If \(L\) is the normal drawn to the parabola \(y^2=8 x\) at the point \(t=\frac{1}{\sqrt{2}}\), then the foot of the perpendicular drawn from the focus of the parabola on to the normal L is

If the straight line $x\cos \alpha +y\sin \alpha =p$ touches the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2$ at the point $\left(a,b\right)$ on it and $\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}$ then $k=$

If the equations
\[
\begin{aligned}
& k x+(k+1) y+(k-1) z=0, \\
& (k-1) x+(k+2) y+k z=0 \text { and } \\
& (k+1) x+k y+(k+2) z=0
\end{aligned}
\]
have a non trivial solution, then the sum of all the possible values of \(k\) is

If \(A=(2,3)\) and \(B=(-4,5)\) are two fixed points, then the locus of a point \(P\) such that the area of \(\triangle P A B\) is 12 square units is

Match the items of List - I with those of the entires of List - II
\(List - I\)
\(\begin{aligned} & \text { (I) } \sin ^2 5^{\circ}+\sin ^2 10^{\circ}+ \\ & \sin ^2 15^{\circ}+\ldots+\sin ^2 90^{\circ}=\end{aligned}\)
\(\begin{aligned} & \text { (II) } \tan ^2 5^{\circ} \cdot \tan ^2 10^{\circ} \text {. } \\ & \tan ^2 15^{\circ} \ldots \tan ^2 85^{\circ}= \\ & \end{aligned}\)
\(\begin{aligned} & \text { (III) } \cos ^2 5^{\circ}+\cos ^2 10^{\circ} \\ & +\cos ^2 15^{\circ}+\ldots+\cos ^2 180^{\circ}=\end{aligned}\)
\(\begin{gathered}\text { (IV) } \cot 5^{\circ}+\cot 10^{\circ}+\cot 15^{\circ} \\ +\ldots .+\cot 175^{\circ}=\end{gathered}\)
\(List - II\)
(A) \(0\)
(B) \(\frac{19}{2}\)
(C) \(18\)
(D) \(1\)
(E) \(-1\)

In \(\triangle A B C, r r_1 \cot \frac{A}{2}+r r_2 \cot \frac{B}{2}+r r_3 \cot \frac{C}{2}=\)

\(\int \frac{x^4+1}{x^6+1} d x=\)

In the reaction of sulphur with concentrated sulphuric acid, the oxidised product is X and reduced product is \(\mathrm{Y} . \mathrm{X}\) and Y are respectively

Equation of tangent to the curve \(y=x+\frac{4}{x^2}\) which is parallel to \(X\)-axis is

\(\int_0^\pi x \sin ^4 x \cos ^6 x d x=\)

If \(f: A \rightarrow B\) and \(g: B \rightarrow C\) are two functions such that \(g \circ f: A \rightarrow C\) is a bijection, then which one of the following is always true?

If \(\mathrm{B}\) is magnetic field and \(\mathrm{q}\) is the charge then the following represents the Gauss's law of magnetism

The circle possessing \(y\)-axis as its tangent at \((0,2)\) and passing through \((-1,0)\), also passes through