If a real valued function
\(f(x)=\left\{\begin{array}{cl}
\frac{x^2+(a+3) x+(a+1)}{x+3} &, \text { when } x \neq-3 \\
-\frac{5}{2} &, \text { when } x=-3
\end{array}\right.\)
is continuous at \(x=-3\), then \(\lim _{x \rightarrow a}\left(x^2+x+1\right)=\)

If \((\alpha, \beta)\) is the external centre of similitude of the circles \(x^2+y^2=3\) and \(x^2+y^2-2 x+4 y+4=0\), then \(\frac{\beta}{\alpha}=\)

In a triangle $ABC,{\left(\tan \frac{A}{2}\tan \frac{B}{2}\tan \frac{C}{2}\right)}^2\le$

If $x$ is chosen at random from the set $\left\{1,2.3,4\right\}$ and $y$ is chosen at random from the set $\left\{5,6,7\right\}$, then the probability that xy will be even is

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

If \(y=\frac{\log x}{x}\), then \(\frac{d^2 y}{d x^2}\) at \(x=1\)

Area of the triangle formed by the lines \(3 x^2-4 x y+y^2=0,2 x-y=6\) is

If one of the lines given by the pair of lines \(3 x^2-2 y^2+a x y=0\) is making an angle \(60^{\circ}\) with x -axis then \(\mathrm{a}=\)

If the normal drawn at the point \(P\) on the curve \(y=x \log x\) is parallel to the line \(2 x-2 y=3\), then \(P=\)

The plane of a dip circle is set in the geographic meridian and the apparent dip is \(\delta_1\). It is then set in a vertical plane perpendicular to the geographic meridian. The apparent dip angle is \(\delta_2\). The declination \(\theta\) at the place is

If the work done during the isothermal reversible expansion of an ideal gas at a pressure of \(10 \mathrm{~atm}\) from \(4 \mathrm{~L}\) to a final volume is \(-184.24 \mathrm{~L}\) atm, the final volume of the gas in \(\mathrm{L}\) is

Analysis shows that nickel oxide has the formula \(\mathrm{Ni}_{0.98} \mathrm{O}_{1.00}\). The fractions \(\mathrm{Ni}^{2+}\) and \(\mathrm{Ni}^{3+}\) ions in the crystal are ......

The area (in sq. units) of the region lying in the first quadrant and enclosed by the \(X\)-axis, the straight line \(x-\sqrt{3} y=0\) and the circle \(x^2+y^2=4\), is