Calculus - Engineering Mathematics - Biomedical Engineering Previous Year Questions

1

2020 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2020

The magnitude of the gradient of the function \(f(x, y) = x^2 + y^2\) at the point (1,1) is ______ (rounded off to two decimal places).

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2

2020 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2020

The value of following double integral is __________ (correct up to three decimal places).
\[ \iint_R xy \, dx \, dy \]
where \( R \) is the first quadrant of the circle with center at the origin and radius of one unit.

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3

2020 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2020

The wavelength of an electron accelerated to a potential of 1 V is __________ nm (rounded off to two decimal places).
Mass of electron = 9.11×10⁻³¹ kg
Planck's constant, h = 6.63×10⁻³⁴ Js
Charge of electron = 1.6 ×10⁻¹⁹ C

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4

2021 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2021

The minimum value, \(f_{min}\), of the function given below is __________. (rounded off to the nearest integer) \(f(x_1, x_2, x_3) = \frac{1}{2}(x_1^2 + x_2^2 + x_3^2) - 2(x_1 + x_2 + x_3)\)

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5

2021 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2021

The radioactivity of a radionuclide with decay constant of \(3.22 \times 10^{-5} s^{-1}\) is 6 mCi at 10:30 AM. The radioactivity of the radionuclide at 4:30 PM on the same day will be __________ mCi. (rounded off to two decimal places)

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6

2022 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2022

Evaluation of the integral \( \int \frac{dx}{\sqrt{2x - x^2}} \) results in

A

\(\sin^{-1}(x-1) + c \)

B

\(\cos^{-1}(x-1) + c \)

C

\(\sin^{-1}\left(\frac{x}{2}\right) + c \)

D

\(\cos^{-1}\left(\frac{x}{2}\right) + c \)

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7

2022 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2022

If \( \vec{V} = a\hat{i} + b\hat{j} + c\hat{k} \), identify the INVALID operation on \( \vec{V} \).

A

\(\nabla \bullet \nabla \times (\vec{V}) \)

B

\(\nabla \times \nabla \bullet (\vec{V}) \)

C

\(\nabla (\nabla \bullet \vec{V}) \)

D

\(\nabla \times \nabla (\vec{V}) \)

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8

2022 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2022

Given \(x\) is real, identify all the even-functions among the following:

A

\(x|x|\)

B

\(\frac{\cos(x)}{x}\)

C

\(\sin x^2\)

D

\(e^{-|x|}\)

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9

2022 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2022

Using divergence theorem, evaluate the integral \(\iint_S \vec{F} \cdot \vec{n} dA\), where \(S\) is the surface of the cone \(x^2 + y^2 \le z^2\), \(0 \le z \le 3\). If \(\vec{F} = 4x\hat{i} + 3z\hat{j} + 5y\hat{k}\) is vector function with outer unit normal vector \(\vec{n}\), the value of the integral is ________ (rounded off to the nearest integer).

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10

2023 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2023

For the function \( f(x, y) = e^x \cos(y) \), what is the value of \( \frac{\partial^2 f}{\partial x \partial y} \) at \( (x = 0, y = \pi/2) \)?

A

0

B

1

C

-1

D

\(e^{\pi/2} \)

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11

2023 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2023

For the function \( f(x) = x^4 - x^2 \), which of the following statements is/are TRUE?

A

The function is symmetric about \(x = 0 \).

B

The minimum value of the function is \(-0.5 \).

C

The function has two minima.

D

The function is an odd function.

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12

2023 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2023

Calculate the reciprocal of the coefficient of \( z^3 \) in the Taylor series expansion of the function \( f(z) = \sin(z) \) around \( z = 0 \). (Provide the answer as an integer.)

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13

2023 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2023

A stone is thrown from an elevation of 2 m above ground level, at an angle of \( 30^\circ \) to the horizontal axis. If the stone hits the ground at a horizontal distance of 6 m from the point of release, at what speed (in m/s) was the stone thrown? Use \( g = 10 \text{ m/s}^2 \) and assume that there is no air resistance. (Round off your answer to one decimal place.)

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14

2024 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2024

For \(\vec{F} = (x + y)\hat{i} + (x + y)\hat{j}\) the value of \(\oint \vec{F} \cdot d\vec{r}\) along the path shown in the figure is ______. Give your answer as an integer.

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15

2025 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2025

For any continuous and differentiable function \(f(x)\) with first derivative \(f'(x) \neq 0\) for all values of \(x\), which one of the following is ALWAYS TRUE for \(a \neq b\)?
Here, \(a\) and \(b\) are finite and real.

A

\(f(a) > f(b)\)

B

\(f(a) = f(b)\)

C

\(f(a) \neq f(b)\)

D

\(f(a) < f(b)\)

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16

2025 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2025

If \( f(x) = x - \frac{1}{x} \), the value of \( \lim_{\Delta x \to 0} \frac{f(1+\Delta x)-f(1)}{\Delta x} \) is __________.

A

0

B

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

C

1

D

2

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17

2026 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2026

Let f be a function of real variables x and y, defined as: \( f(x, y) = x^2 y + 3y^2 x \) The value of \( \frac{\partial^2 f}{\partial x^2} \) at x = 1, y = 1 is ____

A

0

B

1

C

2

D

3

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18

2026 · Biomedical Engineering · Engineering Mathematics · Calculus

Biomedical Engineering (BM) 2026

Let \(f\) be a function of two real variables \(x\) and \(y\), defined as:

\(f(x,y) = x^2 + y^2\)

The value of the line integral \(\int_P^Q \vec{\nabla} f \cdot d\vec{l}\) from point P to point Q along the path \(L\) shown in the figure below is ____

Here \(\vec{\nabla} f = \hat{x} \frac{\partial f}{\partial x} + \hat{y} \frac{\partial f}{\partial y}\) , with \(\hat{x}\) and \(\hat{y}\) as the unit vectors along \(X\) and \(Y\) axes, respectively.

A

0

B

1

C

-1

D

\(\pi\)

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