Match the functions givenin List-I with their relevant characteristics from List-II.
The correct answer is

Match the points on the curve \(2 y^2=x+1\) with the slopes of normals at those points and choose the correct answer.

The numbers \(2,3,5,7,11,13\) are written on six distinct paper chits. If 3 of them are chosen at random, then the probability that the sum of the numbers on the obtained chits is divisible by 3 , is

15 lines are concurrent at a point P. A line L not passing through \(\mathrm{P}\) intersects all the 15 lines and forms triangles with them. Then the number of triangles having \(\mathrm{L}\) as one of its side is

\(\alpha, \beta, \gamma\) are the roots of the equation \(x^3-10 x^2+7 x+8=0\). Match the following and choose the correct answer.

If \(x=3 \sqrt{2} \cos ^3 \theta\) and \(y=4 \tan ^2 \theta\) then \(\left(\frac{d y}{d x}\right)_{\theta=\frac{\pi}{4}}=\)

\(\begin{aligned} & 1+x \log _e a+\frac{x^2}{2 !}\left(\log _e a\right)^2 \ & +\frac{x^3}{3 !}\left(\log _e a\right)^3+\ldots(a>0, x \in R) \text { is equal to }\end{aligned}\)

If \(f: R \rightarrow C\) is defined by \(f(x)=e^{2 i x}\) for \(x \in R\), then \(f\) is (where \(C\) denotes the set of all complex numbers)

If \(\int \frac{\sin x}{\cos x(1+\cos x)} d x=f(x)+c\), then \(f(x)\) is equal to

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

A cell of emf \(10 \mathrm{~V}\) and internal resistance \(3 \Omega\) is connected in parallel with another cell of emf \(7 \mathrm{~V}\) and internal resistance \(\frac{3}{5} \Omega\), such that their positive terminals are joined together and so their are negative terminals. Their positive terminals are joined with the negative terminal and their negative terminal is joined with the positive terminal of a third cell of emf \(20 \mathrm{~V}\) with internal resistance \(2 \Omega\). The combination can be replaced by a battery of emf \(E\) and internal resistance \(r\), then the values of \(E\) and \(R\) are respectively

A soap bubble of initial radius \(R\) is to be blown up. The surface tension of the soap film is \(\mathrm{T}\). The surface energy needed to double the diameter of the bubble is

Let \(M\) be the foot of the perpendicular from a point \(P\) on the parabola \(y^2=8(x-3)\) onto its directrix and let \(S\) be the focus of the parabola. If \(\triangle S P M\) is an equilateral triangle, then \(P\) is equal to