GATE Overflow for GATE ESE - Questions without answers in Mathematics Foundation

GATE ES 2026 | Question: 1

Mon, 23 Feb 2026 13:15:48 +0000

If $2 X+\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 8 \\ 7 & 2\end{array}\right]$, then $X$ is $\_\_\_\_$.

$\left[\begin{array}{ll}2 & 5 \\ 5 & 3\end{array}\right]$
$\left[\begin{array}{cc}-1 & 3 \\ 2 & -1\end{array}\right]$
$\left[\begin{array}{cc}1 & 3 \\ 2 & -1\end{array}\right]$
$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

GATE ES 2026 | Question: 2

Mon, 23 Feb 2026 13:15:46 +0000

The correct integral of $\int \dfrac{\sin x}{\sqrt{1+\cos x}} d x$ (assuming integral constant as $c$ ) is $\_\_\_\_$ .

$-2 \sqrt{2} \sin \dfrac{x}{2}+c$
$-2 \sqrt{2} \cos \dfrac{x}{2}+c$
$-\dfrac{1}{\sqrt{2}} \cos \dfrac{x}{2}+c$
$-\dfrac{1}{\sqrt{2}} \sin \dfrac{x}{2}+c$

GATE ES 2026 | Question: 3

Mon, 23 Feb 2026 13:15:44 +0000

Given $f(x)=3 x^{4}+4 x^{3}-12 x^{2}+6$, the minimum value of $f(x)$ is $\_\_\_\_$.

$-6$
$-36$
$-26$
$-46$

GATE ES 2026 | Question: 4

Mon, 23 Feb 2026 13:15:42 +0000

A box contains $26$ cards with all English alphabets from $\text{A}$ to $\text{Z}$.

The probability of randomly choosing the card with either '$\text{E}$' or '$\text{S}$' from this box is $\_\_\_\_$.

$2 / 26$
$1 / 26$
$3 / 26$
$4 / 26$

GATE ES 2026 | Question: 5

Mon, 23 Feb 2026 13:15:41 +0000

Which ONE of the following options is the CORRECT formula for Chi-Square $\left(\chi^{2}\right)$ test?

Given, $O_{1}, O_{2}, \ldots . O_{n}$ are the observed values, $E_{1}, E_{2}, \ldots . E_{n}$ are the corresponding expected values, $\bar{O}$ is the mean of observed values, $n$ is the number of observations, $\mu$ is the mean of $E_{i}, i=1, \ldots, n, \sigma_{o}$ is the standard deviation of observed values, and $\sigma_{E}$ is the standard deviation of expected values $E_{i}, i=1, \ldots, n$.

$\chi^{2}=\sum_{\mathrm{i}=1}^{\mathrm{n}} \frac{\left(O_{i}-E_{i}\right)^{2}}{E_{i}}$
$\chi^{2}=\sqrt{\frac{\sum_{i=1}^{n}\left(O_{i}-\bar{O}\right)}{n-1}}$
$\chi^{2}=\dfrac{\bar{O}-\mu}{\sigma_{o} \sqrt{n}}$
$\chi^{2}=\dfrac{\left(\sigma_{o}\right)^{2}}{\left(\sigma_{E}\right)^{2}}$

GATE ES 2026 | Question: 24

Mon, 23 Feb 2026 13:15:02 +0000

A $6$-hour rainfall of $6$ cm at a station was found to have a return period of $40$ years.

The probability that a $6$ -hour rainfall of this or larger magnitude will occur at least once in $20$ successive years at that station is $\_\_\_\_$ (rounded off to three decimal places).

GATE ES 2026 | Question: 26

Mon, 23 Feb 2026 13:15:00 +0000

A first-order ordinary differential equation is given as follows:

$$\dfrac{d y}{d x}+x^{2} y=0$$

Which ONE of the following options CORRECTLY represents the characteristics of this equation?

Linear, homogeneous, and exact
Nonlinear, nonhomogeneous, and exact
Linear, homogeneous, and non-exact
Nonlinear, nonhomogeneous, and non-exact

GATE ES 2026 | Question: 27

Mon, 23 Feb 2026 13:14:58 +0000

Consider the matrix $A=\left[\begin{array}{ccc}2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 10\end{array}\right]$.

Which ONE of the following options CORRECTLY lists the eigenvalues of $A$?

