When the origin is shifted to $(-1, 2)$ by the translation of axes, the transformed equation of $x^2+y^2+2x-4y+1=0$ is

If \(\ell, \mathrm{m}, \mathrm{n}\) and \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are direction cosines of two lines then

If the plane passing through the points \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{k}}\) and the origin meets the line passing through the points \(\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) at the points \(A\), then \(A=\)

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets \(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is \(\begin{array}{llll} & \text { List-I } & & \text { List-II } \ \text { (i) } & P & \text { (A) } & (0,0) \ \text { (ii) } & Q & \text { (B) } & (6,0) \ \text { (iii) } & R & \text { (C) } & (-2,1) \ \text { (iv) } & S & \text { (D) }(-6,0) \ & & \text { (E) }(-6,-3) \ & & \text { (F) }(-6,3)\end{array}\) (i) (ii) (iii) (iv)

\(\lim _{x \rightarrow \infty}\left(\frac{2 x^2+3 x+4}{x^2-3 x+5}\right)^{\frac{3|x|+1}{2|x|-1}}=\)

\(\int \frac{1+x+\sqrt{x+x^2}}{\sqrt{x}+\sqrt{1+x}} d x\) is equal to

Coefficient of \(x^{10}\) in the expansion of \((2+3 x) e^{-x}\) is

The common components of photochemical smog are

A monochromatic radiation of wavelength \(\lambda\) is incident on a hydrogen sample in ground state. The sample subsequently emits radiation of six different wavelengths, then the value of \(\lambda\) is [Use ch \(=1242 \mathrm{eV}-\mathrm{m}\) ]

Consider a fluid in a container. Let the density of the fluid at the surface and at depth $H$ be ${\rho }_0$ and $\rho$ respectively. The bulk modulus of the fluid is $B_0.$ If $\rho =\frac{{\rho }_0}{1+\alpha {\rho }_{0}gH}$ then the constant $\alpha$ is
$\left(\mathrm{Assume}\frac{\rho -{\rho }_0}{{\rho }_0}\ll 1\right)$

Which of the following is not correct?

\(x=\log \left(\frac{1}{y}+\sqrt{1+\frac{1}{y^2}}\right) \Rightarrow y\) is equal to

A cannon shell fired breaks into two equal parts at its highest point. One part retraces the path to the cannon with kinetic energy \(E_1\) and kinetic energy of the second part is \(E_2\), relation between \(E_1\) and \(E_2\) is