$A=\left[\begin{array}{ccc}{a}^2& 15& 31\\ 12& b^2& 41\\ 35& 61& c^2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2a& 3& 5\\ 2& 2b& 8\\ 1& 4& 2c-3\end{array}\right]$ are two matrices such that the sum of the principal diagonal elements of both $A$ and $B$ are equal, then the product of the principal diagonal elements of $B$ is

Match the items of List - I with those of List - II
\(\begin{array}{ll}
\text{List - I (Complex number)} & \text{List - II (Polar form)} \\
\text{(i)} \sqrt{3}-i & \text{(a)} 2 \operatorname{cis} \frac{\pi}{6} \\
\text{(ii)} \sqrt{3}+i & \text{(b)} 2 \operatorname{cis} \frac{5 \pi}{6} \\
\text{(iii)} -\sqrt{3}+i & \text{(c)} 2 \operatorname{cis}\left(\frac{-5 \pi}{6}\right) \\
\text{(iv)} -\sqrt{3}-i & \text{(d)} 2 \operatorname{cis}\left(-\frac{\pi}{6}\right) \\
& \text{(e)} 2 \operatorname{cis} \frac{9 \pi}{6} \\
\end{array}\)
The correct matching is

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

Maximum area of the rectangle inscribed in a circle of radius \(10 \mathrm{cms}\) is

\(\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\) \(\times\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \ldots \ldots \infty=\)

If \(X\) is a binomial variate with mean 6 and variance 2 , then the value of \(P(5 \leq X \leq 7)\) is

If \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are mutually perpendicular vectors of the same magnitude, then the cosine of the angle between \(\mathbf{a}\) and \(\mathbf{a}+\mathbf{b}+\mathbf{c}\) is

A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at $30°$ with the horizontal. The horizontal component of the earth's magnetic field at the place is $0.3\mathrm{G}$. Then the magnitude of the earth's magnetic field at the location is

If the function \(y=\sin x(1+\cos x)\) is defined in the interval \([-\pi, \pi]\), then \(y\) is strictly increasing in the interval

Observe the following statements :
a. The thermal stability of hydrides of group 16 elements follow the order :
\(\mathrm{H}_2 \mathrm{O}>\mathrm{H}_2 \mathrm{~S}>\mathrm{H}_2 \mathrm{Se}>\mathrm{H}_2 \mathrm{Te}\)
b. Acidic nature of hydrides of group 16 elements follow the order :
\(\mathrm{H}_2 \mathrm{O}>\mathrm{H}_2 \mathrm{~S}>\mathrm{H}_2 \mathrm{Se}>\mathrm{H}_2 \mathrm{Te}\)
c. The reducing nature of \(\mathrm{H}_2 \mathrm{~S}, \mathrm{H}_2 \mathrm{Se}\) and \(\mathrm{H}_2\) Te follow the order :
\(\mathrm{H}_2 \mathrm{~S} < \mathrm{H}_2 \mathrm{Se} < \mathrm{H}_2 \mathrm{Te}\)
The correct statement are:

When six coins are tossed simultaneously, the probability of getting at least 4 heads is

\(\sin \left(\frac{5 \pi}{3}\right)+\sec \left(\frac{13 \pi}{3}\right)=\)

In a first order reaction, the concentration of the reactant is reduced from \(0.6 \mathrm{~mol} \mathrm{~L}^{-1}\) to \(0.2 \mathrm{~mol} \mathrm{~L}^{-1}\) in \(5 \mathrm{~min}\). What is the rate constant of the reaction? \((\log 3=0.4771)\)