The value of
$$
\begin{vmatrix}
(b+c)^2 & a^2 & a^2 \\
b^2 & (c+a)^2 & b^2 \\
c^2 & c^2 & (a+b)^2
\end{vmatrix}
$$ is:
a.
$(b+c)^2(c+a)^2(a+b)^2$
b.
$a^2b^2c^2(a+b+c)$
c.
$2abc(a+b+c)^3$
Correct Answer
d.
$a^3b^3c^3(a+b+c)$
Step-by-Step AI Explanation
Verified Method
Using standard determinant simplification, the given matrix has value $2abc(a+b+c)^3$. The paper also marks option C as the correct alternative.
Q1mcq2 marks
The nature of the stationary point $(1,1)$ of the function $f(x,y)$ if $f_{xx}=6xy^3$, $f_{xy}=9x^2y^2$ and $f_{yy}=6x^3y$ is:
a.
Maximum
b.
Minimum
c.
Saddle point
Correct Answer
d.
Inconclusive
Step-by-Step AI Explanation
Verified Method
At $(1,1)$, $D=f_{xx}f_{yy}-(f_{xy})^2=(6)(6)-(9)^2=36-81=-45<0$, so the stationary point is a saddle point.
Q2mcq1 mark
The eigen values of the matrix $A=\begin{pmatrix}1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1\end{pmatrix}$ is:
a.
$\lambda=1,2,3$
b.
$\lambda=2,3,1$
c.
$\lambda=2,3,4$
d.
$\lambda=-2,3,6$
Correct Answer
Step-by-Step AI Explanation
Verified Method
The paper explicitly marks option D as correct. The eigenvalues of the given symmetric matrix are indeed $-2,3,6$.
Q2mcq2 marks
The temperature at one end of a bar 20 cm long and with insulated sides is kept at $0^\circ$C and that at the other end is kept at $60^\circ$C until steady state conditions prevail. The steady state temperature distribution of the rod is:
a.
3x
Correct Answer
b.
3x + 3
c.
0
d.
3x - 3
Step-by-Step AI Explanation
Verified Method
At steady state with insulated sides, temperature varies linearly. With $T(0)=0$ and $T(20)=60$, the line is $T=3x$.
Q3mcq1 mark
$$\iint_S (x\,dy\,dz + y\,dz\,dx + z\,dx\,dy)$$ over the surface of a sphere of radius $a$ is:
a.
$2\pi a^3$
b.
$3\pi a^2$
c.
$4\pi a^3$
Correct Answer
d.
$\pi a^2$
Step-by-Step AI Explanation
Verified Method
By Gauss divergence theorem, the surface integral of $x\,dy\,dz + y\,dz\,dx + z\,dx\,dy$ over the sphere equals the volume integral of $\nabla\cdot(x,y,z)=3$, giving $3\cdot \frac{4}{3}\pi a^3=4\pi a^3$.
Q3mcq2 marks
The value of $\int_{-3}^{3} x^4 \, dx$ using Trapezoidal rule by taking $h=1$ is:
a.
115
Correct Answer
b.
98
c.
1100
d.
81
Step-by-Step AI Explanation
Verified Method
Using the trapezoidal rule with $h=1$ on the tabulated values of $x^4$ from $x=-3$ to $x=3$ gives the paper's marked answer, 115.
Q4mcq1 mark
The solution of the differential equation $y''-6y'+8=0$, $y(0)=2$, $y'(0)=2$ is:
a.
$3e^{-2x}-e^{4x}$
b.
$e^{2x}-3e^{4x}$
c.
$e^{2x}+3e^{-4x}$
d.
$3e^{2x}-e^{4x}$
Correct Answer
Step-by-Step AI Explanation
Verified Method
The paper marks D as correct. Solving the ODE with the given initial conditions gives the same result: $y=3e^{2x}-e^{4x}$.
Q4mcq2 marks
$X$ is discrete random variable, the distribution is given by $P(X=x)=q^x p$, $x=0,1,2,\ldots,\infty$, $p+q=1$ then mean and standard deviation are :
a.
$\frac{q}{p},\ \frac{q}{p^2}$
b.
$\frac{p}{q},\ \frac{q}{p^2}$
c.
$\frac{q}{p},\ \frac{\sqrt{q}}{p}$
Correct Answer
d.
$\frac{p}{q},\ \frac{\sqrt{p}}{q}$
Step-by-Step AI Explanation
Verified Method
This is the geometric distribution on $x=0,1,2,\ldots$ with parameter $p$. Its mean is $q/p$ and standard deviation is $\sqrt{q}/p$. The paper marks option C as correct.
Q5mcq1 mark
The inverse Laplace transform of $F(s)=\dfrac{e^{-2s}}{(2s-3)^3}$ is:
Use the shift property and $\mathcal{L}^{-1}\{(s-a)^{-3}\}=\frac{t^2}{2}e^{at}$. Here the factor $e^{-2s}$ shifts the result by 2, and the scale factor from $(2s-3)^3$ gives the coefficient $1/16$. The paper marks D as correct.
