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QUESTION: 1

The perimeters of a circle, a square and an equilateral triangle are equal. Which one ofthe following statements is true?

Solution:

Let us take a circle, square and an equilateral triangle each of equal perimeter say 100m.

πD = 4a = 3s = 100

D=31.84; a=25; s=33.33;

a^{2} = 625

Hence circle has largest area.

QUESTION: 2

The value of the expression

Solution:

QUESTION: 3

Find the missing group of letters in the following series:

BC, FGH, LMNO, _________

Solution:

QUESTION: 4

“The dress ________ her so well that they all immediately _________ her on her appearance.”The words that best fill the blanks in the above sentence are

Solution:

Complement: a thing that contributes extra features to something else in such a way as to improve or emphasize its quality.

Compliment: a polite expression of praise or admiration

QUESTION: 5

“The judge’s standing in the legal community, though shaken by false allegations of wrongdoing, remained __________.”The word that best fills the blank in the above sentence is

Solution:

Even though there were false allegations, but judge’s standing remained same.

Undiminished: Not reduced.

QUESTION: 6

A house has a number which needs to be identified. The following three statements are given that can help in identifying the house number.

i. If the house number is a multiple of 3, then it is a number from 50 to 59.

ii. If the house number is NOT a multiple of 4, then it is a number from 60 to 69.

iii. If the house number is NOT a multiple of 6, then it is a number from 70 to 79.

Q. What is the house number?

Solution:

I. 51,54,57, but none of them is multiple of 4, hence none is valid

II. 61, 62, 63, 65, 66, 67, 69: 63, 66 and 69 are multiple of three hence not valid.

Rest 61, 62, 65, 67 are not a multiple of 6, Hence not valid

III. 70, 71, 73, 74, 75, 76, 77, 79: 75 is multiple of three, hence not valid.

Except 76 none is a multiple of 4, hence only 76 is a valid entry.

QUESTION: 7

Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?

Solution:

Total student = 40

13 − x +16 − x +19 − x + x = 40

Students watches all three movies,

x = 4

QUESTION: 8

An unbiased coin is tossed six times in a row and four different such trials areconducted. One trial implies six tosses of the coin. If H stands for head and T stands fortail, the following are the observations from the four trials:

(1) HTHTHT

(2) TTHHHT

(3) HTTHHT

(4) HHHT ___ ___.

Which statement describing the last two coin tosses of the fourth trial has the highest

probability of being correct?

Solution:

Since the coin is unbiased, the probability of getting heads is equal of tail.

In unbiased coin previous trials does not matter.

Probability of getting both heads =

Probability of getting a head and a tail( HT+TH)

Probability of getting both tails

QUESTION: 9

A wire would enclose an area of 1936 m^{2} , if it is bent into a square. The wire is cut intotwo pieces. The longer piece is thrice as long as the shorter piece. The long and the shortpieces are bent into a square and a circle, respectively. Which of the following choicesis closest to the sum of the areas enclosed by the two pieces in square meters?

Solution:

Area = 1936 m^{2}

a2 = 1936 m^{2}

a = 44 m^{2}

length of wire = 4a

= 4 x 44 = 176 m

part-1 length = 3 x 44 = 132 m

part-2 length = 1 x 44 = 44 m

Long wire is bent in square.

4a = 132

a = 33 m

Area of Square = 33^{2 = }1089 m^{2}

Now, small wire is bent in circle,

So, πD = 44

D = 44

Area of circle

= 153.94m^{2}

Total area enclosed Area of square Area of circle

= 1089 + 153.94

= 1242.97≈1243m^{2}

QUESTION: 10

A contract is to be completed in 52 days and 125 identical robots were employed, each

operational for 7 hours a day. After 39 days, five-seventh of the work was completed.

How many additional robots would be required to complete the work on time, if each

robot is now operational for 8 hours a day?

Solution:

Additional robots required = 131.25 − 125 = 6.25 ≈ 7

QUESTION: 11

A frictionless gear train is shown in the figure. The leftmost 12-teeth gear is given a torque of 100 N-m. The output torque from the 60-teeth gear on the right in N-m is

Solution:

T_{4} = 2000

QUESTION: 12

For an ideal gas with constant properties undergoing a quasi-static process, which one of the following represents the change of entropy from state 1 to 2?

