GATE Civil Engineering

Let $f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}$ where $p$ is a constant. The value of $\frac{d^3}{dx^3} f(x)$ at $x = 0$ is

Suppose $\lambda$ is an eigenvalue of matrix A and x is the corresponding eigenvector. Let x also be an eigenvector of the matrix B = A - 2I, where I is the identity matrix. Then,...

Let $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 3 \\ 1 & -2 & -3 \end{bmatrix}$ and $b = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$. For Ax = b to be solvable, which one of the...

Consider a discrete random variable X whose probabilities are given below. The standard deviation of the random variable is ________ (round off to one decimal place). xi | 1 | 2 |...

Consider the function given below and pick one or more CORRECT statement(s) from the following choices. $$ f(x)=x^3-\frac{15}{2} x^2+18 x+20 $$

Let $y$ be the solution of the initial value problem $y^{\prime}+0.8 y+0.16 y=0$ where $y(0)=3$ and $y^{\prime}(0)=4.5$. Then, $y(1)$ is equal to__________ (rounded off to 1 decima...

A one-way, single lane road has traffic that consists of $30 \%$ trucks and $70 \%$ cars. The speed of trucks (in km/h) is a uniform random variable on the interval ( 30,60 ), and...

Integration of $\ln (x)$ with $x$ i.e. $$ \int \ln (x) d x= $$__________

$X$ is the random variable that can take any one of the values, $0,1,7,11$ and 12 . The probability mass function for $X$ is $$ \begin{aligned} & \mathrm{P}(X=0)=0.4 ; \mathrm{P}(X...

Which of the following equations belong/belongs to the class of second-order, linear, homogeneous partial differential equations:

Consider the differential equation given below. Using the Euler method with the step size (h) of 0.5 , the value of $y$ at $x=1.0$ is equal to _________ (rounded off to 1 decimal p...

The "order" of the following ordinary differential equation is $\qquad$ . $$ \frac{d^3 y}{d x^3}+\left(\frac{d^2 y}{d x^2}\right)^6+\left(\frac{d y}{d x}\right)^4+y=0 $$

For the matrix $[\mathrm{A}]$ given below, the transpose is $\qquad$ . $$ [A]=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{array}\right] $$

Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$. For $\mathrm{Ax}=\mathrm{b}$ to be sol...

Consider a discrete random variable $X$ whose probabilities are given below. The standard deviation of the random variable is _________ (round off to one decimal place). $$ \begin{...

$$ \text { The value of }\mathop {\lim }\limits_{x \to \infty } \left(x-\sqrt{x^2+x}\right) \text { is equal to }$$

Suppose $\lambda$ is an eigenvalue of matrix A and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $\mathrm{B}=\mathrm{A}-2 \mathrm{I}$, where I...

Consider a velocity vector, $\vec{V}$ in ( $\mathrm{x}, \mathrm{y}, \mathrm{z}$ ) coordinates given below. Pick one or more CORRECT statement(s) from the choices given below: $$ \v...

Pick the CORRECT solution for the following differential equation $\frac{d y}{d x}=e^{x-y}$

Pick the CORRECT eigen value(s) of the matrix $[\mathrm{A}]$ from the following choices. $$ [A]=\left[\begin{array}{ll} 6 & 8 \\ 4 & 2 \end{array}\right] $$

The maximum value of the function $h(x)=-x^3+2 x^2$ in the interval $[-1,1.5]$ is equal to _________ . (rounded off to 1 decimal place)

Which one of the following options is the correct Fourier series of the periodic function $f(x)$ described below: $$ f(x)=\left\{\begin{array}{cl} 0 & \text { if }-2

Given that $A$ and $B$ are not null sets, which of the following statements regarding probability is/are CORRECT?

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication. P: (A + B)ᵀ = Aᵀ + Bᵀ Q: (AB)ᵀ = AᵀBᵀ Which one of the following...

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having...

Three vectors $\vec{p}$, $\vec{q}$, and $\vec{r}$ are given as $\vec{p} = \hat{i} + \hat{j} + \hat{k}$ $\vec{q} = \hat{i} + 2\hat{j} + 3\hat{k}$ $\vec{r} = 2\hat{i} + 3\hat{j} + 4\...

The smallest positive root of the equation $$x^5 - 5 x^4 - 10 x^3 + 50 x^2 + 9 x - 45 = 0$$ lies in the range

The second-order differential equation in an unknown function $$u : u(x, y)$$ is defined as $$\frac{\partial^2 u}{\partial x^2}= 2$$ Assuming $$g : g(x)$$, $$f : f(y)$$, and $$h :...

