Calculus - Engineering Mathematics - Petroleum Engineering Previous Year Questions

1

2016 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2016

The value of \( \lim_{x \to 0} \left( \frac{e^x - 1}{\sin x} \right) \) is equal to____.

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2

2016 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2016

The function \( f(x) = \frac{1}{1+|x|} \) is

A

continuous and differentiable.

B

continuous but not differentiable.

C

not continuous but differentiable.

D

not continuous and not differentiable.

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3

2016 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2016

The value of the definite integral \( \int_{1}^{e} (\ln x) dx \) is equal to____.

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4

2016 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2016

The Laplace transform of the function \( e^{-2t} \) is

A

\(\frac{1}{2s} \)

B

\(\frac{2}{s} \)

C

\(\frac{1}{s+2} \)

D

\(e^{-2s} \)

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5

2016 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2016

The divergence of the velocity field \(\vec{V} = (x^2 + y)\hat{i} + (z - 2xy)\hat{j} + (xy)\hat{k}\) at (1, 1, 1) is __________.

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6

2017 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2017

If a vector \(\vec{v}\) has components \(v_x = 1, v_y = 2, v_z = 3\), then its magnitude is ______. (write answer with two decimal places)

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7

2017 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2017

The value of \(\lim_{x \to 0} \frac{(2+x)^4 - 16}{x}\) is ______.

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8

2017 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2017

For a velocity field given by \(\vec{v} = y\hat{i} - x\hat{j} + 0\hat{k}\), calculate the curl of \(\vec{v}\). If the calculated vector is \(a\hat{i} + b\hat{j} + c\hat{k}\), then the value of c is __________.

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9

2018 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2018

The Taylor series expansion of the function, \( f(x) = \frac{1}{1+x} \), around \( x = 0 \) (up to \( 4^{th} \) order term) is:

A

\(1 + x + x^2 + x^3 + x^4 \)

B

\(-1 + x - x^2 + x^3 - x^4 \)

C

\(-1 - x + x^2 - x^3 + x^4 \)

D

\(-1 + x - 2x^2 + 3x^3 - 4x^4 \)

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10

2018 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2018

The line integral of a vector function \( \vec{F}(\vec{r}) \) over a curve \( C \) in a simply connected domain \( D \) in space, is defined by: \( \int_C \vec{F}(\vec{r}) \cdot d\vec{r} = \int_C (F_1 dx + F_2 dy + F_3 dz) \). The line integral is independent of path in \( D \). \( F_1, F_2, \) and \( F_3 \) are continuous, and have continuous first partial derivatives in \( D \). \( C' \) is a closed curve in \( D \). Which one of the following is NOT ALWAYS true in domain \( D \)?

A

\(\nabla \times \vec{F} = \vec{0} \)

B

\(\nabla \cdot \vec{F} = 0 \)

C

\(\oint_{C'} \vec{F}(\vec{r}) \cdot d\vec{r} = 0 \)

D

\(\vec{F} \times \vec{F} = \vec{0} \)

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11

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

The vector that is normal to the surface \( 2xz^2 - 3xy - 4x = 7 \) at the point (1, –1, 2) is

A

2i – 3j + 8k

B

2i + 3j + 4k

C

7i – 3j + 8k

D

7i – 5j + 8k

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12

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

The solution of \( \int_{1}^{a} \int_{1}^{b} \frac{dx}{x} \frac{dy}{y} \) is

A

ln(ab)

B

ln(a / b)

C

ln(a) + ln(b)

D

ln(a) ln(b)

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13

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

The link lengths of a planar four bar mechanism are AB = 100 mm, BC = 25 mm, CD = 75 mm and DA = 90 mm. For achieving the full rotation of both the input (crank) as well as the output (follower) links, the link that needs to be fixed is

A

AB

B

BC

C

CD

D

DA

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14

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

For a classical (Wilson) model of determining economic order quantity (EOQ), the carrying and ordering costs are \(C_r\) and \(C_o\), respectively. For an annual demand \(D\), the minimum yearly total inventory cost is

A

\(\sqrt{DC_oC_r}\)

B

\(\sqrt{5DC_oC_r}\)

C

\(\sqrt{2DC_oC_r}\)

D

\(\sqrt{3DC_oC_r}\)

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15

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

A metallic rod of diameter \( d_o \) is subjected to the tensile test. The engineering stress and the true stress at fracture are 800 MPa and 900 MPa, respectively. The ratio of the rod diameter at fracture \( d_f \) to the initial diameter \( d_o \) is _____ (round off to 2 decimal places)

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16

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

A heat pump is to supply heat at the rate of 10 kW to a building to be maintained at 22 °C. The outside temperature is 2 °C. The minimum power (in kW) required to run the heat pump is _____ (round off to 2 decimal places)

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17

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

One kilogram of air is compressed at constant temperature of 150 °C until its volume is halved. Considering gas constant \( R = 0.287 \) kJ/kg-K for air, magnitude of heat rejected (in kJ) in the compression process is _____ (round off to 2 decimal places)

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18

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

If the Laplace transform of \(e^{\omega t}\) is \(\frac{1}{s-\omega}\), the Laplace transform of \(t\cosh t\) is

A

\(\frac{1+s^{2}}{(s^{2}-1)^{2}}\)

B

\(\frac{s}{(s^{2}-1)^{2}}\)

C

\(\frac{1-s^{2}}{(s^{2}-1)^{2}}\)

D

\(\frac{1+s^{2}}{1-s^{2}}\)

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19

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

A uniform cantilever beam \(ABC\) of length \(L\) is subjected to a point load \(P\) at point \(B\) and a concentrated moment \(M\) at point \(C\) (as shown in figure). Let \(E\) be the Young’s modulus of the beam material and \(I\) be the area moment of inertia of the beam’s cross-section. Assuming the validity of the Euler-Bernoulli theory of slender beams, the downward deflection at point \(C\) is

A

\(\frac{PL^3}{3EI} + \frac{ML^2}{2EI}\)

B

\(\frac{PL^3}{24EI} + \frac{ML^2}{EI}\)

C

\(\frac{PL^3}{48EI} + \frac{ML^2}{2EI}\)

D

\(\frac{5PL^3}{48EI} + \frac{ML^2}{2EI}\)

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20

2019 · Petroleum Engineering · Engineering Mathematics · Calculus

Petroleum Engineering (PE) 2019

Three Carnot engines \(E_1, E_2, E_3\) operate as shown in the figure (\(T_1 > T_2 > T_3\)). The efficiency of the engine \(E_3\) in terms of the efficiencies \(\eta_1\) and \(\eta_2\) of the engines \(E_1\) and \(E_2\), respectively, is

A

\(\eta_1 + \eta_2\)

B

\(\eta_1 + \eta_2 - \eta_1\eta_2\)

C

\(1 - \eta_1 - \eta_2\)

D

\(1 - \eta_1\eta_2\)

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