If \(a x^2+b x+c < 0 \forall x \in \mathbb{R}\) and the expressions \(c x^2+a x+b\) and \(a x^2+b x+c\) have their extreme values at the same point \(x\), then for the expression \(c x^2+a x+b\)

If the intercepts made by a variable circle on the X -axis and Y -axis are 8 and 6 units respectively, then the locus of the centre of the circle is

If \(\lim _{n \rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{1 / n}=k\) then \(\log k=\)

Let \(a \neq 0\) and \(p(x)\) be a polynomial of degree greater than 2. If \(p(x)\) leaves remainders a and \(-a\) when divided respectively by \(x+a\) and \(x-a\), then the remainder when \(p(x)\) is divided by \(x^2-a^2\) is :

Four cards are drawn at random from a pack of 52 playing cards. The probability of getting all four cards of the same suit is

If in the angles of a triangle are in the ratio \(1: 1: 4\), then the ratio of the perimeter of the triangle to its largest side is

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors such that $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=3$ and $\overrightarrow{a}+t\overrightarrow{b}$ and $\overrightarrow{a}-t\overrightarrow{b}$ are perpendicular, where $\text{'}t\text{'}$ is a positive scalar, then

Magnetic moment due to the motion of the electron in \(n\)th energy state of hydrogen atom is proportional to ..........

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

Which of the following are correct?
i. Basic structural unit of silicates is \(-\mathrm{R}_2 \mathrm{SiO}-\)
ii. Silicones are biocompatible
iii. Producer gas contains CO and \(\mathrm{N}_2\)
The correct option is

The respective speeds of the five molecules are 1,2,3,4 and \(5 \mathrm{kms}^{-1}\). Then, the ratio of their rms velocity and the average velocity will be

If \((1, a),(b, 2)\) are conjugate points with respect to the circle \(x^2+y^2=25\), then \(4 a+2 b=\)

Dipole moment of \(\mathrm{HCl}=1.03 \mathrm{D}, \mathrm{HI}=0.38 \mathrm{D}\). Bond length of \(\mathrm{HCl}=1.3 Ã…\) and \(\mathrm{HI}=1.6 Ã…\). The ratio of fraction of electric charge, \(\delta\), existing on each atom in \(\mathrm{HCl}\) and \(\mathrm{HI}\) is