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QUESTION: 1

A rectangle becomes a square when its length and breadth are reduced by 10 m and 5 m, respectively. During this process, the rectangle loses 650 m^{2} . What is the area of the original rectangle in square meters?

Solution:

Let ‘a’ be the side of square, then length and breadth of rectangle are ‘a+10 and ‘a+5’ respectively.

Given that,

Area of rectangle= Area of Square+650

(a + 10)(a + 5) = a^{2} + 650

a^{2} + 15a + 50 = a^{2} + 650

15a = 600

a = 40

Area of rectangle=a^{2} + 650

Area of rectangle=1600+650=2250

QUESTION: 2

A number consists of two digits. The sum of the digits is 9. If 45 is subtracted from the number. Its digits are interchanged. What is the number?

Solution:

Let one’s digit is y and ten’s digit is x

Hence the number becomes ‘10x +y’ and the reverse number will be ‘10y+x’

x + y = 9 … (i)

10x + y− 10y − x = 45

x − y = 5 …. (ii)

Adding (i) and (ii)

x = 7

Subtracting (i) and (ii)

y = 2

Therefore the number is 72.

QUESTION: 3

Seven machines take 7 minutes to make 7 identical toys. At the same rate, how many minutes would it take for 100 machines to make 100 toys?

Solution:

Time taken by a machine to make a toy will be independent of how many machines are making toys in parallel.

From given data, it takes 7 minutes for a machine to make a toy. If hundred such machines are running parallel to make a toy each, the time will remain same 7 minutes.

7 machine → 7 toys → 7 minutes

1 machine → 1 toy → 7 minutes

Because one machine takes 7 minute for making 1 toy.

So, 100 machines will take 7 minute for making 100 toys.

QUESTION: 4

“Her _______ should not be confused with miserliness, she is ever willing to assist those in need.” The word that best fills the bank in the above sentence is :

Solution:

The statement suggests that some weak condition of the person is depicted. And this weak condition should not be taken as being miser.

Miser: A person reluctant to spend.

Frugal: A person who is economically weak.

QUESTION: 5

Going by the __________ that many hands make light work, the school __________ involved all the students in the task.”

Solution:

Principle – Truth/Proposition serving as foundation of person’s belief.

Principal – The most important person in an organization. Here it refers to school Principal

QUESTION: 6

Given that a and b are integers and a + a^{2}b^{3} is odd, which one of the following statementsis correct?

Solution:

Given: a and b are integers

a + a^{2}b^{3} is odd

Taking a common

a(1+ab^{3}) is odd

Odd number is obtained after multiplication if both numbers multiplied are odd.

Hence, a is odd and 1+ab^{3} is also odd.

1+ab^{3 }is odd, so ab^{3} will be even. (odd-1= even quantity)

Because a is odd so for ab^{3} to be even b must be even.

So, a is odd and b is even.

QUESTION: 7

Consider the following three statements:

1. Some roses are red

2. All red flowers fade quickly.

3. Some roses fade quickly.

Q. Which of the following statements can be logically inferred from the above statements?

Solution:

Solving by options

Option A: Even if statement 2^{nd} is false, i.e. All red flowers do not fade quickly, that does not mean that some roses won’t fade quickly.

Option B: There can be a possibility that no rose fade quickly

Option C: This is true in all possibilities.

Option D: There can be a possibility that some roses do fade quickly.

Hence option C is the only correct option.

QUESTION: 8

Which of the following functions describe the graph shows in the below figure.

Solution:

Hence only B option prevails.

QUESTION: 9

For integers a, b and c, what would be the minimum and maximum values respectively of a + b + c if log| a|+ log| b |+ log| c | = 0

Solution:

log| a|+ log| b |+ log| c | = 0

It is possible only,

If | a |,| b | and | c | all are equal to 1.

So, a, b, c may be respectively ‘+1’ or ‘-1’.

For minimum value all three will be negative.

So, minimum value = - 3

For maximum value all three will be positive.

So, maximum value = + 3.

