GATE Electrical Engineering

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

If, for non-zero real variables x, y, and real parameter a > 1, x: y = (a + 1): (a− 1), then, the ratio (x² - y²): (x² + y²) is

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 s...

In the following differential equation, the numerically obtained value of y(t), at t =1, is ________ (Round off to 2 decimal places). $\frac{dy}{dt} = \frac{e^{-at}}{2 + at}$, α =...

Three points in the x-y plane are (-1, 0.8), (0, 2.2) and (1, 2.8). The value of the slope of the best fit straight line in the least square sense is ________ (Round off to 2 decim...

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________....

A quadratic function of two variables is given as f(x1,x2) = x₁² + 2x₂² + 3x₁ + 3x₂ + x₁x₂ + 1 The magnitude of the maximum rate of change of the function at the point (1,1) is ___...

If A = 2xi + 3yj + 4zk and u = x² + y² + z², then div(uA) at (1, 1, 1) is _________.

The probability of a resistor being defective is 0.02. There are 50 such resistors in a circuit. The probability of two or more defective resistors in the circuit (round off to two...

The matrix A = $\begin{bmatrix} 3/2 & 0 & 1/2 \ 0 & -1 & 0 \ 1/2 & 0 & 3/2 \end{bmatrix}$ has three distinct eigenvalues and one of its eigenvectors is $\begin{bmatrix} 1 \ 0 \ 1 \...

An urn contains 5 red balls and 5 black balls. In the first draw, one ball is picked at random and discarded without noticing its colour. The probability to get a red ball in the s...

For a complex number z, $\lim_{z \to i} \frac{z^2+1}{z^3+2z-i(z^2+2)}$ is

The three roots of the equation f(x) = 0 are x = {-2, 0, 3}. What are the three values of x for which f (x - 3) = 0?

Let z(t) = x(t) * y(t), where "*" denotes convolution. Let c be a positive real-valued constant. Choose the correct expression for z(ct).

For what values of k given below is $\frac{(k+2)^2}{k-3}$ an integer?

In a quadratic function, the value of the product of the roots (α, β) is 4. Find the value of $\frac{\alpha^n + \beta^n}{\alpha^{-n} + \beta^{-n}}$

Functions F(a, b) and G(a, b) are defined as follows: F(a,b) = $(a - b)^2$ and G(a,b) = $|a - b|$, where $|x|$ represents the absolute value of x. What would be the value of G(F(1,...

An e-mail password must contain three characters. The password has to contain one numeral from 0 to 9, one upper case and one lower case character from the English alphabet. How ma...

Let f be a real-valued function of a real variable defined as f(x) = x² for x\ge0, and f(x) = -x² for x<0. Which one of the following statements is true?

The value of the directional derivative of the function \Phi(x, y, z) = xy² + yz² + zx² at the point (2, -1, 1) in the direction of the vector p = i + 2j + 2k is

The value of the integral \oint_C \frac{z+1}{z^2-4} dz in counter clockwise direction around a circle C of radius 1 with center at the point z = -2 is

Consider $g(t) = \begin{cases} t - \lfloor t \rfloor, & t \ge 0 \\ t - \lceil t \rceil, & \text{otherwise} \end{cases}$, where $t \in \mathbb{R}$. Here, $\lfloor t \rfloor$ represe...

Let $I = c \iint_R xy^2 \, dxdy$, where $R$ is the region shown in the figure and $c = 6 \times 10^{-4}$. The value of $I$ equals _________ (Give the answer up to two decimal place...

Consider a function f(x,y,z) given by f(x, y, z) = (x² + y² - 2z²) (y² + z²) The partial derivative of this function with respect to x at the point,x = 2, y = 1 and z = 3 is

Let x and y be integers satisfying the following equations 2x² + y² = 34 x + 2y = 11 The value of (x + y) is _________.

Let y² - 2y + 1 = x and √x + y = 5. The value of x + √y equals _________ (Give the answer up to three decimal places)

Let g(x) = { -x, x ≤ 1 and f(x) = { 1-x, x ≤ 0 x+1, x ≥ 1 x², x > 0 Consider the composition of f and g, i.e., (f∘g)(x) = f(g(x)). The number of discontinuities in (f∘g)(x) present...

A function $f(x)$ is defined as $f(x)=\begin{cases} e^x, & x<1 \ \ln x+ax^2+bx, & x\ge1 \end{cases}$, where $x \in \mathbb{R}$. Which one of the following statements is TRUE?

The value of the contour integral in the complex-plane ∫ (z³ - 2z + 3) / (z - 2) dz along the contour |z| = 3, taken counter-clockwise is

Consider the differential equation $(t^2-81)\frac{dy}{dt}+5t y = \sin(t)$ with $y(1) = 2\pi$. There exists a unique solution for this differential equation when $t$ belongs to the...

The eigenvalues of the matrix given below are $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -3 & -4 \end{bmatrix}$$

Consider the line integral $I = \int_C (x^2 +iy^2)dz$, where $z = x + iy$. The line $C$ is shown in the figure below. The value of $I$ is

A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable Y denote the number of heads. The value of var{Y}, wher...

Only one of the real roots of f(x) = x^6 - x - 1 lies in the interval 1 ≤ x ≤ 2 and bisection method is used to find its value. For achieving an accuracy of 0.001, the required min...

Find the smallest number y such that y × 162 is a perfect cube.

The probability that a k-digit number does NOT contain the digits 0, 5, or 9 is

There are 3 red socks, 4 green socks and 3 blue socks. You choose 2 socks. The probability that they are of the same colour is

X is a 30 digit number starting with the digit 4 followed by the digit 7. Then the number X³ will have

The number of roots of e^x + 0.5x² - 2 = 0 in the range [-5, 5] is