If $A, B, C$ are the angles of a triangle then the system of equations $-x+y\cos C+z\cos B=0, x\cos C-y+z\cos A=0, x\cos B+y\cos A-z=0$ have

If the lines \(2 x-y+3=0\) and \(4 x+k y+3=0\) are conjugate with respect to the ellipse \(5 x^2+6 y^2-15=0\), then \(k\) equals

If the sum of the distances of a point \(P\) from two perpendicular lines in a plane is 1 , then the locus of \(P\) is a

If \(f(x)=\cot ^{-1}\left(\frac{x^x-x^{-x}}{2}\right)\), then \(f^{\prime}(1)=\)

For \(k>0\), if \(\mathrm{k} \sqrt{-1}\) is a root of the equation \(x^4+6 x^3-16 x^2+24 x-80=0\) then \(k^2=\)

If \(a\) and \(b\) are respectively the order and degree of a differential equation \(y^2\left(y^{\prime \prime}\right)^2+3 x\left(y^{\prime}\right)^{1 / 3}+x^2 y^2=\sin x\), then

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

If the rate constant for a first order reaction is $2.303\times {10}^{-3}{ \mathrm{s}}^{-1}$, find the time required to reduce $4 \mathrm{g}$ of the reactant to $0.2 \mathrm{g}$.

Identify the incorrect statement regarding Reynold's number $\left(R_e\right):$

A metal crystallizes with a $\mathrm{FCC}$-lattice, the edge of whose unit cell is $\mathrm{xpm}$. The diameter of this metal atom would be $\mathrm{p} \mathrm{m} .$

Identify the correct statements from the following
a) Borax gives blue coloured \(\mathrm{Co}\left(\mathrm{BO}_2\right)_2\) when heated with \(\mathrm{Co}^{2+}\) salt.
b) The correct formula of Borax is \(\mathrm{Na}_2\left[\mathrm{~B}_4 \mathrm{O}_5(\mathrm{OH})_4\right] 8 \mathrm{H}_2 \mathrm{O}\)
c) Trihalides of boron are Lewis bases.

If sucrose is boiled with dilute. \(\mathrm{HCl}\) in alcoholic solution the ratio in which glucose and fructose are formed is

The general solution of the differential equation \(\frac{d x}{d y}+\frac{x}{y}=x^2\) is