$1 ; 8-\sqrt{37} ;-8+\sqrt{37}$
$2 ; 8-\sqrt{37} ;-8+\sqrt{37}$
$1 ; 8+\sqrt{37} ; \quad 8-\sqrt{37}$
$2 ; 8+\sqrt{37} ; \quad 8+\sqrt{37}$

GATE ES 2026 | Question: 32

Mon, 23 Feb 2026 13:14:47 +0000

A system of linear equations is given as $\mathrm{A X}=\mathrm{B}$, where $\mathrm{A}=\left[\begin{array}{cc}2 & 5 \\ 5 & -2\end{array}\right], \quad \mathrm{B}=\left[\begin{array}{l}7 \\ 3\end{array}\right]$, and $\mathrm{X}$ is the solution vector.

Which of the following statements is/are TRUE for this system?

Matrix $\mathrm{A}$ is skew-symmetric.
Eigenvalues of matrix $\mathrm{A}$ are real.
The system is homogeneous.
The system is singular.

GATE ES 2026 | Question: 38

Mon, 23 Feb 2026 13:14:36 +0000

Standard deviation for the particulate matter (PM) concentrations listed in the table is $\_\_\_\_$ $\mu \mathrm{g} / \mathrm{m}^{3}$ (rounded off to one decimal place).
$$\begin{array}{|l|c|c|c|c|c|c|c|}
\hline
\textbf{Location No.} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\begin{array}{l}
\textbf{PM concentration} \\
(\mu \mathrm{g}/\mathrm{m}^{3})
\end{array}
& 45 & 40 & 55 & 65 & 60 & 35 & 50 \\
\hline
\end{array}$$

GATE ES 2026 | Question: 54

Mon, 23 Feb 2026 13:14:21 +0000

The area in the first quadrant bounded by the function $y=(8-x)$ and the coordinate axes is $\_\_\_\_$ square units (answer in integer).

GATE ES 2025 | Question: 1

Thu, 06 Mar 2025 16:22:57 +0000

Assuming $s>\mid a \mid$; Laplace transform of $f(x)=\cosh a x$ is

$\frac{s}{s^{2}+a^{2}}$
$\frac{a}{s^{2}+a^{2}}$
$\frac{s}{s^{2}-a^{2}}$
$\frac{a}{s^{2}-a^{2}}$

GATE ES 2025 | Question: 2

Thu, 06 Mar 2025 16:22:54 +0000

For the ordinary differential equation $\frac{d^{2} y}{d x^{2}}+4 y=0$, the general solution is

$y=c_{1} \cos 2 x+c_{2} \sin 2 x$
$y=c_{1} \cosh 2 x+c_{2} \sinh 2 x$
$y=c_{1} e^{2 x}+c_{2} e^{-2 x}$
$y=c_{1} e^{2 x} \cos 2 x+c_{2} e^{-2 x} \sin 2 x$

GATE ES 2025 | Question: 3

Thu, 06 Mar 2025 16:22:52 +0000

Consider the following two series
$$\begin{array}{l} \mathrm{P}: \sum_{n=1}^{\infty} \frac{1}{n} \\ \mathrm{Q}: \sum_{n=1}^{\infty} \frac{1}{n^{2}} \end{array}$$

Choose the correct option from the following

$P$ is convergent series; $Q$ is divergent series
$P$ is divergent series; $Q$ is convergent series
Both $P$ and $Q$ are convergent series
Both $P$ and $Q$ are divergent series

GATE ES 2025 | Question: 14

Thu, 06 Mar 2025 16:22:22 +0000

Consider the following statements

Environmental pollutant concentration is generally modeled using lognormal distribution.
Environmental pollutant concentration is generally modeled using Poisson distribution.
The weekly rate of exceedance of environmental pollutant concentration with regards to a given standard is generally modeled using lognormal distribution.
The weekly rate of exceedance of environmental pollutant concentration with regards to a given standard is generally modeled using Poisson distribution.

Choose the correct option(s) from the following

$(i)$ and $(iv)$ are correct
$(i)$ is correct and $(iii)$ is incorrect
$(ii)$ and $(iv)$ are correct
$(ii)$ and $(iv)$ are incorrect

GATE ES 2025 | Question: 24

Thu, 06 Mar 2025 16:21:51 +0000

$\lim _{x \rightarrow 0}\left(\frac{\ln (1+x)}{2 \sin x}\right)$ is $\_\_\_\_\_\_$ (rounded off to two decimal places).

GATE ES 2025 | Question: 37

Thu, 06 Mar 2025 16:21:16 +0000

A residential family is considering two cities for relocation. The data related to pollutant exposure and associated health cost per year are given in the following figure.

