GATE Electronics & Communication
2,058 questions · 40 years · 19 subjects
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Real numbers y, p, and n (all greater than 1) satisfy $(\log_{p^{1/n}} y)(\log_{y^{1/n}} p) = 16$, where the logarithms are taken to the bases $p^{1/n}$ and $y^{1/n}$. The value of...
Let Pk represent the perimeter of a square with sides of length k. The value of the expression P₁ + P₂ + P₃ + ⋯ + P₁₀ is
A stick of length one meter is broken at two locations at distances of b₁ and b₂ from the origin (0), as shown in the figure. Note that 0 < b₁ < b₂ < 1. Which one of the following...
Consider the following series: (i) $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ (ii) $\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$ (iii) $\sum_{n=1}^{\infty} \frac{1}{n!}$ Choose the correct...
A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement. What is the probability that the two balls drawn have different col...
Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$, defined as $f(x) = 2x^3 - 3x^2 - 12x + 1$. Which of the following statements is/are correct? (Here, $\mathbb{R}$ is th...
Consider a non-negative function $f(x)$ which is continuous and bounded over the interval [2, 8]. Let $M$ and $m$ denote, respectively, the maximum and the minimum values of $f(x)$...
For a real number x > 1, $\frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} = 1$. The value of x is
The greatest prime factor of (3^199 – 3^196) is
Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that X≥Y is _________.
Consider the Earth to be a perfect sphere of radius R. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is __...
The number of 4-digit numbers formed by using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 such that the product of the digits is 0 is ______.
The number of ways to arrange the letters of the word 'EQUATION' such that all vowels come together is ______.
The number of ways to choose 4 cards from a pack of 52 playing cards such that all four cards are of the same suit is ______.
The number of ways to choose 3 men and 2 women from a group of 7 men and 5 women is ______.
The number of terms in the expansion of $(x+y+z)^n$ is ______.
The coefficient of $x^3$ in the expansion of $(1+x)^5$ is ______.
The value of $\sum_{r=0}^{n} \binom{n}{r}$ is ______.
The value of $\binom{n}{r} + \binom{n}{r-1}$ is ______.
The number of diagonals in a polygon of $n$ sides is ______.
The number of ways to distribute 10 identical items among 3 distinct persons such that each person gets at least one item is ______.
The number of ways to choose 5 balls from a bag containing 6 red balls and 4 blue balls such that at least 3 red balls are chosen is ______.
The number of ways to seat 5 men and 5 women in a row such that no two women sit together is ______.
The number of ways to arrange 7 books on a shelf such that two particular books are always together is ______.
The number of ways to form a committee of 3 members from 6 men and 4 women such that the committee has at least one woman is ______.
The number of distinct permutations of the letters of the word 'MISSISSIPPI' is ______.
The number of ways to choose 2 distinct numbers from the set {1, 2, 3, ..., 10} such that their sum is even is ______.
The number of ways to arrange 3 boys and 3 girls in a row such that no two boys sit together is ______.
The number of ways to choose 4 items from a set of 10 distinct items is ______.
The number of ways to form a 3-digit number using the digits 1, 2, 3, 4, 5 without repetition is ______.
The number of ways to arrange 4 distinct objects in a circle is ______.
The number of ways to choose 3 students from a class of 10 students for a competition is ______.
The number of ways to arrange the letters of the word 'APPLE' is ______.
The number of ways to choose 2 boys and 2 girls from a group of 4 boys and 3 girls is ______.
The number of ways to distribute 5 identical balls into 3 distinct boxes is ______.
The number of ways to choose a committee of 5 members from 10 people such that a particular person is always included is ______.
The number of ways to choose a committee of 5 members from 10 people such that a particular person is always excluded is ______.
The number of ways to arrange 6 distinct books on a shelf is ______.
The number of ways to choose 3 items from a set of 7 distinct items is ______.
The number of ways to form a 4-letter word using the letters of the word 'ROSE' without repetition is ______.
The number of ways to arrange 5 distinct objects in a row is ______.
The value of the contour integral $\frac{1}{2\pi j} \oint (z + \frac{1}{z})^2 dz$ evaluated over the unit circle $|z| = 1$ is ________.
The number of distinct eigenvalues of the matrix $A = \begin{bmatrix} 2 & 2 & 3 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 2 \end{bmatrix}$ is equal to ________.
The value of the integral $\int_0^\pi \int_y^\pi \frac{\sin x}{x} dx dy$, is equal to _________.
Consider the homogeneous ordinary differential equation $x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0$, $x > 0$ with $y(x)$ as a general solution. Given that $y(1) = 1$ and $y...