GATE CSE & IT

In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and line QR. What is...

In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of...

Consider the given function $f(x)$. $f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^3 + x^2 + 1 & \text{for } x \ge 1 \end{cases}$ If the function is differentiable everywher...

A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. Suppose from the se...

For positive non-zero real variables p and q, if log (p² + q²) = log p + log q + 2 log 3, then, the value of $\frac{p^4+q^4}{p^2q^2}$ is

A rectangular paper sheet of dimensions 54 cm x 4 cm is taken. The two longer edges of the sheet are joined together to create a cylindrical tube. A cube whose surface area is equa...

Let f(x) be a continuous function from ℝ to ℝ such that f(x) = 1 − f(2 – x) Which one of the following options is the CORRECT value of $\int_0^2 f(x)dx$?

When six unbiased dice are rolled simultaneously, the probability of getting all distinct numbers (i.e., 1, 2, 3, 4, 5, and 6) is

If $f(x) = R \sin (\frac{\pi x}{2}) + S$, $f' (\frac{1}{2}) = \sqrt{2}$ and $\int_0^1 f(x)dx = \frac{2R}{\pi}$, then the constants R and S are, respectively

Let $P = \begin{bmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{bmatrix}$ and $Q = \begin{bmatrix} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{bmatrix}$ be two matrices. The...

P and Q are considering to apply for a job. The probability that P applies for the job is $\frac{1}{4}$, the probability that P applies for the job given that Q applies for the job...

If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)^2] equals _________.

If the characteristic polynomial of a 3 × 3 matrix M over R (the set of real numbers) is λ³ – 4λ² + αλ + 30, a ∈ R, and one eigenvalue of M is 2, then the largest among the absolut...

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 $e^x + 0.5x^2 - 2 = 0$ in the range $[-5, 5]$ is