The pollutant exposure is characterized in high, mild and low exposure categories with respective probability values. The difference in expected value of health cost of City $1$ with respect to that of City $2$ is $\_\_\_\_\_\_$ lakhs/year. (rounded off to two decimal places)

GATE ES 2025 | Question: 38

Thu, 06 Mar 2025 16:21:14 +0000

The following is a system of linear equations
$$\begin{array}{c} x-2 y+z=34 \\ 2 x+y+z=102 \\x+y-3 z=17\end{array}$$

The value of $(x+y+z)$ is $\_\_\_\_\_\_$ (rounded off to two decimal places)

GATE ES 2025 | Question: 39

Thu, 06 Mar 2025 16:21:14 +0000

The value of $\int_{0}^{\infty} \frac{\sin 4 x}{\pi x} d x$ is $\_\_\_\_\_\_$ (rounded off to two decimal places)

GATE ES 2025 | Question: 40

Thu, 06 Mar 2025 16:21:13 +0000

A tank has inflow, outflow and stirring mechanism. Initially, the tank holds $500 \: L$ of a brine solution of concentration $200 \mathrm{~g} / \mathrm{L}$. At $t=0$, an inflow of another brine solution of concentration $100 \mathrm{~g} / \mathrm{L}$ starts entering the tank at the rate of $15 \mathrm{~L} /$ minute. At the same time the outflow of thoroughly stirred mixture also takes place at the same rate so that the volume of brine in the tank remains constant. The brine concentration $C(\mathrm{~g} / \mathrm{L})$ in the tank at any time $t$ (minute) can be expressed by the following differential equation
$$\frac{d C}{d t}+0.03 C=3$$

The brine concentration in the tank at $t=1.5$ hour is $\_\_\_\_\_\_ \: \mathrm{g/L}$. (rounded off to two decimal places)

GATE ES 2024 | GA Question: 7

Mon, 19 Feb 2024 10:29:49 +0000

The probability of a boy or a girl being born is $1 / 2$. For a family having only three children, what is the probability of having two girls and one boy?

$3 / 8$
$1 / 8$
$1 / 4$
$1 / 2$

GATE ES 2024 | Question: 1

Mon, 19 Feb 2024 10:24:14 +0000

Ten cards in a pack are numbered as $1,2,3, \ldots 10$. The probability of drawing a card with an even number or a number which is a multiple of 5 from the pack is

$4 / 10$
$6 / 10$
$2 / 10$
$3 / 10$

GATE ES 2024 | Question: 14

Mon, 19 Feb 2024 10:24:11 +0000

The eigen values of the matrix $\left[\begin{array}{ll}4 & 3 \\ 3 & 4\end{array}\right]$ are

GATE ES 2024 | Question: 15

Mon, 19 Feb 2024 10:24:11 +0000

If $\mathbf{X}$ is a vector, and $\mathbf{A}$ and $\mathbf{B}$ are linear operators; then the correct mathematical relationship(s) is/are

$(\mathbf{A}+\mathbf{B}) \mathbf{X}=\mathbf{A X}+\mathbf{B X}$
$(\lambda \mathbf{A}) \mathbf{X}=\lambda(\mathbf{A X})$
$(\mathbf{A B}) \mathbf{X}=\mathbf{A}(\mathbf{B X})$
$(\mathbf{A}+\mathbf{B}) \mathbf{X}=\mathbf{A}^{\mathrm{T}}\mathbf{X}+\mathbf{B}^{\mathrm{T}} \mathbf{X}$

GATE ES 2024 | Question: 23

Mon, 19 Feb 2024 10:24:10 +0000

Consider the equation $\frac{d y}{d x}-x^{2}+e^{x}=0$; with $y=1$ at $x=0$.
The value of $y$ at $x=1$ is ________ $\text{(rounded off to 2 decimal places)}$.

Take the value of $e$ $\text{(base of natural logarithm)}$ as $2.7$.