QUESTION: 10

From the time the front of a train enters a platform, it takes 25 seconds for the back ofthe train to leave the platform, while travelling at a constant speed of 54 km/h. At thesame speed, it takes 14 seconds to pass a man running at 9 km/h in the same directionas the train. What is the length of the train and that of the platform in meter,respectively?

Solution:

Train speed = 54 km/h

Time = 25 sec for travelling length of train and length of platform

Man speed = 9 km/h

Relative speed of train with respect to man = 45 km/h

Time = 14 sec

So, length of train = time x speed

Length of train = 35 x 5 m = 175 m

Length of platform + length of train = speed x time

Length of platform 375 -175 = 200 m

QUESTION: 11

For a two-dimensional incompressible flow field given by where A > 0, whichone of the following statements is FALSE?

Solution:

This has to be solved through options

Continuity equation for 2D incompressible flow

A − A = 0

It satisfies continuity equation.

As y → velocity vector field will not be defined along y axis.

So flow will be along x-axis i.e. 1-D flow.

⇒ Stream line equation for 2D

*Answer can only contain numeric values

QUESTION: 12

An ideal gas undergoes a process from state 1 (T_{1} = 300 K,p_{1} = 100 kPa) to state

( ) 2(T_{2} = 600k,p_{2} = 500 kPa . The specific heats of the ideal gas are: cp=1 kJ/kg-k = and c_{v} 0.7 kJ/kg-K. The change in specific entropy of the ideal gas from state 1 to state 2 (in kJ/kg-K) is ________ (correct to two decimal places)

Solution:

Ideal gas

State 1 : T_{1} = 300 K, P_{1} = 100 kPa

State 2 : T_{2} = 600 K,P_{2} = 500 kPa

Change in specific entropy

QUESTION: 13

A bar of uniform cross section and weighing 100 N is held horizontally using two massless and inextensible strings S_{1} and S_{2} as shows in the figure.

Tension in the springs are

Solution:

Comments: This can also be inferred from fact that if T_{1} is finite then the bar will not remain horizontal.

*Answer can only contain numeric values

QUESTION: 14

For a Pelton wheel with a given water jet velocity, the maximum output power from the

Pelton wheel is obtained when the ratio of the bucket speed to the water jet speed is_______ (correct to two decimal places).

Solution:

In Pelton wheel turbine for maximum efficiency,

QUESTION: 15

A six- faced fair dice is rolled five times. The probability (in%)of obtaining “ONE” at leastfour times is

Solution:

A dice is rolled 5 times

Probability of getting 1 ‘at least’ 4 times will include probability of getting 1- “4

times+5times”

The probability distribution is binomial distribution.

QUESTION: 16

Using the Taylors tool life equation with exponent n = 0.5, if the cutting speed is reducedby 50% the ratio of new tool life to original tool life is

Solution:

Taylor’s Tool life equation is VT^{n}=Constant

Let subscript 1 represent the initial conditions and subscript 2 represent the final conditions.

QUESTION: 17

A grinding ratio of 200 implies that the

Solution:

Grinding ratio is defined as ratio of Volume of work material removed to the ratio of wheel material removed.

Grinding wheel wears 1/200 (= 0.005) times the volume of material removed.

QUESTION: 18

The number of atoms per unit cell and the number of slip systems, respectively, for aface-centered cubic (FCC) crystal are

Solution:

QUESTION: 19

The type of weld represented by the shaded region in the figure is

Solution:

The figure represents fillet weld on T joint.

*Answer can only contain numeric values

QUESTION: 20

The height (in mm) for a 125 mm sine bar to measure a taper of 27°32’ on a flat work

piece is __________ (correct to three decimal places).

Solution:

Where H= height of specimen and L= Length of Sine bar (length between center of two rollers)

QUESTION: 21

Interpolator in a CNC machine

Solution:

As interpolator provides two functions:

1. It computes individual axis velocities to drive the tool along the programmed path at given feed rate.

2. It generates intermediate coordinate positions along the programmed path.

*Answer can only contain numeric values

QUESTION: 22

If the wire diameter of a compressive helical spring is increased by 2% the change in

spring stiffness (in %) is __________ (correct to two decimal places)

Solution:

Stiffness of helical spring

Where, d=Spring Wire diameter

D= Mean Coil diameter

n= Number of coils

G= Shear Modulus

d increased by 2% i.e. d’=1.02d

*Answer can only contain numeric values

QUESTION: 23

A four bar mechanism is made up of links of length 100,200,300 and 350 mm. if the 350

mm link is fixed, the number of links that can rotate fully is _________

Solution:

Grashoff’s law is satisfied.

