GATE Civil Engineering

If a positive real x satisfies the following equation log₂ x + log√₂ x = 48, then the value of x is

At how many points will the curves y = x² and y = -x² – 2x - 1 intersect in the real (x, y) plane?

In the given figure, $\overline{PQ}$ is the diameter of a circle with center $O$. Two points $R$ and $S$ are chosen on the circle such that $\angle ROS = 80^{\circ}$. When $\overli...

P and Q are two positive integers such that P² = Q² + 13. The product of the numbers P and Q is _________

Periodic function $f(x)$ is given below. $f(x) = \begin{cases} -1, & \text{when} & -\pi < x < 0 \\ 1, & \text{when} & 0 < x < \pi \end{cases} ; f(x + 2\pi) = f(x)$ The CORRECT opti...

Bag I contains 4 white and 6 black balls. Bag II contains 4 white and 3 black balls. One ball is drawn at random from any one of the two bags and it is found to be a black ball. Th...

A matrix is given as: $$\begin{bmatrix} 9 & 15 \\ 15 & 50 \end{bmatrix}$$ By performing Cholesky decomposition, $|L_{22}|$ of the lower triangular matrix is ________ (in integer).

It is given that x and y are integers in the following equation: $(x + y - 7)^2 + (y + 3x - 13)^2 = 0$ The value of $(x^3 + y^3)$ is ________ (in integer).

A partial differential equation is given below. $\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0$ Possible solution(s) is/are:

The eigenvalues of $[A] = \begin{bmatrix} 2 & -3.5 & 6 \\ 3.5 & 5 & 2 \\ 8 & 1 & 8.5 \end{bmatrix}$ are $\lambda_1 = -1.547$, $\lambda_2 = 12.330$, and $\lambda_3 = 4.711$. The abs...

Let $f(x)$ be a continuous function defined in $[0,2] \rightarrow \mathbb{R}$ and satisfying the equation $\int_{0}^{2} f(x)[x - f(x)]dx = \frac{2}{3}$. The value of $f(1)$ is

An ordinary differential equation is given below. $x^2 \frac{d^2y}{dx^2} = 6y$ Considering $a$ and $b$ as arbitrary constants, the general solution of the equation is

Starting with the first approximation as x = 0.5, the second approximation for the root of the following function by the Newton-Raphson method is ________ (rounded off to two decim...

Values of y for different values of x are tabulated below. If a second-degree interpolating polynomial P_2(x) is used to represent y, the value of P_2(0) is ________ (rounded off t...

The sum of the following infinite series is: $\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + ...$

A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts requir...

A circle with center at (x, y) = (0.5, 0) and radius = 0.5 intersects with another circle with center at (x, y) = (1, 1) and radius = 1 at two points. One of the points of intersec...

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

Consider the function given below and pick one or more CORRECT statement(s) from the following choices. f(x) = x³ − (15/2)x² + 18x + 20

Pick the CORRECT eigen value(s) of the matrix [A] from the following choices. [A] = [[6, 8], [4, 2]]

Let y be the solution of the initial value problem y'' + 0.8y' + 0.16y = 0, where y(0) = 3 and y'(0) = 4.5. Then, y(1) is equal to ________ (rounded off to 1 decimal place).

The maximum value of the function h(x) = −x³ + 2x² in the interval [-1,1.5] is equal to ________ (rounded off to 1 decimal place).

A student was supposed to multiply a positive real number p with another positive real number q. Instead, the student divided p by q. If the percentage error in the student's answe...

For positive integers p and q, with $\frac{p}{q} \neq 1$, $(\frac{p}{q})^{\frac{p}{q}} = p^{(\frac{p}{q}-1)}$. Then,

If the sum of the first 20 consecutive positive odd numbers is divided by 20^2, the result is

The smallest positive root of the equation x^5 - 5x^4 - 10x^3 + 50x^2 + 9x - 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: h(y), the general s...

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...

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

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 –...

If x satisfies the equation $4^{8x} = 256$, then x is equal to ________.

Let $a = 30!, b = 50!$, and $c = 100!$. Consider the following numbers: $\log_a c, \log_c a, \log_b a, \log_a b$ Which one of the following inequalities is CORRECT?

A square of side length 4 cm is given. The boundary of the shaded region is defined by one semi-circle on the top and two circular arcs at the bottom, each of radius 2 cm, as shown...

For the integral $I = \int_{-1}^{1} \frac{1}{x^2} dx$, which of the following statements is TRUE?

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...

For the matrix [A] = [1 2 3] [3 2 1] [3 1 2] which of the following statements is/are TRUE?

For the function f(x) = e⁻⁸|sin x|; x ∈ R, which of the following statements is/are TRUE?

The matrix M is defined as $M = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}$ and has eigenvalues 5 and -2. The matrix Q is formed as $Q = M^3 - 4M^2 - 2M$ Which of the following i...

P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct?