The probability that a student passes only in Mathematics is $\frac{1}{3}$. The probability that the student passes only in English is $\frac{4}{9}$. The probability that the stude...

For the following partial differential equation, $x \frac{\partial^2 f}{\partial x^2} + y \frac{\partial^2 f}{\partial y^2} = \frac{x^2 + y^2}{2}$ which of the following option(s)...

Consider the data of $f(x)$ given in the table. $i$ $0$ $1$ $2$ $x_i$ $1$ $2$ $3$ $f(x_i)$ $0$ $0.3010$ $0.4771$ The value of $f(1.5)$ estimated using second-order Newton’s interpo...

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ ?

A vector field $\vec{p}$ and a scalar field $r$ are given by: $\vec{p} = (2x^2 - 3xy + z^2) \hat{i} + (2y^2 - 3yz + x^2) \hat{j} + (2z^2 - 3xz + x^2) \hat{k}$ $r = 6x^2 + 4y^2 - z^...

The return period of a large earthquake for a given region is 200 years. Assuming that earthquake occurrence follows Poisson’s distribution, the probability that it will be exceede...

A 2 m × 2 m tank of 3 m height has inflow, outflow and stirring mechanisms. Initially, the tank was half-filled with fresh water. At $ t = 0 $, an inflow of a salt solution of conc...

A partial differential equation $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$ is defined for the two-dimensional field $T: T(x, y)$, inside a planar...

The statements P and Q are related to matrices A and B , which are conformable for both addition and multiplication. P: $(A + B)^T = A^T + B^T$ Q: $(AB)^T = B^T A^T$ Which one of t...

The second derivative of a function $f$ is computed using the fourth-order Central Divided Difference method with a step length $h$. The CORRECT expression for the second derivativ...

The function $f(x) = x^3 - 27x + 4$, $1 \leq x \leq 6$ has

Consider two Ordinary Differential Equations (ODEs): P: $ \dfrac{dy}{dx} = \dfrac{x^4 + 3x^2 y^2 + 2y^4}{x^3 y} $ Q: $ \dfrac{dy}{dx} = -\dfrac{y^2}{x^2} $ Which one of the followi...

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having...

Three vectors $\overrightarrow{p}$, $\overrightarrow{q}$, and $\overrightarrow{r}$ are given as $ \overrightarrow{p} = \hat{i} + \hat{j} + \hat{k}$ $ \overrightarrow{q} = \hat{i} +...

The expression for computing the effective interest rate $(i_{eff})$ using continuous compounding for a nominal interest rate of 5% is $i_{eff} = \lim\limits_{m \to \infty} \left(1...

Consider two matrices $A = \begin{bmatrix}2 & 1 & 4 \\ 1 & 0 & 3\end{bmatrix}$ and $B = \begin{bmatrix}-1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}$. The determinant of the matrix $AB$ i...

If M is an arbitrary real n x n matrix, then which of the following matrices will have non-negative eigenvalues?

The following function is defined over the interval [-L, L]: f(x) = px^4 + qx^5. If it is expressed as a Fourier series, f(x) = a_0 + \sum_{n=1}^{\infty} \{a_n \sin(\frac{\pi x}{L}...

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100 °C and the other thr...

The solution of the differential equation $\frac{d^3y}{dx^3} - 5.5 \frac{d^2y}{dx^2} + 9.5 \frac{dy}{dx} - 5 y = 0$ is expressed as $y = C_1e^{2.5 x} + C_2e^{\alpha x} + C_3e^{\bet...

Two vectors $[2 \ 1 \ 0 \ 3]^T$ and $[1 \ 0 \ 1 \ 2]^T$ belong to the null space of a $4 \times 4$ matrix of rank 2. Which one of the following vectors also belongs to the null spa...

For the integral $\rm I=\displaystyle\int^1_{-1}\frac{1}{x^2}dx$ which of the following statements is TRUE?

If M is an arbitrary real n × n matrix, then which of the following matrices will have non-negative eigenvalues?

The following function is defined over the interval [-L, L]: f(x) = px 4 + qx 5 . If it is expressed as a Fourier series, $\rm f(x)=a_0 +\displaystyle\sum^\infty_{n=1} \left\{a_n \...

The probabilities of occurrences of two independent events A and B are 0.5 and 0.8, respectively. What is the probability of occurrence of at least A or B (rounded off to one decim...