350 mm link is fixed, i.e. adjacent to shortest link is fixed.

The mechanism obtained will be crank rocker mechanism, it means only one link (crank) will fully rotate.

QUESTION: 24

Four red balls, four green balls and four blue balls are put in a box. Three balls are pulledout of the box at random one after another without replacement. The probability thatall the three balls are red is

Solution:

Probability that all the three balls are red is

Alter:

The required probability can also be obtained as,

Probability=

QUESTION: 25

The rank of the matrix is

Solution:

Converting the matrix into Echelon form,

Now since the matrix is in Echelon

Rank=Number of non-zero rows=2

QUESTION: 26

According to the Mean Value Theorem for a continuous function f (x) in the interval [a,b], there exists a value

in this interval such that

Solution:

This implies that within interval [a,b] there exists a value such that

*Answer can only contain numeric values

QUESTION: 27

A flat plate of width L = 1 m is pushed down with a velocity U = 0.01 m/s towards a

wall resulting in the drainage of the fluid between the plate and the wall as shown in

the figure. Assume two-dimensional incompressible flow and that the plate remains

parallel to the wall. The average velocity, U_{avg} of the fluid (in m/s) draining out at the

instant shows in the figure is ___________ (correct to three decimal places).

Solution:

Assuming length of plate as B

Let in infinitely small time ‘dt’ the plate is displaced with ‘dh’

So,

As per continuity,

Rate of mass displaced through plate= Rate of mass displaced between plates and wall

Since the flow is 2D the velocity of flow in direction perpendicular to plane of paper is

zero.

QUESTION: 28

F (z) is a function of the complex variable z = x + iy given by

For what value of k will f(z) satisfy the Cauchy-Riemann equations?

Solution:

Where u = (kx − y) and v = (x + y)

According to Cauchy Riemann Equations,

From first condition k=1.

QUESTION: 29

If σ_{1} and σ_{3} are the algebraically largest and smallest principal stresses respectively the

value of the maximum shear stress is.

Solution:

Maximum shear stress at a point is given by =

QUESTION: 30

The time series forecasting method that gives equal weightage to each of the M mostrecent observation is

Solution:

‘Simple moving average method’ is generally also called ‘Moving average method’ which gives equal weightage to all data points for period ‘M’ on which it is defined.

QUESTION: 31

In a linearly hardening plastic material. The true stress beyond initial yielding

Solution:

Given that the strain hardening is linear, the true stress will increase linearly after yield point.

*Answer can only contain numeric values

QUESTION: 32

A steel column of rectangular section (15 mm x10 mm) and length 1.5 m is simply supported at both ends. Assuming modulus of elasticity, E = 200GPa for steel, the critical axial load (in kN) is _______ (correct to two decimal places)

Solution:

For Simply Supported column at both ends,

QUESTION: 33

Which one of the following statement is correct for a superheated vapour?

Solution:

P_{1}, T_{1} represent the state of superheated vapours.

Let P_{0} be the saturation pressure at T_{1} temperature.

Conclusions (According to options)

A. Pressure will be less than saturation Pressure at given temperature.

B. Temperature will be higher than saturation temperature at given pressure (superheated state).

C. Volume is more than the volume of the saturated vapour at a given temperature (Volume is more in superheated state at same temperature).

D. Its enthalpy is higher than enthalpy of the saturated vapour at a given pressure (superheated vapours have more enthalpy than saturated vapours).

Hence only option A is correct.

QUESTION: 34

The equation of motion for a spring-mass system excited by a harmonic force is

Where M is the mass, K is the spring stiffness, F is the force amplitude and ω is the

angular frequency of excitation. Resonance occurs when is equal to.

Solution:

Resonance is when

QUESTION: 35

For an Oldham coupling used between two shafts, which among the following

statements are correct?