GATE ES 2024 | Question: 27

Mon, 19 Feb 2024 10:24:09 +0000

Which one is the solution $y(x)$ for the following ordinary differential equation and the specified boundary conditions?

\[
\frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y=2 e^{-x} ; y(0)=2 ; \quad\left(\frac{d y}{d x}\right)_{x=0}=1
\]

$y(x)=\frac{1}{3} e^{-x}-2 e^{x}-\frac{1}{3} e^{2 x}$
$y(x)=\frac{1}{3} e^{x}+2 e^{x}-\frac{1}{3} e^{2 x}$
$y(x)=\frac{1}{3} e^{-x}+2 e^{-x}-\frac{1}{3} e^{2 x}$
$y(x)=\frac{1}{3} e^{-x}+2 e^{x}-\frac{1}{3} e^{2 x}$

GATE ES 2024 | Question: 26

Mon, 19 Feb 2024 10:24:09 +0000

A population $\text{(with mean $\mu$)}$ follows normal distribution. Ten samples $(N)$ are drawn at random with a mean value of " $x$ " and standard deviation of " $S$ ". Following table provides the confidence limits, $\mathrm{C}(\mathrm{t})$ of the cumulative probability function for Student's $t$ - distribution two-tailed test with degree of freedom, $\text{D}$

Which one of the following expression is correct for testing the null hypothesis $H_{0}: \mu=0$ at $10 \%$ significance level?

$\text{D}$	$\text{C(t)}$
$\text{D}$	$0.9$	$0.95$	$0.975$
$9$	$1.38$	$1.83$	$2.26$
$10$	$1.37$	$1.81$	$2.23$
$11$	$1.36$	$1.80$	$2.20$

$-1.81<\frac{x}{\frac{S}{\sqrt{N-1}}}<1.81$
$-1.83<\frac{x}{\frac{S}{\sqrt{N-1}}}<1.83$
$-1.37<\frac{x}{\frac{S}{\sqrt{N-1}}}<1.37$
$-2.23<\frac{x}{\frac{S}{\sqrt{N-1}}}<2.23$

GATE ES 2024 | Question: 32

Mon, 19 Feb 2024 10:24:08 +0000

Consider the function $f(x)=\ln (\sin (x))$

Expand $f(x+h)$ using Taylor's series. In this context, the correct statement(s) is/are

Second term in the Taylor's series i.e., the term which includes $h$ is: $h \cdot \ln (\sin (x))$
First term is $\ln (\sin (x))$
Third term in the Taylor's series i.e., the term which includes $h^{2}$ is: $\frac{-h^{2}}{2(\sin (x))^{2}}$
Third term in the Taylor's series i.e., the term which includes $h^{2}$ is: $\frac{2 h^{2}}{(\sin (x))^{2}}$

GATE ES 2024 | Question: 39

Mon, 19 Feb 2024 10:24:07 +0000

Consider the equation for a curve, $y=f(x)=x^{2}+x$.

The area enclosed by the curve, the $x$-axis $\text{( $y=0$ line)}$; the vertical lines passing through $x=1$ and $x=2$ is ________ $\text{(rounded off to $2$ decimal places)}$

GATE ES 2023 | GA Question: 3

Tue, 23 May 2023 03:15:39 +0000

Given a fair six-faced dice where the faces are labelled $'1'$, $'2'$, $'3'$, $'4'$, $'5'$, and $'6'$, what is the probability of getting a $'1'$ on the first roll of the dice and a $'4'$ on the second roll?

$\frac{1}{36}$
$\frac{1}{6}$
$\frac{5}{6}$
$\frac{1}{3}$

GATE ES 2023 | Question: 1

Tue, 23 May 2023 03:12:50 +0000

Given are two ordinary differential equations

$\mathrm{P}: \frac{d y}{d x}+x=x \sin y$

$\mathrm{Q}: \frac{d y}{d x}+x y=e^{x} y$

The correct choice is

$\mathrm{P}$ is linear; $\mathrm{Q}$ is nonlinear
$\mathrm{P}$ is nonlinear; $\mathrm{Q}$ is linear
Both $\mathrm{P}$ and $\mathrm{Q}$ are linear
Both $\mathrm{P}$ and $\mathrm{Q}$ are nonlinear

GATE ES 2023 | Question: 2

Tue, 23 May 2023 03:12:49 +0000

$\mathbf{P}$ and $\mathbf{Q}$ are square matrices. Consider the following

$\mathrm{X}:\left(\mathbf{P}^{-\mathbf{1}}\right)^{-1}=\mathbf{P}$

Y: Symmetric if $\mathbf{Q}=-\mathbf{Q}^{\mathrm{T}}$

The correct choice is

$\mathrm{X}$ is TRUE; $\mathrm{Y}$ is FALSE
$\mathrm{X}$ is FALSE; $\mathrm{Y}$ is TRUE
Both $\mathrm{X}$ and $\mathrm{Y}$ are TRUE
Both $\mathrm{X}$ and $\mathrm{Y}$ are FALSE

