Differential Equations - Engineering Mathematics - Biotechnology Previous Year Questions

1

2013 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2013

The solution to \(\frac{dy}{dx} + y \cot x = \cosec x\) is

A

\(y = (c+x) \cot x\)

B

\(y = (c+x) \cosec x\)

C

\(y = (c + x) \cosec x \cot x\)

D

\(y = (c+x) \frac{\cosec x}{\cot x}\)

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2

2013 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2013

Select the pair that best expresses a relationship similar to that expressed in the pair:
water: pipe:

A

cart: road

B

electricity: wire

C

sea: beach

D

music: instrument

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3

2014 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2014

The concentration profile of a chemical at a location x and time t, denoted by c(x,t), changes as per the following equation,
\[c(x, t) = \frac{c_0}{\sqrt{2\pi D t}} \exp\left[-\frac{x^2}{2 D t}\right]\]
where D and c₀ are assumed to be constant. Which of the following is correct?

A

\(\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}\)

B

\(\frac{\partial c}{\partial t} = \frac{D}{2} \frac{\partial^2 c}{\partial x^2}\)

C

\(\frac{\partial^2 c}{\partial t^2} = D \frac{\partial^2 c}{\partial x^2}\)

D

\(\frac{\partial^2 c}{\partial t^2} = \frac{D}{2} \frac{\partial^2 c}{\partial x^2}\)

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4

2016 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2016

d²y/dx² − y = 0. The initial conditions for this second order homogeneous differential equation are y(0) = 1 and dy/dx = 3 at x = 0.
The value of y when x = 2 is __________.

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5

2017 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2017

For y = f(x), if \(\frac{d^2 y}{dx^2} = 0\), \(\frac{dy}{dx} = 0\) at x = 0, and y = 1 at x =1, the value of y at x = 2 is ______.

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6

2019 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2019

Which one of the following equations represents a one-dimensional wave equation?

A

\(\frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2}\)

B

\(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\)

C

\(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial u}{\partial x}\)

D

\(\frac{\partial^2 u}{\partial t^2} + \frac{\partial^2 u}{\partial x^2} = 0\)

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7

2019 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2019

What is the solution of the differential equation \(\frac{dy}{dx} = \frac{x}{y}\), with the initial condition, at \(x = 0\), \(y = 1\)?

A

\(x^2 = y^2 + 1\)

B

\(y^2 = x^2 + 1\)

C

\(y^2 = 2x^2 + 1\)

D

\(x^2 - y^2 = 0\)

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8

2022 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2022

What is the order of the differential equation given below? \[ \frac{d^2 y}{dx^2} - 6x = 3x^4 - 2x^3 + 2 \]

A

1

B

2

C

3

D

4

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9

2024 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2024

The solution of the differential equation \( \frac{dy}{dx} = y + e^{-x} \) that satisfies \( y(0) = -\frac{1}{2} \) is ______.

A

\(-\frac{1}{2} e^{-\frac{x}{2}} \)

B

\(-\frac{1}{2} e^{x} \)

C

\(-\frac{1}{2} e^{-x} \)

D

\(-\frac{1}{2} e^{\frac{x}{2}} \)

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10

2024 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2024

Let \(y(x) = x^2 \ln x\) for \(x > 0\), be a solution of \(x^2 \frac{d^2 y}{dx^2} + 4y = \alpha x \frac{dy}{dx}\). Then the value of \(\alpha\) is ______.

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11

2025 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2025

Let \( y(t) \) be a bacterial population whose growth is given by \[ \frac{dy}{dt} = \lambda (y + 2) \] where \( \lambda \) is the growth rate constant. If \( y(0) = 1 \) and \( y(1) = 4 \), then the value of \( \lambda \) is

A

ln 2

B

ln 3

C

ln 4

D

ln 6

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12

2025 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2025

A thermometer measuring body temperature follows a first-order response with a time constant of 40 seconds. The instrument will reach 95% of its steady-state output at ______ seconds.
(Round off to the nearest integer)

A

60

B

80

C

120

D

160

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13

2025 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2025

The output \(y(t)\) of a first-order process is governed by the following differential equation
\[\tau_p \frac{dy}{dt} + y = K_p f(t)\]
where \(\tau_p\) is a non-zero time constant, \(K_p\) is the gain and \(f(t)\) is the input with \(f(0) = 0\).
Assume \(y(0) = 0\). The transfer function for this process is (consider \(s\) as the independent variable in the Laplace domain)

A

\(\frac{K_p}{\tau_p s + 1}\)

B

\(\frac{\tau_p}{K_p s + 1}\)

C

\(\frac{\tau_p}{K_p(s + 1)}\)

D

\(\frac{K_p}{\tau_p(s + 1)}\)

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14

2025 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2025

Let \(m\) and \(n\) be fixed real numbers. If the function \(y(t) = C_1 e^t + C_2 e^{-t}\) is a solution of
\[\frac{d^2y}{dt^2} + m \frac{dy}{dt} + ny = 0\]
for any constants \(C_1\) and \(C_2\), then \(m + n\) is equal to

A

-2

B

-1

C

0

D

1

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15

2026 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2026

The equation \frac{d^{2}y}{dx^{2}} - y = 0 has a solution of the form y = e^{Ax}. The value(s) of A satisfying this is/are:

A

0

B

1

C

-1

D

-∞

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16

2026 · Biotechnology · Engineering Mathematics · Differential Equations

Biotechnology (BT) 2026

A heat exchanger during operation in a bioprocess has a steady temperature of 90 °C. After completion of its operation, it was shut down and it was observed that the rate of decrease of temperature at any time was directly proportional to the difference \(T(t) - 30\) °C, where \(T(t)\) denotes temperature at time \(t\). It was observed that it took 30 min for the temperature to drop to 70 °C. The temperature after 51.5 min will be _______ °C. (rounded off to the nearest integer)

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