In the differential equation $\frac{dy}{dx}+\alpha\ x\ y =0, \alpha$ is a positive constant. If y = 1.0 at x = 0.0, and y = 0.8 at x = 1.0, the value of α is ________ (rounded off...

For the matrix $[A]= \begin{bmatrix}1&2&3\\\ 3&2&1\\\ 3&1&2 \end{bmatrix} $ which of the following statements is/are TRUE?

For the function f(x) = e x |sin x|; x ∈ ℝ, which of the following statements is/are TRUE?

The differential equation, $\rm \frac{du}{dt}+2tu^2=1,$ is solved by employing a backward difference scheme within the finite difference framework. The value of 𝑢 at the (𝑛 − 1)...

Let 𝜙 be a scalar field, and 𝒖 be a vector field. Which of the following identities is true for div(𝜙𝒖)?

A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000. Out of the total vehicles, 30% are without pollution clearance certificate. Furt...

For the matrix $\rm [A]=\begin{bmatrix}1&-1&0\\\ -1&2&-1\\\ 0&-1&1\end{bmatrix}$ which of the following statements is/are TRUE?

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100°C and the other thre...

The solution of the differential equation $\rm \frac{d^3y}{dx^3}-5.5\frac{d^2y}{dx^2}+9.5\frac{dy}{dx}-5y=0$ is expressed as 𝑦 = 𝐶 1 𝑒 2.5𝑥 + 𝐶 2 𝑒 𝛼𝑥 + 𝐶 3 𝑒 𝛽𝑥 , wher...

Two vectors [2 1 0 3] 𝑇 and [1 0 1 2] 𝑇 belong to the null space of a 4 × 4 matrix of rank 2. Which one of the following vectors also belongs to the null space?

Cholesky decomposition is carried out on the following square matrix [𝐴]. $\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$ Let 𝑙 ij and 𝑎ij be the (i, j) th elements of m...

Let y be a non-zero vector of size 2022 × 1. Which of the following statement(s) is/are TRUE?

Consider the following expression: z = sin(y + it) + cos(y $$-$$ it) where z, y, and t are variables, and $$i = \sqrt { - 1} $$ is a complex number. The partial differential equati...

For the equation $${{{d^3}y} \over {d{x^3}}} + x{\left( {{{dy} \over {dx}}} \right)^{3/2}} + {x^2}y = 0$$ the correct description is

The matrix M is defined as $$M = \left[ {\matrix{ 1 & 3 \cr 4 & 2 \cr } } \right]$$ and has eigenvalues 5 and $$-$$2. The matrix Q is formed as Q = M 3 $$-$$ 4M 2 $$-$$ 2M Which of...

Consider the following recursive iteration scheme for different values of variable P with the initial guess x 1 = 1: $${x_{n + 1}} = {1 \over 2}\left( {{x_n} + {P \over {{x_n}}}} \...

The Fourier cosine series of a function is given by : $$f(x) = \sum\limits_{n = 0}^\infty {{f_n}\cos nx} $$ For f(x) = cos 4 x, the numerical value of (f 4 + f 5 ) is _________. (r...

The Cartesian coordinates of a point P in a right-handed coordinate system are (1, 1, 1). The transformed coordinates of P due to a 45$$^\circ$$ clockwise rotation of the coordinat...

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statements is/are TRUE about the function f(x) = max{3 $$-$$ x, x $$-$$ 1}?

Consider the differential equation $${{dy} \over {dx}} = 4(x + 2) - y$$ For the initial condition y = 3 at x = 1, the value of y at x = 1.4 obtained using Euler's method with a ste...

A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below. x 5 2 4 3 y 16 10 13 12 Based on the data, the coefficient a of the l...

Let y be a non-zero vector of size 2022 $$\times$$ 1. Which of the following statements is/are TRUE?

A pair of six-faced dice is rolled twice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is _________ (round off to three d...

Consider the polynomial f(x) = x 3 $$-$$ 6x 2 + 11x $$-$$ 6 on the domain S, given by 1 $$\le$$ x $$\le$$ 3. The first and second derivatives are f'(x) and f''(x). Consider the fol...

The components of pure shear strain in a sheared are given in the matrix form: $$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]$$ Here, Trace ($$\varepsilon $$...

The function f(x, y) satisfies the Laplace equation $$\Delta$$ 2 f(x, y) = 0 on a circular domain of radius r = 1 with its center at point P with coordinates x = 0, y = 0. The valu...

P and Q are two square matrices of the same order. Which of the following statements is/are correct?