfall in a catchment of area 1000 km^{2} , during a 6 h storm, is 19 cm. The surface runoff due to this storm computed from triangular direct runoff hydrograph is 1 x 10^{8} m^{3}. The or this storm (in cm/h, up to one decimal place) is ________
A schematic flow diagram of a completely mixed biological reactor with provision for recycling of solids is shown in the figure.
S_{0},S = readily biodegradable soluble BOD, mg/L
Q_{r}, Q_{r}, Q_{w} = flow rates,
X_{0} , X_{r}, X_{e}, X_{u} = microorganism concentrations (mixedliquor volatile suspended solids or MLVSS), mg/L
The mean cell residence time (in days, up to one decimal place) is ___________
A 6 m long simply supported beam is prestressed as shown in the figure.
The beam carries a uniformly distributed load 6 kN/m over its entire span. If the effective flexural rigididty EI = 2´ 10^{4} kNm^{2} and the effective prestressing force is 200 kN, the net increase in length of the prestressing cable (in mm, up to two decimal places) is ___________
Span of PSC beam = 6 m
(a) Slope of beam due to Pforce
(b) Slope of beam due to UDL
(c) Net slope of beam
(d) Total net increase in length
Two rigid bodies of mass 5 kg and 4 kg are at rest on a frictionless surface until acted upon by a force of 36 N as shown in the figure. The contact force generated between the two bodies is
As after action of 36 N, both blocks will move with same acceleration so considering 5 kg and 4 kg together in a system and applying Newton’s 2nd law.
N = ma {Newton's 2nd law}
N = 4 x 4 = 16 N
The rank of the following matrix is
No. of non zero rows = 2
rank of A = 2
The Laplace transform F (s ) of the exponential function, f (t ) = e^{at }when where a is a constant and ( s −a)> 0, is
Four bolts P, Q, R and S of equal diameter are used for a bracket subjected to a load of 130 kN as shown in the figure.
The force in bolt P is
In a 5 m wide rectangular channel, the velocity μ distribution in the vertical direction y is given by u The distance y is measured from the channel bed. If the flow depth is 2 m, the discharge per unit width of the channel is
Consider an element dy at distance y from channel bed.
The space mean speed (kmph) and density (vehicles/km) of a traffic stream are linearly related. The free flow speed and jam density are 80 kmph and 100 vehicles/km respectively. The traffic flow (in vehicles/h, up to one decimal place) corresponding to a speed of 40 kmph is _____________.
V_{f} = Free mean speed = 50 kmph
k_{j} = Jam density = 100 veh/km
As per linear model (greenshield)
⇒ Density @ speed = 40 kmph
Traffic flow @ density 50 veh/km @ speed 40 kmph
q = Vk
= 2000 veh/km
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