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

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

The value of $\lim_{x\to\infty} \frac{x \ln(x)}{1+x^2}$ is

The rank of matrix $\begin{bmatrix} 1 & 2 & 2 & 3 \\ 3 & 4 & 2 & 5 \\ 5 & 6 & 2 & 7 \\ 7 & 8 & 2 & 9 \end{bmatrix}$ is

The rank of the matrix $\begin{bmatrix} 5 & 0 & -5 & 0 \\ 0 & 2 & 0 & 1 \\ -5 & 0 & 5 & 0 \\ 0 & 1 & 0 & 2 \end{bmatrix}$ is

If $P = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $Q = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ then $Q^T P^T$ is

The unit normal vector to the surface X²+Y²+Z²-48=0 at the point (4, 4, 4) is

The shape of the cumulative distribution function of Gaussian distribution is

If A is a square matrix then orthogonality property mandates

The value (round off to one decimal place) of $\int_{-1}^{1} x e^{|x|} dx$ is

A function is defined in Cartesian coordinate system as $f(x, y) = xe^y$. The value of the directional derivative of the function (in integer) at the point (2, 0) along the directi...

The area of an ellipse represented by an equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

For the Ordinary Differential Equation $\frac{d^2x}{dt^2} - 5\frac{dx}{dt} + 6x = 0$, with initial conditions $x(0) = 0$ and $\frac{dx}{dt}(0) = 10$, the solution is

A continuous function f(x) is defined. If the third derivative at $x_i$ is to be computed by using the fourth order central finite-divided-difference scheme (with step length = h),...

If C represents a line segment between (0,0,0) and (1,1,1) in Cartesian coordinate system, the value (expressed as integer) of the line integral $\int_C [(y+z)dx+(x+z)dy+ (x + y)dz...

Consider the system of equations $\begin{bmatrix} 1 & 3 & 2 \\ 2 & 2 & -3 \\ 4 & 4 & -6 \\ 2 & 5 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix}...

Which one of the following is correct?

For a small value of h, the Taylor series expansion for f(x+h) is

The value of the function f(x) is given at n distinct values of x and its value is to be interpolated at the point x*, using all the n points. The estimate is obtained first by the...

Which one of the following is NOT a correct statement?

Consider two functions: x=ψlnφ and y=φlnψ. Which one of the following is the correct expression for ∂ψ/∂x?

Consider the ordinary differential equation $x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0$. Given the values of y(1) = 0 and y(2) = 2, the value of y(3) (round off to 1 decima...

The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?

Consider the following simultaneous equations (with c₁ and c₂ being constants): 3 x₁ + 2 x₂ = c₁ 4 x₁ + x₂ = c₂ The characteristic equation for these simultaneous equations is

Which one of the following matrices is singular?

For the given orthogonal matrix Q, Q = [3/7 2/7 6/7; -6/7 3/7 2/7; 2/7 6/7 -3/7]. The inverse is

Let w = f (x, y), where x and y are functions of t. Then, according to the chain rule, dw/dt is equal to

At the point x = 0, the function f(x) = x³ has

Let x be a continuous variable defined over the interval (-∞, ∞), and f (x) = e^-x - e^-x. The integral g(x) = ∫ f(x) dx is equal to

Which of the following function(s) is an accurate description of the graph for the range(s) indicated? (i) y = 2x + 4 for -3 ≤ x ≤ -1 (ii) y = |x-1| for -1 ≤ x ≤ 2 (iii) y = ||x|-1...

Consider a sequence of numbers $a_1, a_2, a_3, \dots, a_n$ where $a_n = \frac{1}{n} - \frac{1}{n+2}$, for each integer $n > 0$. What is the sum of the first 50 terms?

Given that $\frac{\log P}{y-z} = \frac{\log Q}{z-x} = \frac{\log R}{x-y} = 10$ for $x \neq y \neq z$, what is the value of the product PQR?

Probability (up to one decimal place) of consecutively picking 3 red balls without replacement from a box containing 5 red balls and 1 white ball is _________

The divergence of the vector field V = x² i + 2y³ j + z⁴ k at x = 1, y = 2, z = 3 is

A two-faced fair coin has its faces designated as head (H) and tail (T). This coin is tossed three times in succession to record the following outcomes: H, H, H. If the coin is tos...

Consider the following partial differential equation: $3\frac{\partial^2\phi}{\partial x^2} + B\frac{\partial^2\phi}{\partial x\partial y} + 3\frac{\partial^2\phi}{\partial y^2} +...

The quadratic equation $2x^2-3x+3=0$ is to be solved numerically starting with an initial guess as $x_0 = 2$. The new estimate of $x$ after the first iteration using Newton-Raphson...

lim (tanx / (x^2 - x)) is equal to

The tangent to the curve represented by $y = x \ln x$ is required to have 45° inclination with the x-axis. The coordinates of the tangent point would be

For the function f(x)=a+bx, 0 ≤ x ≤ 1, to be a valid probability density function, which one of the following statements is correct?