GATE Mechanical Engineering

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

P and Q play chess frequently against each other. Of these matches, P has won 80% of the matches, drawn 15% of the matches and lost 5% of the matches. If they play 3 more matches,...

The divergence of the curl of a vector field is

If two unbiased coins are tossed, then what is the probability of having at least one head?

In the closed interval [0,3], the minimum value of the function f given below is f(x) = 2x³ - 9x² + 12x

The real variables x, y, z, and the real constants p, q, r satisfy $\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$ Given that the denominators are non-zero, the valu...

Take two long dice (rectangular parallelepiped), each having four rectangular faces labelled as 2, 3, 5, and 7. If thrown, the long dice cannot land on the square faces and has 1/4...

How many combinations of non-null sets A, B, C are possible from the subsets of {2, 3, 5} satisfying the conditions: (i) A is a subset of B, and (ii) B is a subset of C?

Let f(z) be an analytic function, where z = x + iy. If the real part of f(z) is cosh x cos y, and the imaginary part of f(z) is zero for y = 0, then f(z) is

Consider the system of linear equations x + 2y + z = 5 2x + ay + 4z = 12 2x + 4y + 6z = b The values of a and b such that there exists a non-trivial null space and the system admit...

If the value of the double integral $\int_{x=3}^{4}\int_{y=1}^{2}\frac{dydx}{(x+y)^2}$ is $\log_e (a/24)$, then $a$ is ________ (answer in integer).

If $x(t)$ satisfies the differential equation $t\frac{dx}{dt} + (t - x) = 0$ subject to the condition $x(1) = 0$, then the value of $x(2)$ is ________ (rounded off to 2 decimal pla...

Let X be a continuous random variable defined on [0,1] such that its probability density function f(x) = 1 for 0 ≤ x ≤ 1 and 0 otherwise. Let Y = loge(X + 1). Then the expected val...

Consider the following inequalities p² - 4q < 4 3p + 2q < 6 where p and q are positive integers. The value of (p + q) is _________.

How many pairs of sets (S,T) are possible among the subsets of {1, 2, 3, 4, 5, 6} that satisfy the condition that S is a subset of T?

The smallest perimeter that a rectangle with area of 4 square units can have is ________ units. (Answer in integer)

Consider the second-order linear ordinary differential equation $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = 0, x \geq 1$ with the initial conditions $y(x = 1) = 6, \frac{dy}{dx}...

If y(x) satisfies the differential equation (sin x) dy/dx + y cos x = 1, subject to the condition y(π/2) = π/2, then y(π/6) is

The value of lim (x→0) (1-cos x)/x^2 is

In the above figure, O is the center of the circle and, M and N lie on the circle. The area of the right triangle MON is 50 cm². What is the area of the circle in cm² ?

The mean and variance, respectively, of a binomial distribution for n independent trials with the probability of success as p, are

Consider a binomial random variable X. If X1, X2, ..., Xn are independent and identically distributed samples from the distribution of X with sum Y = $\Sigma_{i=1}^n X_i$, then the...

The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is _____

Consider a square sheet of side 1 unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is agai...

Value of $\int_{4}^{5.2} \ln x \, dx$ using Simpson's one-third rule with interval size 0.3 is

Value of $(1 + i)^8$, where $i = \sqrt{-1}$, is equal to

The set of equations x + y + z = 1 ax - ay + 3z = 5 5x - 3y + az = 6 has infinite solutions, if a =

A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x² - 2y² + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is

The variable x takes a value between 0 and 10 with uniform probability distribution. The variable y takes a value between 0 and 20 with uniform probability distribution. The probab...

The probability that a part manufactured by a company will be defective is 0.05. If 15 such parts are selected randomly and inspected, then the probability that at least two parts...

The value of the following definite integral is _________ (round off to three decimal places) $\int_{1}^{e} (x \ln x) dx$

Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is

The product of eigenvalues of the matrix P is P = [[2, 0, 1], [4, -3, 3], [0, 2, -1]]

The divergence of the vector -yi+x j is

The value of lim_{x\to 0} \frac{x^3 - \sin(x)}{x} is

Consider the following partial differential equation for u(x, y) with the constant c > 1: \frac{\partial u}{\partial y} + c \frac{\partial u}{\partial x} = 0 Solution of this equat...

The determinant of a 2×2 matrix is 50. If one eigenvalue of the matrix is 10, the other eigenvalue is

The differential equation \frac{d^2y}{dx^2} + 16y = 0 for y(x) with the two boundary conditions \frac{dy}{dx}|_{x=0} = 1 and \frac{dy}{dx}|_{x=\frac{\pi}{2}} = -1 has

A sample of 15 data is as follows: 17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. The mode of the data is

The perimeters of a circle, a square and an equilateral triangle are equal. Which one of the following statements is true?

The value of the expression $\frac{1}{1+\log_u vw} + \frac{1}{1+\log_v wu} + \frac{1}{1+\log_w uv}$ is

A six-face fair dice is rolled a large number of times. The mean value of the outcomes is _________

The Laplace transform of $te^t$ is

Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and n...

Which of the following functions describe the graph shown in the below figure?

If A = $\begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 1 \end{bmatrix}$ then det(A$^{-1}$) is ________ (correct to two decimal places).

The arrival of customers over fixed time intervals in a bank follow a Poisson distribution with an average of 30 customers/hour. The probability that the time between successive cu...

The surface integral $\iint_S \mathbf{F} \cdot \mathbf{n}\,dS$ over the surface $S$ of the sphere $x^2 + y^2 + z^2 = 9$, where $\mathbf{F}=(x+y)\mathbf{i}+(x+z)\mathbf{j}+(y+z)\mat...

Consider the matrix P = $\begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{bmatrix}$. Which one of the f...

Consider the differential equation $3y''(x)+27y(x)=0$ with initial conditions $y(0)=0$ and $y'(0)=2000$. The value of y at x = 1 is _________.

For the vector $\vec{V} = 2yz \hat{i} + 3xz \hat{j} + 4xy \hat{k}$, the value of $\vec{V} \cdot (\nabla \times \vec{V})$ is _________

Consider the matrix A = $\begin{bmatrix} 50 & 70 \\ 70 & 80 \end{bmatrix}$ whose eigenvectors corresponding to eigenvalues $\lambda_1$ and $\lambda_2$ are $\mathbf{x}_1 = \begin{bm...

A parametric curve defined by $x = \cos\left(\frac{\pi u}{2}\right)$, $y = \sin\left(\frac{\pi u}{2}\right)$ in the range $0 \le u \le 1$ is rotated about the X- axis by 360 degree...

If $f(z) = (x^2 + ay^2) + ibxy$ is a complex analytic function of $z = x + iy$, where $i = \sqrt{-1}$, then

P (0, 3), Q (0.5, 4), and R (1, 5) are three points on the curve defined by $f(x)$. Numerical integration is carried out using both Trapezoidal rule and Simpson's rule within limit...

The minimum value of 3x + 5y such that: 3x + 5y ≤ 15 4x + 9y ≤ 8 13x + 2y ≤ 2 x ≥ 0, y ≥ 0 is ________.

A right-angled cone (with base radius 5 cm and height 12 cm), as shown in the figure below, is rolled on the ground keeping the point P fixed until the point Q (at the base of the...

In a company with 100 employees, 45 earn Rs. 20,000 per month, 25 earn Rs. 30,000, 20 earn Rs. 40,000, 8 earn Rs. 60,000, and 2 earn Rs. 150,000. The median of the salaries is