Solution:

Standard Equation for change of entropy from state 1 to 2

QUESTION: 13

Select the correct statement for 50% reaction stage in a steam turbine.

Solution:

In Parsons Reaction Turbine (50% reaction), α_{2} = β_{1}

QUESTION: 14

In a single degree of freedom underdamped spring-mass-damper system as shown in

the figure, an additional damper is added in parallel such that the system still remains

underdamped. Which one of the following statements is ALWAYS true?

Solution:

Transmissibility can increase or decrease depending upon the frequency of oscillation.

But due to under-damping, frequencydecreases, hence time period of free oscillations will increase.

QUESTION: 15

The divergence of the vector fieldis

Solution:

*Answer can only contain numeric values

QUESTION: 16

Fatigue life of a material for a fully reversed loading condition is estimated from

σ = 1100 N^{-0.15},

Where, σ_{a} is the stress amplitude in MPa and is the failure life in cycles. The maximum

allowable stress amplitude (in MPa) for a life of 1 x 10^{5 }cycles under the same loading

condition is __________ (correct to two decimal places).

Solution:

For a completely reversed loading, amplitude of stress is maximum stress.

σ = 1100 × (10^{5})−0.15

σ = 195.61 MPa

*Multiple options can be correct

QUESTION: 17

Denoting L as liquid and M as solid in a phase-diagram with the subscripts representing different phases, a eutectoid reaction is described by

Solution:

A and D both

Option A represents cooling, while option D represents heating.

QUESTION: 18

Metal removal in electric discharge machining takes place through

Solution:

Metal removal in electric discharge machining takes place through melting and vaporization.

QUESTION: 19

Match the following products with the suitable manufacturing process

Solution:

QUESTION: 20

Pre-tensioning of a bolted joint is used to

Solution:

Pre-tensioning of bolt increases the stiffness of bolted joint.

QUESTION: 21

The peak wavelength of radiation emitted by a black body at a temperature of 2000 K is 1.45 um. If the peak wavelength of emitted radiation changes to 2.90 um, then the temperature (in K) of the black body is

Solution:

λ_{m}T = constant

2000 × 1.45 = 2.9 × T

T=1000

*Answer can only contain numeric values

QUESTION: 22

A hollow circular shaft of inner radius 10 mm, outer radius 20 mm and length 1 m is to be used as a torsional spring. If the shear modulus of the material of the shaft is 150 GPa, the torsional stiffness of the shaft (in kN-m/rad) is __________ (correct to two decimal places).

Solution:

QUESTION: 23

The Fourier cosine series for an even function is given by

The value of the coefficient a_{2} for the function

Solution:

QUESTION: 24

- If y is the solution of the differential equation the value of y (−1)

is

Solution:

*Answer can only contain numeric values

QUESTION: 25

If A = then det (A^{-1}0 is _________ (correct to two decimal places).

Solution:

QUESTION: 26

Consider a function u which depends on position x and time t. The partial differential equation

is known as the

Solution:

The above equation is known as heat equation.

*Answer can only contain numeric values

QUESTION: 27

The viscous laminar flow of air over a flat plate results in the formation of a boundary layer. The boundary layer thickness at the end of the plate of length L is . When the plate length is increased to twice its original length, the percentage change in laminar boundary layer thickness at the end of the plate (with respect to ) is (correct to two

decimal places).

Solution:

Percentage change=

QUESTION: 28

Feed rate in slab milling operation is equal to

Solution:

Feed rate in slab milling= f_{t}ZN

Where, f_{t}=feed/tooth

Z= number of teeth

N= rpm

QUESTION: 29

The minimum axial compressive load, P, required to initiate buckling for a pinnedpinned

slender column with bending stiffness EI and length L is

Solution:

Pinned-pinned column means hinged on both sides. And the minimum load required to buckle is given by

QUESTION: 30

During solidification of a pure molten metal, the grains in the casting near the mouldwall are

Solution:

Grain orientation is random at time of solidification because of many different

nucleation sites. Coarse grain structure is obtained after solidification at slower rates.

QUESTION: 31

A ball is dropped from rest from a height of 1 m in a frictionless tube as shown in the figure. If the tube is approximated by two straight lines (ignoring the curved portion), the total distance travelled (in m) by the ball is ___________ (correct to two decimal places).

