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Linear Algebra(9)Calculus(18)Probability and Statistics(9)

Linear Algebra

9 Questions

GATE→2026→Mathematics Foundation→Linear Algebra

Q1

MCQ[1 Mark]

If $2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}$, then $X$ is

(A)

\begin{bmatrix} 2 & 5 \\ 5 & 3 \end{bmatrix}

(B)

\begin{bmatrix} -1 & 3 \\ 2 & -1 \end{bmatrix}

(C)

\begin{bmatrix} 1 & 3 \\ 2 & -1 \end{bmatrix}

(D)

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Explanation

GATE→2026→Mathematics Foundation→Linear Algebra

Q2

MCQ[2 Marks]

Consider the matrix $A = \begin{bmatrix} 2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 10 \end{bmatrix}$. Which ONE of the following options CORRECTLY lists the eigenvalues of A?

(A)

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

(B)

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

(C)

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

(D)

$2; 8+\sqrt{37}; 8+\sqrt{37}$

Explanation

GATE→2026→Mathematics Foundation→Linear Algebra

Q3

MSQ[2 Marks]

A system of linear equations is given as $AX = B$, where $A = \begin{bmatrix} 2 & 5 \\ 5 & -2 \end{bmatrix}$, $B = \begin{bmatrix} 7 \\ 3 \end{bmatrix}$ and $X$ is the solution vector. Which of the following statements is/are TRUE for this system?

(A)

Matrix A is skew-symmetric.

(B)

Eigenvalues of matrix A are real.

(C)

The system is homogeneous.

(D)

The system is singular.

Explanation

GATE→2024→Mathematics Foundation→Linear Algebra

Q4

MSQ[1 Mark]

The eigenvalues of the matrix $A = \begin{bmatrix} 4 & 3 \\ 3 & 4 \end{bmatrix}$ are

(A)

1

(B)

2

(C)

7

(D)

0

Explanation

GATE→2024→Mathematics Foundation→Linear Algebra

Q5

MSQ[1 Mark]

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

(A)

$(A+B)X = AX + BX$

(B)

$(\lambda A)X = \lambda(AX)$

(C)

$(AB)X = A(BX)$

(D)

$(A+B)X = A^T X + B^T X$

Explanation

GATE→2023→Mathematics Foundation→Linear Algebra

Q6

MCQ[1 Mark]

P and Q are square matrices. Consider the following: $X: (P^{-1})^{-1} = P$ $Y: \text{Symmetric if } Q = -Q^T$ The correct choice is

(A)

X is TRUE; Y is FALSE

(B)

X is FALSE; Y is TRUE

(C)

Both X and Y are TRUE

(D)

Both X and Y are FALSE

Explanation

GATE→2023→Mathematics Foundation→Linear Algebra

Q7

NAT[2 Marks]

Consider two matrices $P = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ and $Q = \begin{bmatrix} 5 & 4 \\ 0 & 2 \end{bmatrix}$. If $R = (PQ)^T$ then $\det R$ is _____ (in integer).

Explanation

GATE→2022→Mathematics Foundation→Linear Algebra

Q8

MCQ[1 Mark]

Given P is $m \times n$ matrix, Q is $n \times l$ matrix, R and S are $n \times n$ matrices, and superscript T denotes transpose. Consider: Relationship 1: $(PQ)^T = Q^T P^T$ Relationship 2: $(RS)^{-1} = S^{-1} R^{-1}$ Which one of the following is correct?

(A)

Relationship 1 is false; Relationship 2 is false

(B)

Relationship 1 is true; Relationship 2 is false

(C)

Relationship 1 is true; Relationship 2 is true

(D)

Relationship 1 is false; Relationship 2 is true

Explanation

GATE→2022→Mathematics Foundation→Linear Algebra

Q9

MCQ[2 Marks]

Eigenvalues of the matrix $\begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix}$ are

(A)

1 and 2

(B)

1 and 3

(C)

1 and -2

(D)

0 and 3

Explanation

Calculus

18 Questions

GATE→2026→Mathematics Foundation→Calculus

Q10

MCQ[1 Mark]

The correct integral of $\int \frac{\sin x}{\sqrt{1+\cos x}} dx$ (assuming integral constant as $c$) is

(A)

$-2\sqrt{2} \sin\left(\frac{x}{2}\right) + c$

(B)

$-2\sqrt{2} \cos\left(\frac{x}{2}\right) + c$

(C)

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

(D)

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

Explanation

GATE→2026→Mathematics Foundation→Calculus

Q11

MCQ[1 Mark]

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

(A)

-6

(B)

-36

(C)

-26

(D)

-46

Explanation

GATE→2026→Mathematics Foundation→Calculus

Q12

MCQ[2 Marks]

A first-order ordinary differential equation is given as follows: $\frac{dy}{dx} + x^2y = 0$. Which ONE of the following options CORRECTLY represents the characteristics of this equation?