I. Torsional load is transferred along shaft axis.

II. A velocity ratio of 1:2 between shafts is obtained without using gears

III. Bending load is transferred transverse to shaft axis.

IV. Rotation is transferred along shaft axis.

Solution:

Oldham’s coupling is used to transmit power between two shafts having small angular misalignment. There is no bending load or reduction/increase of speed criterion.

Hence only I and IV statements are correct.

*Answer can only contain numeric values

QUESTION: 36

The minimum value of 3x + 5y such that:

is __________.

Solution:

Z= 3x+5y

Z = 3x + 5y

Feasible region will be as shown in figure.

Since the minimum value of objective function is to be found and both the coefficients in objective function are zero.

Z will be zero at point (0,0).

*Answer can only contain numeric values

QUESTION: 37

A bar is compressed to half of its original length. The magnitude of true strain produced

in the deformed bar is ________ (correct to two decimal places).

Solution:

As examiner mentioned “magnitude” only magnitude will be given 0.693.

QUESTION: 38

An epicyclic gear train is shown in the figure below. The number of teeth on the gearsA, B and D are 20, 30 and 20, respectively. Gear C has 80 teeth on the inner surface and100 teeth on the outer surface. If the carrier arm AB is fixed and the sun gear A rotatesat 300 rpm in the clockwise direction, then the rpm of D in the clockwise direction is

Solution:

Arm is fixed, no epicyclic nature. Taking clockwise direction as positive.

*Answer can only contain numeric values

QUESTION: 39

An engine working on air standard Otto cycle is supplied with air at 0.1 MPa and 35oC.

The compression ratio is 8. The heat supplied is 500 kJ/kg. Property data for air:The maximum temperature (in K) of the cycle is _________ (correct to one decimal place).

Solution:

For process 1-2

*Answer can only contain numeric values

QUESTION: 40

A solid block of 2.0 kg mass slides steadily at a velocity V along a vertical wall as shown in the figure below. A thin oil film of thickness h=0.15mm provides lubrication between the block and the wall. The surface area of the face of the block in contact with the oilfilm is 0.04 m^{2} . The velocity distribution within the oil film gap is linear as shown in the figure. Take dynamic viscosity of oil as 7×10−3 Pa-s and acceleration due to gravity as 10 m/s^{2} Neglect weight of the oil. The terminal velocity V (in m/s) of the block is _________ (correct to one decimal place).

Solution:

Terminal velocity is a constant velocity i.e. the net acceleration is zero.

QUESTION: 41

A self-aligning ball bearing has a basic dynamic load rating ( C_{10} , for 10^{6} revolutions) of

35 kN. If the equivalent radial load on the bearing is 45 kN, the expected life (in 10^{6} revolutions) is

Solution:

QUESTION: 42

The maximum reduction in cross-sectional area per pass (R) of a cold wire drawing process is

where n represents the strain hardening coefficient. For the case of a perfectly plastic

material, R is

Solution:

Perfectly plastic material implies that strain hardening coefficient n is zero.

Hence

QUESTION: 43

The value of integral

over the closed surface S bounding a volume, whereis the position vector

and is the normal to the surface S, is

Solution:

By Gauss Divergence Theorem

*Answer can only contain numeric values

QUESTION: 44

The percentage scrap in a sheet metal blanking operation of a continuous strip of sheet

metal as shown in the figure ______ (correct to two decimal places)

Solution:

This rectangle ABCD will be repeated again and again.

Sides of this rectangle are (D+D/5) and (D+2D/5)

*Answer can only contain numeric values

QUESTION: 45

An orthogonal cutting operations is being carried out in which uncut thickness is 0.010 mm, cutting speed is 130 m/min, rake angle is 15o and width of cut is 6 mm. It is observed that the chip thickness is 0.015 mm, the cutting force is 60 N and the thrust force is 25 N. The ratio of friction energy to total energy is __________ (correct to two

decimal places)

Solution:

Where F represents frictional force

Ratio of frictional energy to total energy

QUESTION: 46

Let 1 2 X , X be two independent normal random variables with means and standard

deviations respectively. Consider Then,

Solution:

X_{1} and X_{2} are two independent random variables

Since X_{1} and X_{2} are independent variables

*Answer can only contain numeric values

QUESTION: 47

An electrochemical machining (ECM) is to be used to cut a through hole into a 12 mm thick aluminum plate. The hole has a rectangular cross-section, 10 mm × 30 mm The ECM operation will be accomplished in 2 minutes, with efficiency of 90%. Assuming specific removal rate for aluminum as 3.44 x10^{-2} mm^{3} /(As), the current (in A) required

is __________(correct to two decimal places).