GATE ES 2023 | Question: 3

Tue, 23 May 2023 03:12:48 +0000

Given are two infinite series

$\qquad P: \sum \frac{n^{2}+1}{n^{2}}$

$\qquad Q: \sum\left(1+\frac{1}{n}\right)^{-n}$

The correct choice is

$\text{P}$ is convergent series; $\text{Q}$ is divergent series
$\text{P}$ is divergent series; $\text{Q}$ is convergent series
Both $\text{P}$ and $\mathrm{Q}$ are convergent series
Both $\text{P}$ and $\text{Q}$ are divergent series

GATE ES 2023 | Question: 20

Tue, 23 May 2023 03:12:23 +0000

A researcher compiled the following information about the performance of a kit in an outbreak

Infection state	Kit response
Disease (probability =0.002)	Positive response (probability =0.98)
No Disease	Positive response (probability =0.03 )

The probability of detecting an infection for a positive result through the kit would be ___________ (rounded off to three decimal places).

GATE ES 2023 | Question: 23

Tue, 23 May 2023 03:12:20 +0000

The value of $\frac{4}{\pi} \int_{0}^{\pi / 2} \sin ^{2} x d x$ is ___ ____ $\text{(rounded off to two decimal places)}$.

GATE ES 2023 | Question: 26

Tue, 23 May 2023 03:12:18 +0000

$\text{W1, W2, W3...W9}$ represent the holding times of $9$ water samples, which follow a normal distribution with mean $=8.33$ and standard deviation $=4.472$.$\text{M}$ represents the sample mean value of holding times, which also has a normal distribution. Assuming $\mathrm{Z}$ has a standard normal distribution $\text{(mean $=0$ and standard deviation $=1$)}$, select the correct statement which describes the expression for calculating the value of type $1$ error where
$$
\begin{array}{ll}
\text { null hypothesis }\left(\mathrm{H}_0\right): & \mathrm{M}>6 \\
\text { alternate hypothesis }\left(\mathrm{H}_{\mathrm{a}}\right): & \mathrm{M} \leq 6
\end{array}
$$

$\mathrm{P}\{\mathrm{Z}<(-1.565)\}$
$\mathrm{P}\{\mathrm{Z}<1.565\}$
$\mathrm{P}\{\mathrm{Z}>(-1.565)\}$
$\mathrm{P}\{\mathrm{Z}>1.565\}$

GATE ES 2023 | Question: 37

Tue, 23 May 2023 03:12:01 +0000

Second order ordinary differential equation $\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0$ has values $y=2$ and $\frac{d y}{d x}=1$ at $x=0$. The value of y at $x=1$ is _____________ $\text { off to three decimal places).}$

GATE ES 2023 | Question: 38

Tue, 23 May 2023 03:12:01 +0000

Consider two matrices $\mathbf{P}=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]$ and $\mathbf{Q}=\left[\begin{array}{ll}5 & 4 \\ 0 & 2\end{array}\right]$. If $\mathbf{R}=(\mathbf{P Q})^{\mathrm{T}}$ then $\operatorname{det} \mathbf{R}$ is (in integer).

GATE ES 2023 | Question: 39

Tue, 23 May 2023 03:12:00 +0000

For the function } f(x)=x \sqrt{4-x^{2}}, the maximum value in the range $-2 \leq x \leq 2$ is ____________ $\text { (rounded off to two decimal places). }$

GATE ES 2021 | Question: 1

Tue, 14 Mar 2023 03:44:52 +0000

A tangent is drawn on the curve of the function $y=x^{2}$ at the point $(x, y)=(3,9)$. The slope of the tangent is

$3$
$6$
$9$
$12$

GATE ES 2021 | Question: 2

Tue, 14 Mar 2023 03:44:52 +0000

$\lim _{x \rightarrow 0} \frac{x^{2}}{\sin x}=$

$0$
$1$
$2$
$-1$

GATE ES 2021 | Question: 20

Tue, 14 Mar 2023 03:44:29 +0000

The ordinary differential equation
\[
\frac{d y}{d x}=x^{2} y
\]
has $y$ as the dependent variable and $x$ as the independent variable. Which of the following classification(s) is/are applicable to the equation?