(A)

Linear, homogeneous, and exact

(B)

Nonlinear, nonhomogeneous, and exact

(C)

Linear, homogeneous, and non-exact

(D)

Nonlinear, nonhomogeneous, and non-exact

Explanation

GATE→2026→Mathematics Foundation→Calculus

Q13

NAT[2 Marks]

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

Explanation

GATE→2024→Mathematics Foundation→Calculus

Q14

NAT[1 Mark]

$f(x) = x^3 - 4.5x^2 - 12x$ has a local maximum at $x =$ _____ (an integer value) in the range $x = -2$ to $+2$.

Explanation

GATE→2024→Mathematics Foundation→Calculus

Q15

NAT[1 Mark]

Consider the equation $\frac{dy}{dx} - x^2 + e^x = 0$; with $y = 1$ at $x = 0$. The value of $y$ at $x = 1$ is _____ (rounded off to 2 decimal places). (Take $e = 2.7$)

Explanation

GATE→2024→Mathematics Foundation→Calculus

Q16

MCQ[2 Marks]

Which one is the solution $y(x)$ for the following ordinary differential equation and the specified boundary conditions? $\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 2e^{-x}$; $y(0) = 2$; $\left(\frac{dy}{dx}\right)_{x=0} = 1$

(A)

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

(B)

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

(C)

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

(D)

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

Explanation

GATE→2024→Mathematics Foundation→Calculus

Q17

MSQ[2 Marks]

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

(A)

Second term in the Taylor's series i.e., the term which includes $h$ is: $h \cdot \ln(\sin(x))$

(B)

First term is $\ln(\sin(x))$

(C)

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

(D)

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

Explanation

GATE→2024→Mathematics Foundation→Calculus

Q18

NAT[2 Marks]

Consider the equation for a curve, $y = f(x) = x^2 + x$. The area enclosed by the curve, the $x$-axis ($y=0$ line); the vertical lines passing through $x = 1$ and $x = 2$ is _____ (rounded off to 2 decimal places).

Explanation

GATE→2023→Mathematics Foundation→Calculus

Q19

MCQ[1 Mark]

Given are two ordinary differential equations: $P: \frac{dy}{dx} + x = x \sin y$ $Q: \frac{dy}{dx} + xy = e^xy$ The correct choice is

(A)

P is linear; Q is nonlinear

(B)

P is nonlinear; Q is linear

(C)

Both P and Q are linear

(D)

Both P and Q are nonlinear

Explanation

GATE→2023→Mathematics Foundation→Calculus

Q20

MCQ[1 Mark]

Given are two infinite series: $P: \sum \frac{n^2 + 1}{n^2}$ $Q: \sum \left(1 + \frac{1}{n}\right)^{-n}$ The correct choice is

(A)

P is convergent series; Q is divergent series

(B)

P is divergent series; Q is convergent series

(C)

Both P and Q are convergent series

(D)

Both P and Q are divergent series

Explanation

GATE→2023→Mathematics Foundation→Calculus

Q21

NAT[1 Mark]

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

Explanation

GATE→2023→Mathematics Foundation→Calculus

Q22

NAT[2 Marks]

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

Explanation

GATE→2023→Mathematics Foundation→Calculus

Q23

NAT[2 Marks]

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

Explanation

GATE→2022→Mathematics Foundation→Calculus

Q24

MCQ[1 Mark]

Assuming $s > 0$; Laplace transform for $f(x) = \sin ax$ is

(A)

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

(B)

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

(C)

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

(D)

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

Explanation

GATE→2022→Mathematics Foundation→Calculus

Q25

NAT[1 Mark]

$\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$ is _____ (rounded off to one decimal place).

Explanation

GATE→2022→Mathematics Foundation→Calculus

Q26

MCQ[2 Marks]

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

(A)

initial value problem having solution $y = x$

(B)

boundary value problem having solution $y = x$

(C)

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

(D)

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

Explanation

GATE→2022→Mathematics Foundation→Calculus

Q27

NAT[2 Marks]

The area of the region (rounded off to one decimal place) enclosed between the curves $y = x$ and $y = 3\sqrt{x}$ and between the lines $x = 0$ and $x = 1$ is _____ units.

