\[
\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r+2}{r(r+1)(r+3)}=
\]

\(\begin{aligned}
& \text { If }\left[1-\cos \left(\frac{\pi}{2}+\alpha\right)+\sin \left(\frac{3 \pi}{2}+\alpha\right)\right]^2+ \\
& {\left[1-\sin \left(\frac{3 \pi}{2}-\alpha\right)-\cos \left(\frac{3 \pi}{2}+\alpha\right)\right]^2=a+b \sin ^2\left(\frac{\pi}{4}+\alpha\right)}
\end{aligned}\)
then \(a^2+b^2=\)

If \(E_1\) and \(E_2\) are two events of a random experiment such that \(P\left(E_1\right)=\frac{1}{8}, P\left(E_1 \mid E_2\right)=\frac{1}{3}\), \(P\left(E_2 \mid E_1\right)=\frac{1}{4}\), then match the items of List-I with the items of List-II.
\(\begin{array}{lllc}
\hline & \text { List-I } & & \text { List-II } \\
\hline \text { (A) } & P\left(E_2\right) & \text { I. } & \frac{3}{16} \\
\hline \text { (B) } & P\left(E_1 \cup E_2\right) & \text { II. } & \frac{3}{29} \\
\hline \text { (C) } & P\left(\bar{E}_1 \mid \bar{E}_2\right) & \text { III. } & \frac{3}{32} \\
\hline \text { (D) } & P\left(E_1 \mid \bar{E}_2\right) & \text { IV. } & \frac{26}{29} \\
\hline & & \text { V. } & \frac{13}{32} \\
\hline
\end{array}\)
The correct match is

The area (in sq units) of the region bounded by \(x=-1, x=2, y=x^2+1\) and \(y=2 x-2\) is

If \(\int x^3 e^{2 x} d x=\frac{e^{2 x}}{8} f(x)+c\), then the sum of all the complex roots of \(f(x)=1\) is

\(\lim _{n \rightarrow \infty}\left[\begin{array}{l}\frac{1}{n}+\frac{n^2}{(n+1)^3}+\frac{n^2}{(n+2)^3}+\frac{n^2}{(n+3)^3} \\ +\ldots+\frac{1}{125 n}\end{array}\right]=\)

Let \(\mathrm{PQR}\) be a right angled isosceles triangle, right angled at \(P(2,1)\). If the equation of the side \(Q R\) is \(2 x+y=3\), then the equation of one of the sides other than \(Q R\) is

If a complex number \(z=x+i y\) represents a point \(P\) on the Argand plane and \(\operatorname{Arg}\left(\frac{z-3+2 i}{z+2-3 i}\right)=\frac{\pi}{4}\), then the locus of \(P\) is a

In \(\triangle A B C\), if \(a \cos ^2 \frac{C}{2}+c \cos ^2 \frac{A}{2}=\frac{3 b}{2}\), then

If $S$ and $S^{\text{'}}$ are the focii of an ellipse, $B$ is one end of the minor axis and $\angle SBS\text{'}=90°$, then the eccentricity of that ellipse is

A line L passing through the point \((2,0)\) makes an angle \(60^{\circ}\) with the line \(2 x-y+3=0\). If L makes an acute angle with the positive X -axis in the anticlockwise direction. then the Y -intercept of the line L is

An electron having kinetic energy of \(100 \mathrm{eV}\) circulates in a path of radius \(10 \mathrm{~cm}\) in a magnetic field. The magnitude of magnetic field \(|\mathbf{B}|\) is approximately [Mass of electron \(=0.5 \mathrm{MeV} \mathrm{c}^{-2}\), where \(\mathrm{c}\) is the velocity of light].

The mid-point of the line segment joining the centroid and the orthocentre of the triangle whose vertices are \((a, b),(a, c)\) and \((d, c)\), is