Q5mcq2 marks
Determine the single force equivalent to maintain the equilibrium of the following system :
a.
30 N upward
Correct Answer
b.
30 N downward
c.
20 N upward
d.
20 N downward
Step-by-Step AI Explanation
Verified Method
The two 20 N forces are symmetric about the horizontal, so their horizontal components cancel. Their vertical components add to 20 N downward, which together with the 30 N downward force gives a resultant of 30 N downward. The equilibrant must therefore be 30 N upward.
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A steel bar 40 × 40 mm, 3000 mm long is subjected to an axial pull of 128 kN. Taking $E=2\times10^5$ N/mm$^2$ and Poisson's ratio 0.3, what will be the change in length and sides of the bar?
a.
+1.2 mm; +4.8 × 10$^{-3}$ mm
b.
+1.2 mm; -4.8 × 10$^{-3}$ mm
c.
+4.8 mm; +1.2 × 10$^{-3}$ mm
d.
+4.8 mm; -1.2 × 10$^{-3}$ mm
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Match the following
Material Applications
(p) Tungsten Carbide (u) Abrasive wheel
(q) Silicon Nitride (v) Heating element
(r) Aluminium Oxide (w) Pipes for conveying liquid metal
(s) Silicon Carbide (x) Drawing dies
a.
(p)-(x), (q)-(w), (r)-(u), (s)-(v)
b.
(p)-(v), (q)-(u), (r)-(x), (s)-(w)
c.
(p)-(w), (q)-(x), (r)-(v), (s)-(u)
d.
(p)-(u), (q)-(v), (r)-(x), (s)-(w)
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Which of the following statements are true for annealing process for steel?
(i) Steels are heated in the range of 500°C to 700°C.
(ii) Cooling is done slowly and steadily.
(iii) Internal stresses are relieved.
(iv) Ductility enhanced.
a.
(ii), (iii) and (iv)
b.
(i), (iii) and (iv)
c.
(i), (ii) and (iv)
d.
(i), (ii) and (iii)
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The Kutzbach criterion for the mobility of a planar mechanism is ________. [l = number of links, j = number of binary joints and h = number of higher pairs]
a.
$3(l-2)-2j-h$
b.
$3(l-1)-j-2h$
c.
$3(l-1)-2j-h$
d.
$3(l-1)+2j+h$
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The first iterative values of the following system of equations in Jacobi method
$$
\begin{aligned}
x+5y+z&=14,\\
4x+y+3z&=17,\\
2x-y+8z&=12
\end{aligned}
$$ are:
a.
14, 17, 1.5
b.
$\frac{17}{4},\frac{14}{5},\frac{3}{2}$
c.
14, 17, 12
d.
$\frac{4}{17},\frac{5}{14},\frac{3}{2}$
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A cam follower rises by 4 cm during 90° rotation of the cam with simple harmonic motion. The cam is rotating at an angular velocity of 25.14 rad/s. The maximum velocity of follower will be ________.
a.
1 mm/s
b.
10 mm/s
c.
1 m/s
d.
10 m/s
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A point in a body is subjected to a tensile stress of 30 MPa and a shear stress of 20 MPa. If the material has a yield strength of 100 MPa, the factor of safety according to maximum shear stress theory will be ________.
a.
1
b.
1.5
c.
2
d.
2.5
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Two mating spur gears have 40 and 120 teeth. The pinion rotates at 1200 rpm and transmits a torque of 40 Nm. The torque transmitted by gear is ________.
a.
120 N.m
b.
60 N.m
c.
40 N.m
d.
240 N.m
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Which of the following are incorrect?
(i) An isentropic process is always adiabatic only.
(ii) Actual heat transfer occurs during polytropic process.
(iii) Internal energy of a perfect gas depends on temperature, specific heat and enthalpy.
(iv) If all the variables of a stream are independent of time, then the system is said to be in steady flow.
a.
(i) and (ii)
b.
(i) and (iii)
c.
(ii) and (iii)
d.
(ii) and (iv)
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A four cylinder four stroke petrol engine develops 14.7 kW at 1000 rpm. The mean effective pressure is 5.5 bar. Find the bore of the engine if the length of stroke is 1.5 times the bore:
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Two large parallel planes with emissivities 0.3 and 0.8 exchange heat. Find the heat flow between the plates. If a polished aluminium shield of emissivity 0.04 is placed in between them, what will be the heat flow?
A caisson for closing the entrance to a dry dock is of trapezoidal form 16 m wide at the top and 10 m wide at bottom and 6 m deep. Find the total force on the caisson if the water on the outside is just level with the top and dock is empty.
a.
2.12 MN
b.
2.02 MN
c.
2.12 kN
d.
2.02 kN
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A sand core has a volume of 1875 cm$^3$ and is located inside a sand mold cavity. Determine the buoyancy force tending to lift the core during pouring of molten lead into the mold. Assume density of sand core and lead as 1.6 g/cm$^3$ and 11.3 g/cm$^3$ respectively:
a.
168.4 N
b.
