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

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

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

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

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

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

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 \(X_1^TX_2=\)

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

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

Correlation coefficient is independent of

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 \(A\) 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?

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.

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

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

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