Linear
Non-linear
First order
Second order

GATE ES 2021 | Question: 27

Tue, 14 Mar 2023 03:44:23 +0000

A biased die has six faces numbered as $k=1,2,3,4,5$, and $6$. On rolling the die, the probability of the number k appearing is proportional to $k^{2}$. What is the probability that an even number will appear on rolling the die?

$\frac{35}{91}$
$\frac{56}{91}$
$\frac{12}{21}$
$\frac{9}{21}$

GATE ES 2021 | Question: 42

Tue, 14 Mar 2023 03:43:58 +0000

Consider a function $y=f(x)$ which satisfies the following equation:
$\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0$
As $x \rightarrow-\infty, y=1$, and at $x=0, y=2$.

The value of $\frac{d y}{d x}$ at $x=0$ is _________________ $\text{(answer in integer)}$

GATE ES 2022 | Question: 1

Fri, 17 Feb 2023 07:06:16 +0000

A student reported following arsenic concentrations in water samples:
$$\begin{array}{|l|l|l|l|l|l|l|l|}\hline \text{Arsenic concentration (mg/L)} & 0.10 & 0.12 & 0.20 & 0.05 & 0.40 & 0.30 & 0.35 \\\hline \end{array}$$
Which one of the following is a correct statement about arsenic concentration distribution?

Arsenic concentration distribution is symmetric
Arsenic concentration distribution is positively skewed
Arsenic concentration distribution is negatively skewed
Arsenic concentration is following Weibull distribution

GATE ES 2022 | Question: 2

Fri, 17 Feb 2023 07:06:15 +0000

$\mathrm{X}$ is normally distributed with following data $(25.8,36.6,26.3,21.8,27.2)$. Select the correct statement about $X\left(\mathrm{t}_{\text {crit }, \alpha}=0.05,4=2.132\right)$ :

Population mean $\leq 25$ with $95 \%$ confidence
Population mean $\leq 25$ with $100 \%$ confidence
Population mean $>25$ with $95 \%$ confidence
Population mean $>25$ with $100 \%$ confidence

GATE ES 2022 | Question: 4

Fri, 17 Feb 2023 07:06:14 +0000

Given $\mathbf{P}$ is $m \times n$ matrix, $\mathbf{Q}$ is $n \times l$ matrix, $\mathbf{R}$ and $\mathbf{S}$ are $n \times n$ matrices, and superscript $\mathrm{T}$ denotes transpose. Consider

Relationship $1$: $(\mathbf{P Q})^{\mathrm{T}}=\mathbf{Q}^{\mathrm{T}} \mathbf{P}^{\mathrm{T}}$

Relationship $2$: $(\mathbf{R S})^{\mathrm{-1}}=\mathbf{S}^{\mathrm{-1}} \mathbf{R}^{\mathrm{-1}}$

Which one of the following is correct?

Relationship $1$ is false; Relationship $2$ is false
Relationship $1$ is true; Relationship $2$ is false
Relationship $1$ is true; Relationship $2$ is true
Relationship $1$ is false; Relationship $2$ is true

GATE ES 2022 | Question: 6

Fri, 17 Feb 2023 07:06:13 +0000

Ten litres of the sample was filtered through a membrane filter and the filter was transferred to solid agar media which supports growth of coliform organisms. After incubation at $37{ }^{\circ} \mathrm{C}$ for $48$ hours, the agar plates showed an average of $64$ colonies per plate. What was the average concentration of coliform organisms (in Colony Forming Units per unit volume) in the original water sample?

$64 \;\mathrm{CFU} / \mathrm{ml}$
$640 \; \mathrm{CFU} / \mathrm{ml}$
$6.4 \times 10^{-3} \; \mathrm{CFU} / \mathrm{ml}$
$64 \times 10^{-3} \; \mathrm{CFU} / \mathrm{ml}$

GATE ES 2022 | Question: 24

Fri, 17 Feb 2023 07:05:53 +0000

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$ is (rounded off to one decimal place)

GATE ES 2022 | Question: 26

Fri, 17 Feb 2023 07:05:52 +0000

Given, $y=f(x) ; \frac{d^{2} y}{d x^{2}}+4 y=0 ; y(0)=0 ; \frac{d y}{d x}(0)=1$. The problem is a/an

initial value problem having solution $\mathrm{y}=x$
boundary value problem having solution $\mathrm{y}=x$
initial value problem having solution $\mathrm{y}=\frac{1}{2} \sin 2 x$
boundary value problem having solution $\mathrm{y}=\frac{1}{2} \sin 2 x$