Solution:

Volume of metal to be removed

Ideal energy required

*Answer can only contain numeric values

QUESTION: 48

Steam flows through a nozzle at mass flow rate of kg/s with a heat loss of 5 kW.The enthalpies at inlet and exit are 2500 kJ/kg and 2350 kJ/kg, respectively. Assuming negligible velocity at inletthe velocityof steam (in m/s) at the nozzle exit is _________ (correct to two decimal places)

Solution:

*Answer can only contain numeric values

QUESTION: 49

A simply supported beam of width 100 mm, height 200 mm and length 4 m is carrying a uniformly distributed load of intensity 10 kN/m. The maximum bending stress (in MPa) in the beam is _________ (correct to one decimal place)

Solution:

Maximum bending moment will be at centre.

(L = 4)

Maximum Bending Stress

QUESTION: 50

The state of stress at a point, for a body in place stress, is shown in the figure below. If

the minimum principal stress is 10 kPa, then the normal stress σy (in kPa) is

Solution:

Minimum principal stress

By squaring

*Answer can only contain numeric values

QUESTION: 51

A sprinkler shown in the figure rotates about its hinge point in a horizontal plane due

to water flow discharged through its two exit nozzles.

The total flow rate Q through the sprinkler is 1 litre/sec and the cross-sectional area of

each exit nozzle is 1cm^{2}. Assuming equal flow rate through both arms and a frictionless

hinge, the steady state angular speed of rotation (rad/s) of the sprinkler is _________

(correct to two decimal places).

Solution:

Relative velocities of water with sprinkler

Absolute velocity from B side

Absolute velocity from A side

The external torque to the sprinkler is zero.

So,

*Answer can only contain numeric values

QUESTION: 52

A slider crank mechanism is shown in the figure. At some instant, the crank angle is 45° and a force of 40 N is acting towards the left on the slider. The length of the crank is 30 mm and the connecting rod is 70 mm. Ignoring the effect of gravity, friction and inertial forces, the magnitude of the crankshaft torque (in Nm) needed to keep the mechanism in equilibrium is ________(correct to two decimal places).

Solution:

QUESTION: 53

A carpenter glues a pair of cylindrical wooden logs by bonding their end faces at an

angle of as shown in the figure.

The glue used at the interface fails if

Criterion 1 : the maximum normal stress exceeds 2.5 MPa

Criterion 2 : the maximum shear stress exceeds 1.5 MPa

Assume that the interface fails before the logs fail. When a uniform tensile stress of 4

MPa is applied, the interface

Solution:

General equation learned for

Normal stress on inclined plane

And shear stress for inclined plane

when is measured clockwise from left face.

Hence must be replaced by (- ) in above equations and

Since both the stress exceeds the given limits, answer is option (C).

*Answer can only contain numeric values

QUESTION: 54

The schematic of an external drum rotating clockwise engaging with a short shoe is shown in the figure. The shoe is mounted at point Y on a rigid lever XYZ hinged at point X. A force F = 10 N is applied at the free end of the lever as shown. Given that the coefficient of friction between the shoe and the drum is 0.3, the braking torque (in Nm)

applied on the drum is ____________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 55

Processing times (including step times) and due dates for six jobs waiting to be processed at a work centre are given in the table. The average tardiness (in days) using shortest processing time rule is _________(correct to two decimal places).

Solution:

By SPT Rule

Total tardiness = 38

Average tardiness per job

*Answer can only contain numeric values

QUESTION: 56

An explicit forward Euler method is used to numerically integrate the differential

equation

Using a time step of 0.1. With the initial condition y (0) = 1, the value of y(1) computed

by this method is ___________ (correct to two decimal places).