Solution:

Since the tube is frictionless, the ball will travel to and fro, up-down motion without being stopped, hence the total distance travelled by ball is infinite.

*Answer can only contain numeric values

QUESTION: 32

An engine operates on the reversible cycle as shown in the figure. The work output from the engine (in kJ/cycle) is _________ (correct to two decimal places).

Solution:

Work Output from cycle= Area under PV curve.

=62.5

QUESTION: 33

The preferred option for holding an odd-shaped workpiece in a centre lathe is

Solution:

A four jaw chuck is preferred to handle odd-shaped work piece in centre lathe, because the motion of 4 different jaws are not dependent on each other, hence most of the odd shapes can be fit in.

QUESTION: 34

A local tyre distributor expects to sell approximately 9600 steel belted radial tyres nextyear. Annual carrying cost in Rs. 16 per tyre and ordering cost is Rs. 75. The economicorder quantity of the tyres is

Solution:

*Answer can only contain numeric values

QUESTION: 35

The arrival of customers over fixed time intervals in a bank follow a Poisson distribution with an average of 30 customers/hour. The probability that the time between successive customer arrivals is between 1 and 3 minutes is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 36

A bar is subjected to a combination of a steady load of 60 kN and a load fluctuating between -10 kN and 90 kN. The corrected endurance limit of the bar is 150 MPa, the yield strength of the material is 480 MPa and the ultimate strength of the material is600 MPa. The bar cross-section is square with side a. If the factor of safety is 2, the value

of a (in mm), according to the modified Goodman’s criterion, is __________ (correct to two decimal places).

Solution:

Solution by Goodman Equation,

Solution by Langar euation,

Hence final answer by modified Goodman’s Griterion is 31.62 mm.

*Answer can only contain numeric values

QUESTION: 37

A thin-walled cylindrical can with rigid end caps has a mean radius R = 100 mm and a wall thickness of t = 5 mm. The can is pressurized and an additional tensile stress of 50 MPa is imposed along the axial direction as shown in the figure. Assume that the state of stress in the wall is uniform along its length. If the magnitudes of axial and

circumferential components of stress in the can are equal, the pressure (in MPa) inside the can is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 38

The true stress (in MPa) versus true strain relationship for a metal is given byThe cross-sectional area at the start of test (when the stress and strain values are equal to zero) is 100 mm^{2} . The cross-sectional area at the time of necking (in mm^{2}) is __________ (correct to two decimal places).

Solution:

At necking, true strain equals the strain hardening exponent.

*Answer can only contain numeric values

QUESTION: 39

Air is held inside a non-insulated cylinder using a piston (mass M=25 kg and area A=100 cm^{2}) and stoppers (of negligible area), as shown in the figure. The initial pressure i P and temperature T_{i} of air inside the cylinder are 200 kPa and 400°C, respectively. The ambient pressure P_{∞} and temperature T_{∞} are 100 kPa and 27°C, respectively. The temperature of the air inside the cylinder (°C) at which the piston will begin to move is __________ (correct to two decimal places).

Solution:

Using pressure balance, Pressure at which the piston will start to move,

Since volume and mass remains constant.

*Answer can only contain numeric values

QUESTION: 40

A force of 100 N is applied to the centre of a circular disc, of mass 10 kg and radius 1 m, resting on a floor as shown in the figure. If the disc rolls without slipping on the floor, the linear acceleration (in m/s^{2}) of the centre of the disc is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 41

Steam in the condenser of a thermal power plant is to be condensed at a temperature of 30°C with cooling water which enters the tubes of the condenser at 14°C and exits at 22°C.Overall heat transfer coefficient is 2000W/m^{2}K and the total surface area of the tubes is 50 m^{2}. Net heat transfer (in MW) is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 42

A steel wire is drawn from an initial diameter (d_{i} ) of 10 mm to a final diameter (d_{f}) of 7.5 mm. The half cone angle (α) of the die is 5° and the coefficient of friction (μ ) between the die and the wire is 0.1. The average of the initial and final yield stress is 350 MPa. The equation for drawing stress σ_{f} , (in MPa) is given as:

The drawing stress (in MPa) required to carry out this operation is _________ (correct to two decimal places).