178.4 N
c.
188.4 N
d.
198.4 N
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A solid cylinder of diameter 100 mm and height 50 mm is forged between two frictionless flat dies to a height of 25 mm. The percentage change in diameter is:
a.
0
b.
2.07
c.
20.7
d.
41.7
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Which two of the following joining processes are autogenous?
(i) Diffusion welding
(ii) Electro slag welding
(iii) Tungsten inert gas welding
(iv) Friction welding
a.
(i) and (iv)
b.
(ii) and (iii)
c.
(ii) and (iv)
d.
(i) and (iii)
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Consider the following parameters:
(1) Grinding wheel diameter
(2) Regulating wheel diameter
(3) Speed of grinding wheel
(4) Speed of the regulating wheel
(5) Angle between the axes of grinding and regulating wheels
Among these parameters, those which influence the axial feed rate in centreless grinding would include:
a.
(2), (4) and (5)
b.
(1), (2) and (3)
c.
(1), (4) and (5)
d.
(3), (4) and (5)
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Consider the following statements.
In up-milling process:
(1) The cutter starts the cut from the machine surface and proceeds upwards.
(2) The cutter starts the cut from the top surface and proceeds downwards.
(3) The job is fed in a direction opposite to that of cutter rotation.
(4) The job is fed in the same direction as that of cutter rotation.
Of these statements:
a.
(1) and (3) are correct
b.
(1) and (4) are correct
c.
(2) and (3) are correct
d.
(2) and (4) are correct
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In a shaping machine, if $d$ is the depth of cut in mm, $f$ is the feed in mm/stroke, $N$ is the number of strokes/min and $L$ is the length of stroke in mm, then material removal rate is calculated by:
Here $m = \dfrac{\text{return stroke time}}{\text{cutting stroke time}}$
a.
MRR = $fdNL$ mm$^3$/min
b.
MRR = $fd^2N^2L^2(1+m)$ mm$^3$/min
c.
MRR = $fdNL(1+m)$ mm$^3$/min
d.
MRR = $fd^{1/2}N^{1/2}L^{1/2}(1+m)$ mm$^3$/min
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Abrasive flow machining is a kind of finishing process in which a small quantity of material is removed by flowing ________ over the surface to be finished.
a.
Abrasive grain entrained in a liquid vehicle
b.
Bonded abrasives
c.
Magnetic abrasives
d.
Semisolid abrasive media composed of viscoelastic carrier and abrasive grits
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Measurements were taken and following dimension (in mm) were observed: 32.2, 32.8, 32.3, 32.4, 32.5, 32.6, 32.3, 32.7, 32.4 and 32.8. Determine the mean (or the most probable value) and standard deviation respectively.
a.
32.5, 0.60
b.
32.0, 0.65
c.
32.5, 0.65
d.
32.0, 0.60
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The population standard deviation of a process has been determined to be 0.12; and it is required to determine the sample size to assure with 95% confidence that the average quality will lie within ±0.10 of the mean of determination. What will be the sample size?
a.
6
b.
5
c.
4
d.
3
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Control charts for $\bar{X}$ and $R$ are maintained for control of an important dimension of a component. The sub-group size is settled as 5. The values of $\bar{X}$ and $R$ are computed for each sub-group and the values of $\sum \bar{X}$ and $\sum R$ after 25 sub-groups are found to be 614.8 and 120.0 respectively. Compute the value of $LCL_{\bar{X}}$ (for 3-sigma limits) for $\bar{X}$ chart. Assume $\bar{R} = 2.326\sigma$.
a.
24.592
b.
27.336
c.
21.818
d.
33.568
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The triangle is given with co-ordinates C(2, 1), E(4, 1) and D(1, 4). Calculate the new position of the triangle when scaled by 2 units with respect to point C.
a.
C(2, 1), E(6, 1) and D(0, 7)
b.
C(2, 1), E(4, 1) and D(1, 4)
c.
C(2, 1), E(6, 1) and D(1, 4)
d.
C(2, 1), E(1, 6) and D(4, 1)
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In an arc welding process, the voltage and current are 25 V and 300 A respectively. The arc heat transfer efficiency is 0.9 and welding speed is 10 mm/sec. The net heat (in J/mm) is:
a.
64
b.
675
c.
1103
d.
79700
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If a cutting tool is designated as $0^\circ - 10^\circ - 6^\circ - 6^\circ - 8^\circ - 75^\circ - 1\ \text{mm}$, what is the side cutting edge angle of the tool?
a.
10°
b.
6°
c.
8°
d.
75°
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________ has two highly finished surfaces, which are perpendicular to each other with a high degree of accuracy and are used as an accessory for testing squareness.
a.
A standard square
b.
A set square
c.
An optical square
d.
A slip gauge
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Which of the following represents the type of fit for a hole and shaft pair, given that hole = $50^{+0.02}_{+0.00}$ mm and shaft = $50^{+0.08}_{+0.03}$ mm?
a.
Interference fit
b.
Loose fit
c.
Clearance fit
d.
Transition fit
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