List of top Engineering Mathematics Questions asked in TS PGECET

The solution of \[ \frac{dy}{dx}+y=f(x), \] where \[ f(x)= \begin{cases} x, & 0\le x 0, & x\ge1, \end{cases} \] with the conditions \[ y(0)=0,\qquad y(1)=e^{-1}, \] is

\[ \lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}= \]

If \[ \vec{v}=(x+y+1)\hat{i}+\hat{j}+(-x-y)\hat{k}, \] then the value of \[ \vec{v}\cdot(\nabla\times\vec{v}) \] is

Let the matrix \[ A=\begin{bmatrix} 3& 1 -1& 2 \end{bmatrix} \] and \[ I=\begin{bmatrix} 1& 0 0& 1 \end{bmatrix}, \] then \[ A^{2}-5A+7I= \]

If \[ f(x,y,z,w)=x^{2}e^{2y+3z}\cos(4w), \] then \[ \frac{\partial f}{\partial z} \] at \((2,0,-2,1)\) is

In the binomial distribution \(B(n,p)\), the variance \((\sigma^{2})\) and the mean \((\mu)\) are related by

Assertion (A): If \(X\sim N(\mu,\sigma^{2})\), then \[ P(X<\mu)=0.5. \] Reason (R): Normal distribution is symmetric about its mean.

If \[ \vec{v}=(x+y+1)\hat{i}+\hat{j}+(-x-y)\hat{k}, \] then the value of \[ \vec{v}\cdot(\nabla\times\vec{v}) \] is

\[ \lim_{n\rightarrow\infty}\frac{5n^{2}+4}{7n^{2}+6n}= \]

The solution of \[ \frac{dy}{dx}+y=f(x), \] where \[ f(x)= \begin{cases} x, & 0\le x\\ 0, & x\ge1, \end{cases} \] with the conditions \[ y(0)=0,\qquad y(1)=e^{-1}, \] is

If \(y_{1}=x^{2}\) is a known solution of \[ x^{2}y^{\prime\prime}-3xy^{\prime}+4y=0, \] then the second independent solution \(y_{2}\) is

Consider the differential equation \[ \frac{dy}{dx}=x^{2}-y,\qquad y(0)=1. \] Using the Simple Euler's Method with a step size of \(h=0.1\), the value of \(y(0.1)\) is

Let the matrix \[ A=\begin{bmatrix} 3& 1 -1& 2 \end{bmatrix} \] and \[ I=\begin{bmatrix} 1& 0 0& 1 \end{bmatrix}, \] then \[ A^{2}-5A+7I= \]

If \[ f(x,y,z,w)=x^{2}e^{2y+3z}\cos(4w), \] then \[ \frac{\partial f}{\partial z} \] at \((2,0,-2,1)\) is

Let \(A\) and \(B\) be any two \(n\times n\) matrices and \[ tr(A)=\sum_{i=1}^{n}a_{ii}, \qquad tr(B)=\sum_{i=1}^{n}b_{ii}. \] Consider the following statements: \begin{itemize
• Statement-I : \(tr(AB)=tr(BA)\).
• Statement-II : \(tr(A+B)=tr(A)+tr(B)\).
• Statement-III : If \(tr(A)=5,\; tr(A^{2})=13\), then \[ tr(A-I)^2=6, \] where \(A_{3\times3}\) and \(I_{3\times3}\) are matrices. Choose the correct option. }

If $X \sim N(\mu,\sigma^2)$ then:

Assertion (A): Spearman rank correlation coefficient is used for ordinal data.
Reason (R): It is based on ranks rather than actual values.

If $f(x)=e^{-x}$ in $(-1,1)$ and the Fourier series of $f(x)$ is given by \[ f(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx), \] then the Fourier coefficient $a_0$ is:

If $e^{\sin x}$ is an integrating factor of \[ \frac{dy}{dx}+(y-1)\cos x=e^{-\sin x}\cos x, \] then the general solution is:

Find $L\{t\,u(t-2)\}$.

Using Runge-Kutta method, when $\frac{dy}{dx}=xy+y^{2}$, $y(0)=1$ is solved by taking $h=0.1$, then in the usual notation the value of $k_{1}$ is:

If \[ \vec{F}=(x+2y+az)\hat{i}+(bx-3y-z)\hat{j}+(4x+cy+2z)\hat{k} \] is irrotational, then $(a,b,c)$ is:

Which of the following series is divergent?

The system of equations \[ x-2y+3z=6,\quad 3x+y-4z=-7,\quad 5x-3y+2z=5 \] has:

If \[ \begin{bmatrix} 1 1 \end{bmatrix} \] is an eigenvector of the matrix \[ A= \begin{bmatrix} m& 3 \\ 3& m \end{bmatrix}, \] then the corresponding eigenvalue is: