List of top Engineering Mathematics Questions asked in GATE CE

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having a heart attack by 50%. Medicine X is taken by 50 of these 100 patients. The probability that a randomly selected patient, out of the 100 patients, takes medicine X and has a heart attack is

A partial differential equation
\(\frac{∂^2T}{∂x^2}+\frac{∂^2T}{∂y^2}=0\)
is defined for the two-dimensional field T: T(x, y), inside a planar square domain of size 2 m x 2 m. Three boundary edges of the square domain are maintained at value T = 50, whereas the fourth boundary edge is maintained T = 100.
The value of T at the center of the domain is

Consider two matrices \(A=\begin{bmatrix} 2 & 1 & 4 \\ 1 & 0 & 3 \end{bmatrix}\) and \(B=\begin{bmatrix} -1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}\)
The determinant of the matrix AB is _____ (in integer).

A storm with a recorded precipitation of 11.0 cm, as shown in the table, produced a direct run-off of 6.0 cm.

Time from start (hours)	1	2	3	4	5	6	7	8
Recorded cumulative precipitation (cm)	0.5	1.5	3.1	5.5	7.3	8.9	10.2	11.0

The Ø-index of this storm is _________ cm/hr (rounded off to 2 decimal places).

The expression for computing the effective interest rate (i_eff) using continuous compounding for a nominal interest rate of 5% is
\(i_{eff}=\lim\limits_{m\rightarrow\infin}(1+\frac{0.05}{m})^m-1\)
The effective interest rate (in percentage) is ________ (rounded off to 2 decimal places).

Three vectors \(\vec{p},\vec{q}\ \text{and}\ \vec{r}\) are given as
\(\vec{p}=\hat{i}+\hat{j}+\hat{k}\)
\(\vec{q}=\hat{i}+2\hat{j}+3\hat{k}\)
\(\vec{r}=2\hat{i}+3\hat{j}+4\hat{k}\)
Which of the following is/are CORRECT ?

The function f(x) = x³ - 27x + 4, 1 ≤ x ≤ 6 has

Consider two Ordinary Differential Equations (ODEs) :
\(P:\frac{dy}{dx}=\frac{x^4+3x^2y^2+2y^4}{x^3y}\)
\(Q:\frac{dy}{dx}=\frac{-y^2}{x^2}\)
Which one of the following options is CORRECT ?

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication.
P : (A + B)^T = A^T + B^T
Q: (AB)^T = A^TB^T
Which one of the following options is CORRECT ?

The second derivative of a function f is computed using the fourth-order Central Divided Difference method with a step length h.
The CORRECT expression for the second derivative is

The return period of a large earthquake for a given region is 200 years. Assuming that earthquake occurrence follows Poisson's distribution, the probability that it will be exceeded at least once in 50 years is _____ % (rounded off to the nearest integer).

What are the eigenvalues of the matrix \(\begin{bmatrix} 2 & 1 & 1 \\[0.3em] 1 & 4 & 1 \\[0.3em] 1 & 2 & 2 \end{bmatrix}\)?

A vector field \(\vec p\) and a scalar field \(r\) are given by
\(\vec p=(2x^2-3xy+z^2)\hat i+(2y^2−3yz+x^2)\hat j+(2z^2−3xz+x^2)\hat k\)
\(r=6x^2+4y^2 - z^2-9xyz - 2xy + 3xz-yz\)
Consider the statements P and Q.
P: Curl of the gradient of the scalar field r is a null vector.
Q: Divergence of curl of the vector field p is zero.
Which one of the following options is CORRECT?

For the following partial differential equation,
\(x\frac {∂^2f}{dx^2}+y\frac {∂^2f}{dy^2}=\frac {x^2+y^2}{2}\)
which of the following option(s) is/are CORRECT?

Consider the data of f (x) given in the table.

i	0	1	2
xi	1	2	3
f(xi)	0	0.3010	0.4771

The value of ƒ(1.5) estimated using second-order Newton's interpolation formula is ____ . (rounded off to 2 decimal places).

The three-dimensional state of stress at a point is given by
\(\sigma =\begin{pmatrix} 10 & 0 & 0\\  0 & 40 & 0\\  0 & 0 & 0  \end{pmatrix} \)
The maximum shear stress at the point is

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 student passes in both of these subjects \(\frac{1}{6}\).The probability that the student will pass in at least one of these two subjects 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:\(\frac{\partial^2u }{\partial x^2 } =2\)
Assuming g: g(x), f: f(y), and h: h(y), the general solution of the above differential equation is

Find the value of \(ϕ\) for critical stability. Bita \(\beta\) was given as 30°, \(\gamma\) set is given as 20, \(γ_w = 10\). Find \(\phi\) minimum for stability?