Recent questions tagged gatexe-2022

Let $\mathbb{C}=\{z=x+i y: x$ and $y$ are real numbers, $i=\sqrt{-1}\}$ be the set of complex numbers. Let the function $f(z)=u(x, y)+i v(x, y)$ for $z=x+i y \in \mathbb{...

The value of$\lim _{x \rightarrow 0} \frac{1}{x} \int_{2}^{2+x}\left(t+\sqrt{t^{2}+5}\right) d t$$0$$4$$5$$6$

If the partial differential equation$$(x+2) \frac{\partial^2 u}{\partial x^2}+2(x+y) \frac{\partial^2 u}{\partial x \partial y}+2(y-1) \frac{\partial^2 u}{\partial y^2}-3...

Let $y_{1}(x)$ and $y_{2}(x)$ be two linearly independent solutions of\[x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0, \quad x>0\]Let $W\left(y_{1}, y_{2}\right...

Let $\Gamma$ be the positively oriented circle $x^2+y^2=9$ in the $x y$-plane. If$$\oint_{\Gamma}\left(3 y+e^{x \sin x}\right) d x+\left(7 x+\sqrt{e^y+2}\right) d y=\alph...

Let $A=\left[\begin{array}{rrrr}2 & 0 & 1 & 1 \\ 1 & 2 & 5 & -5 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3\end{array}\right]$Then the sum of the geometric multiplicities of the di...

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be the continuous function such that $f(x)=2+\frac{g(x)}{x}$ for all $x>0$, where $g(x)=\int_{1}^{x} f(t) d t$ for all $x>0$. T...

In a cosmopolitan city, the population comprises of $30 \%$ female and $70 \%$ male. Suppose that $5 \%$ of female and $30 \%$ of male in the population are foreigners. A...

Let $A$ and $B$ be $n \times n$ matrices with real entries.Consider the following statements:P: If $A$ is symmetric then $\operatorname{rank}(A)=$ Number of nonzero eigen...

If the quadrature rule$$\int_{-1}^1 f(x) d x \approx f(\alpha)+\gamma f(\beta),$$where $\alpha, \beta$ and $\gamma$ are real constants, is exact for all polynomials of de...

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by $f(x, y)=4 x y-2 x^{2}-y^{4}+1$. The number of critical points where $f$ has local maximum is equal to

A heavy horizontal cylinder of diameter $D$ supports a mass of liquid having density $\rho$ as shown in the figure. Find out the vertical component of force exerted by th...

The figure shows the developing zone and the fully developed region in a pipe flow where the steady flow takes place from left to right. The wall shear stress in the sect...

The left hand column lists some non-dimensional numbers and the right hand column lists some physical phenomena. Indicate the correct combination1. Reynolds numberi. Wave...

Which of the following statement(s) regarding a venturimeter is/are correct?In the direction of flow, it consists of a converging section, a throat, and a diverging secti...

Which of the following statement(s) is/are true for streamlines in a steady incompressible flow?Two streamlines cannot intersect each other.Flow rate increases between tw...

A flow has a velocity potential given by $\phi=A x^{3}$ where ' $A$ ' is a non-zero constant. Which of the following statement(s) is/are true about the flow?The flow is i...

A $10 \mathrm{~kg}$ mass placed on an infinitely long horizontal massless flat platform is to be supported by a steady vertical water jet as shown in the figure. The diam...

A boundary layer develops due to a two-dimensional steady flow over a horizontal flat plate. Consider a vertical line away from the leading edge which extends from the wa...