Explanation

Probability and Statistics

9 Questions

GATE→2026→Mathematics Foundation→Probability and Statistics

Q28

MCQ[1 Mark]

A box contains 26 cards with all English alphabets from A to Z. The probability of randomly choosing the card with either 'E' or 'S' from this box is

(A)

2/26

(B)

1/26

(C)

3/26

(D)

4/26

Explanation

GATE→2026→Mathematics Foundation→Probability and Statistics

Q29

MCQ[1 Mark]

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

(A)

$\chi^2 = \sum_{i=1}^{n}\frac{(O_i-E_i)^{2}}{E_i}$

(B)

$\chi^2 = \sqrt{\frac{\sum_{i=1}^{n}(O_i-\overline{O})^{2}}{n-1}}$

(C)

$\chi^2 = \frac{\overline{O}-\mu}{\sigma_o\sqrt{n}}$

(D)

$\chi^2 = \frac{(\sigma_o)^{2}}{(\sigma_E)^{2}}$

Explanation

GATE→2024→Mathematics Foundation→Probability and Statistics

Q30

MCQ[1 Mark]

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

(A)

4/10

(B)

6/10

(C)

2/10

(D)

3/10

Explanation

GATE→2024→Mathematics Foundation→Probability and Statistics

Q31

MCQ[2 Marks]

A population follows normal distribution. Ten samples ($N=10$) are drawn at random with a mean value of $\overline{x}$ and standard deviation of $S$. Following table provides confidence limits $C(t)$ for Student's $t$-distribution two-tailed test with degrees of freedom $D$. Which one of the following expression is correct for testing the null hypothesis $H_0: \mu = 0$ at $10\%$ significance level?

(A)

Option (A)

(B)

Option (B)

(C)

Option (C)

(D)

Option (D)

Explanation

GATE→2023→Mathematics Foundation→Probability and Statistics

Q32

NAT[1 Mark]

A researcher compiled the following information about the performance of a kit in an outbreak: - Disease (probability = $0.002$), Positive response given disease = $0.98$ - No Disease (probability = $0.998$), Positive response given no disease = $0.03$ The probability of detecting an infection for a positive result through the kit would be _____ (rounded off to three decimal places).

Explanation

GATE→2023→Mathematics Foundation→Probability and Statistics

Q33

MCQ[2 Marks]

$W_1, W_2, \dots, W_9$ represent the holding times of 9 water samples, which follow a normal distribution with mean $\mu = 8.33$ and standard deviation $S = 4.472$. $M$ represents the sample mean value of holding times. Assuming $Z$ has a standard normal distribution, select the correct statement which describes the expression for calculating the value of type 1 error where null hypothesis $H_0: \mu = 0$ and alternate hypothesis $H_a: M \le 6$.

(A)

P\{Z < (-1.565)\}

(B)

P\{Z < 1.565\}

(C)

P\{Z > (-1.565)\}

(D)

P\{Z > 1.565\}

Explanation

GATE→2022→Mathematics Foundation→Probability and Statistics

Q34

MCQ[1 Mark]

A student reported following arsenic concentrations in water samples (mg/L): $0.10, 0.12, 0.20, 0.05, 0.40, 0.30, 0.35$. Which one of the following is a correct statement about arsenic concentration distribution?

(A)

Arsenic concentration distribution is symmetric

(B)

Arsenic concentration distribution is positively skewed

(C)

Arsenic concentration distribution is negatively skewed

(D)

Arsenic concentration is following Weibull distribution

Explanation

GATE→2022→Mathematics Foundation→Probability and Statistics

Q35

MCQ[1 Mark]

X is normally distributed with following data ($25.8, 36.6, 26.3, 21.8, 27.2$). Select the correct statement about X:

(A)

Population mean $\le 25$ with 95% confidence

(B)

Population mean $\le 25$ with 100% confidence

(C)

Population mean $> 25$ with 95% confidence

(D)

Population mean $> 25$ with 100% confidence

Explanation

GATE→2022→Mathematics Foundation→Probability and Statistics

Q36

NAT[2 Marks]

An individual has four different email accounts. $60\%$ of emails come into his corporate account, $30\%$ come into his gmail account and remaining $10\%$ are equally divided into his yahoo and zoho accounts. Only $1\%$ of the emails in his corporate accounts are spam, whereas corresponding percentages for gmail, yahoo and zoho accounts are $2\%$, $3\%$ and $3\%$, respectively. Assuming that the same spam filter is used in all the four email accounts, the probability (in percentage, rounded off to one decimal place) of having a randomly selected email as spam is _____.

Explanation