List of top Engineering Mathematics Questions asked in TS PGECET

If the initial guess of the solution of \[ 1+x-x^3=0 \] is \(x_0=1\), then the \(2^{\text{nd}}\) approximate root of \[ 1+x-x^3=0 \] using Newton-Raphson method is

In a Poisson distribution, \[ P(X=x+k)=\frac{\lambda^k}{f(x)}P(X=x), \] then \(f(x)=\)

Let \(X\) be a discrete random variable, \(\mu\) be its mean, and \(P(X=x)\) be the probability function. The variance of \(X\) is

If \[ \frac{dy}{dx}=xy+x,\; x\in\mathbb{R} \] and \(y(0)=1\), then \(y(0.2)\) with step size \(h=0.1\) using Euler's method is

Let \[ f(t)= \begin{cases} \sin t, & 0\le t\le \pi,\\ 0, & t>\pi. \end{cases} \] The Laplace transform of \(f(t)\) is

If the Laplace transform of \(f(t)\) and \(f'(t)\) exist and \(F(s)\) is the Laplace transform of \(f(t)\), then \[ \lim_{s\to\infty}sF(s) \] is

The system of equations \[ x+4y+6z=20,\qquad x+y+z=6,\qquad x+\lambda y+6z=20 \] has

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

Let \[ \vec{F}=x^2yz\,\hat{i}+y^2z\,\hat{j}+zx^3\,\hat{k} \] be a vector point function. Let \(S\) be the surface of the sphere \[ x^2+y^2+z^2=a^2. \] Then \[ \iint_{S}(\nabla\times\vec{F})\cdot\vec{N}\,dS \] is equal to (\(\vec{N}\) is any unit outward normal vector to \(S\)).

Let \(A\) be a real symmetric matrix. \(\lambda_1,\lambda_2\;(\lambda_1\neq\lambda_2)\) be two eigen values of \(A\) and \(X_1,X_2\) are respectively the eigen vectors of \(A\) corresponding to \(\lambda_1\) and \(\lambda_2\), then \(X_1^TX_2=\)

If variance of \(X\) is \(4\), then the variance of \(Y=4X+2\) is

One of the normal equations for fitting a straight line \(y=a+bx\) is

Correlation coefficient is independent of

If \(A\) and \(B\) are independent random variables, then which of the following is correct?

If the observed frequencies exactly match expected frequencies, then what is \(\chi^2\) (Chi-square) value?

A random variable \(X\) has the probability density function \[ f(x)=Kx,\qquad 0\lt x\lt 2, \] then the ratio between \(K\) and the mean of \(X\) is

If \[ A= \begin{bmatrix} 2026 & 2025 & 2024\\ 2025 & 2024 & 2023\\ 2024 & 2023 & 2022 \end{bmatrix}, \] then \(\left|A^{2026}-A^{2025}\right|=\)

If \({}^nC_r={}^{\,n}C_{r-1}\) and \({}^{\,n}P_{r-1}=9\,{}^{\,r}P_r\), then \((n,r)=\)

If the sum of the three numbers which are in A.P. is \(27\) and the product of the first and last is \(77\), then the numbers are

If \(\alpha,\beta\) are the roots of \(ax^2+bx+c=0\), then \[ \left(\frac{\alpha}{\beta}-\frac{\beta}{\alpha}\right)^2 = \]

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:
• 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.

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.

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

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