Solution:

QUESTION: 43

A bimetallic cylindrical bar of cross sectional area 1 m^{2} is made by bonding Steel (Young’s modulus = 210 GPa) and Aluminium (Young’s modulus = 70 GPa) as shown in the figure. To maintain tensile axial strain of magnitude 10^{−6} in Steel bar and compressive axial strain of magnitude10^{−6} in Aluminum bar, the magnitude of the

required force P (in kN.) along the indicated direction is

Solution:

QUESTION: 44

Let z be a complex variable. For a counter-clockwise integration around a unit circle C,

centred at origin,

the value of A is

Solution:

QUESTION: 45

In a rigid body in plane motion, the point R is accelerating with respect to point P at m/s . o is zero If the instantaneous acceleration of point Q is zero, the acceleration (in m/s^{2}) of point R is

Solution:

Since acceleration of point Q is zero, so body PQR is hinged at Q.

*Answer can only contain numeric values

QUESTION: 46

Given the ordinary differential equation

the value of y (1) is _________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 47

A circular hole of 25 mm diameter and depth of 20 mm is machined by EDM process. The material removal rate (in mm^{3}/min) is expressed as

where I = 300 A and the melting point of the material .T =1600°C The time (in minutes) for machining this hole is __________ (correct to two decimal places).

Solution:

QUESTION: 48

In a cam-follower, the follower rises by h as the cam rotates by (radians) at constant angular velocity ω (radians/s). The follower is uniformly accelerating during the first half of the rise period and it is uniformly decelerating in the latter half of the rise period.

Assuming that the magnitudes of the acceleration and deceleration are same, the maximum velocity of the follower is

Solution:

Since the cam is having uniform velocity, first half of rise will be accelerating, other half will be decelerating. Hence velocity will be maximum at height=h/2

QUESTION: 49

Let X_{1} and X_{2} be two independent exponentially distributed random variables with means 0.5 and 0.25, respectively. Then is

Solution:

QUESTION: 50

A bar of circular cross section is clamped at ends P and Q as shown in the figure. A torsional moment T = 150 Nm is applied at a distance of 100 mm from end P. The torsional reactions (T_{P},T_{Q}) in Nm at the ends P and Q respectively are

Solution:

Since, G and are constant for bar,

T_{1}L_{1}=T_{2}L_{2}

T_{1}.100=T_{2}.200

T_{1}=2T_{2}

And, T_{1} + T_{2} = T

Hence, T_{1} = 100nm T_{2} = 50Nm

QUESTION: 51

Air flows at the rate of 1.5 m^{3} /s through a horizontal pipe with a gradually reducing cross-section as shown in the figure. The two cross-sections of the pipe have diameters of 400 mm and 200 mm. Take the air density as 1.2kg/m^{3 }and assume inviscid incompressible flow. The change in pressure (p_{2} - p1) (in kPa) between sections 1 and 2 is

Solution:

QUESTION: 52

The problem of maximizing z= x_{1} - x_{2} subject to constraints and x_{2} ≤ 5 has

Solution:

The problem has one solution

*Answer can only contain numeric values

QUESTION: 53

A standard vapor compression refrigeration cycle operating with a condensing temperature of 35^{o}C and an evaporating temperature of -10^{o}C develops 15 kW of cooling. The p-h diagram shows the enthalpies at various states. If the isentropic efficiency of the compressor is 0.75, the magnitude of compressor power (in kW) is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 54

A 0.2 m thick infinite black plate having a thermal conductivity of 3.96 W/m-K is exposed to two infinite black surfaces at 300 K and 400 K as shown in the figure. At steady state, the surface temperature of the plate facing the cold side is 350 K. The value of Stefan-Boltzmann constant, σ, is 5.67 x 10^{-8 }W/m^{2 }K^{4} . Assuming 1-D heat conduction, the magnitude of heat flux through the plate (in w/m2) is __________ (correct to two decimal places).

Solution:

Since there is steady state, the heat transfer between plate and surface is equal to heat transfer between the two surfaces, which is equal to heat transferred through the plate.

Heat flux between plate and surface, q" =

∈_{1}=∈_{2}= 1, Surface are black

q" = 391.612 W/m^{2}

*Answer can only contain numeric values

QUESTION: 55

Following data correspond to an orthogonal turning of a 100 mm diameter rod on a lathe. Rake angle: +15^{o}; Uncut chip thickness: 0.5 mm; nominal chip thickness after the cut: 1.25 mm. The shear angle (in degrees) for this process is _________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 56

Taylor’s tool life equation is used to estimate the life of a batch of identical HSS twist drills by drilling through holes at constant feed in 20 mm thick mild steel plates. In test 1, a drill lasted 300 holes at 150 rpm while in test 2, another drill lasted 200 holes at 300 rpm. The maximum number of holes that can be made by another drill from the above batch at 200 rpm is __________ (correct to two decimal places).

Solution:

Here tool life is represented by number of holes (H) and speed is represented by rpm (N)

Taylors Tool life equation:

Since all the conditions are same for different batches, except rpm and number of holes,

QUESTION: 57

A rigid rod of length 1 m is resting at an angle θ = 45^{0} as shown in the figure. The end P is dragged with a velocity of U = 5 m/s to the right. At the instant shown, the magnitude of the velocity V (in m/s) of point Q as it moves along the wall without losing contact is

Solution:

*Answer can only contain numeric values

QUESTION: 58

A frictionless circular piston of area 10^{-2} m and mass 100 kg sinks into a cylindrical container of the same area filled with water of density 3 1000 kg/m as shown in the figure. The container has a hole of area 10^{-3} m^{2} at the bottom that is open to the atmosphere. Assuming there is no leakage from the edges of the piston and considering water to be incompressible, the magnitude of the piston velocity (in m/s) at the instant shown is __________ (correct to two decimal places).

Solution:

Total gauge pressure at top

Total gauge pressure at bottom = 0

*Answer can only contain numeric values

QUESTION: 59

A test is conducted on a one-fifth scale model of a Francis turbine under a head of 2 m and volumetric flow rate of 1 m^{3} /s at 450 rpm. Take the water density and the acceleration due to gravity as 10^{3} kg/m^{3} and 10 m/s^{2} , respectively. Assume no losses both in model and prototype turbines. The power (in MW) of a full sized turbine while working under a head of 30 m is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 60

A welding operation is being performed with voltage = 30 V and current = 100 A. The cross-sectional area of the weld bead is 20 mm^{2} . The work-piece and filler are of titanium for which the specific energy of melting is 14 J/mm^{3}. Assuming a thermal efficiency of the welding process 70%, the welding speed (in mm/s) is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 61

The arc lengths of a directed graph of a project are as shown in the figure. The shortest path length from node 1 to node 6 is ____________.

Solution:

There was ambiguity in question. In some textbooks it is mentioned that critical path is the shortest path. But the question intended to ask the shortest time taking path. The shortest time taking path is of 7 days and critical path is of 11 days.

*Answer can only contain numeric values

QUESTION: 62

Ambient air is at a pressure of 100 kPa, dry bulb temperature of 30^{o}C and 60% relative humidity. The saturation pressure of water at 30^{o}C is 4.24 kPa. The specific humidity of air (in g/kg of dry air) is __________ (correct to two decimal places).

Solution:

QUESTION: 63

For a position vector the norm of the vector can be defined as Given a function its gradient is

Solution:

*Answer can only contain numeric values

QUESTION: 64

A vehicle powered by a spark ignition engine follows air standard Otto cycle (γ = 1.4 .) The engine generates 70 kW while consuming 10.3 kg/hr of fuel. The calorific value of fuel is 44,000 kJ/kg. The compression ratio is __________ (correct to two decimal places). Answer: 7.61

Solution:

Amount of heat supplied per second= 10.3×44000/3600 kJ/s

=125.88kW

*Answer can only contain numeric values

QUESTION: 65

For sand-casting a steel rectangular plate with dimensions 80mm x 120mm x 20mm, a cylindrical riser has to be designed. The height of the riser is equal to its diameter. The total solidification time for the casting is 2 minutes. In Chvorinov’s law for the estimation of the total solidification time, exponent is to be taken as 2. For a solidification time of 3 minutes in the riser, the diameter (in mm) of the riser is __________ (correct to two decimal places).

Solution:

### CBSE Past Year Paper Session (2017), Math Class 12

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