Solution:

QUESTION: 57

A tank open at the top with a water level of 1 m, as shown in the figure, has a hole at a

height of 0.5 m. A free jet leaves horizontally from the smooth hole. The distance X (in

m) where the jet strikes the floor is

Solution:

Let free jet velocity is x displacement

Velocity of jet coming from the orifice in X direction,

To calculate time ‘t’, using equation of motion for y direction,

By second law of motion equation

For free fall u=0, s=0.5m

is the time of free fall of object

By equation (i) and (ii)

x=1 m.

*Answer can only contain numeric values

QUESTION: 58

A machine of mass m = 200 kg is supported on two mounts, each of stiffness k = 10 kN/m. The machine is subjected to an external force (in N) F (t) = 50 cos 5t. Assuming only vertical translator motion, the magnitude of the dynamic force (in N) transmitted from each mount to the ground is _________(correct to two decimal places).

Solution:

Transmissibility

*Answer can only contain numeric values

QUESTION: 59

A plane slab of thickness L and thermal conductivity k is heated with a fluid on one side (P) , and the other side (Q) is maintained at a constant temperature, TQ of 25°C, as shown in the figure. The fluid is at 45°C and the surface heat transfer coefficient, h, is 2 10 W/mK. The steady state temperature. TP (in °C) of the side which is exposed to the fluid is _________ (correct to two decimal places).

Solution:

Assuming steady state conditions

Thermal circuit

Assuming steady state, heat transferred from fluid will be equal to heat transferred from wall.

*Answer can only contain numeric values

QUESTION: 60

F (s) is the Laplace transform of the function

f (t) = 2t^{2}e^{-t}

F(1) is __________ (correct to two decimal places).

Solution:

*Answer can only contain numeric values

QUESTION: 61

Block P of mass 2 kg slides down the surface and has a speed 20 m/s at the lowest point,

Q where the local radius of curvature is 2 m as shown in the figure. Assuming 2 g = 10 m/s , the

normal force (in N) at Q is ________ (correct to two decimal places).

Solution:

QUESTION: 62

In a Lagrangian system, the position of a fluid particle in a flow is described asandwhere t is the time whileand k are constants. The flow is

Solution:

x direction scalar of velocity field,

y direction scalar of velocity field

u & v are non zero scalar t 0 so it is 2D flow.

Continuity equation for steady flow,

0 + 0 = 0 Continuity is satisfied.

Hence flow is 2D and steady.

*Answer can only contain numeric values

QUESTION: 63

The true stress (σ ) , true strain (ε ) diagram of a strain hardening material is shown in figure. First, there is loading up to point A, i.e. up to stress of 500 MPa and strain of 0.5. then from point A, there is unloading up to point B, i.e. to stress of 100 MPa, Given that the Young’s modulus E = 200 GPa, the natural strain at point B (ε_{B} ) ________ (correct to two decimal places).

Solution:

BC is the elastic recovery related strain.

QUESTION: 64

A point mass is shot vertically up from ground level with a velocity of 4 m/s at time, t = 0. It loses 20% of its impact velocity after each collision with the ground. Assuming that the acceleration due to gravity is 2 10 m/s and that air resistance is negligible, the mass stops bouncing and comes to complete rest on the ground after a total time (in

seconds) of

Solution:

So, t,t ',t " are forming a GP series

So, total time

*Answer can only contain numeric values

QUESTION: 65

A tank of volume 0.05 m^{3} contains a mixture of saturated water and saturated steam at 200°C. The mass of the liquid present is 8 kg. The entropy (in kJ/kg/K) of the mixture is __________ (correct of two decimal places)

Property data for saturated steam and water are:

At o 200°C, Psat = 1.5538 MPa

Solution:

Total volume of tank (V) = 0.05 m^{3}

Means of liquid (m_{2} ) = 8 kg

Volume occupied by liquid in tank= mxV_{f}

V_{L}=8×0.001157 m^{3}

V_{L}=0.009256

V_{v}=V-VL

V_{v}=0.05-0.009256=4.990744

Specific entropy of mixture (s)

### CBSE Past Year Paper Session (2018), Math Class 12

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