The system of equations
\(\begin{aligned} & x-3 y-8 z=-10 \\ & 2 x+5 y+\lambda z=13 \\ & 3 x+y-4 z=0\end{aligned}\)
has infinite number of solutions if the value of \(\lambda\) is equal to:

If \(A=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]\) and \(A^4=\left[\begin{array}{ll}1 & 0 \\ k & 1\end{array}\right]\), then the value of \(k\) is :

If \(\frac{1}{x^2}-\frac{1}{x}\) > 0, then \(x\) lies in the interval

The integrating factor of the linear differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is:

If \(x=t^3\) and \(y=t^2\), then \(\frac{d^2 y}{d x^2}\) is equal to :

If \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right], a \neq 0\) then \(|\operatorname{adj} A|\) is equal to

The solution of LPP :
Min. \(z=100 x+400 y\)
s.t. \(x+2 y \geq 8\)
\(2 x+5 y \leq 20\)
\(x \geq 0, y \geq 0\)
occurs at :

What is the magnification of simple microscope having focal length 5 mm?

Which one of the following reactions is given by ozone (\(O_3\))?

Helium gas is filled in a transparent balloon which is kept in air. The balloon behaves as:

Read the passage carefully and answer the questions
Ethers with the general formula \(C _n H _{2 n+2} O\) and functional group C-O-C may be divided as symmetrical and unsymmetrical.
Lower symmetrical ether may be obtained by the dehydration of alcohols at low temperatures in the presence of protic acids,
such as \(H _2 SO _4\) and \(H _3 PO _4\) whereas at high temperature intramolecular dehydration takes place.
Formation of ether from primary alcohols is also accompanied by the formation of small amount of alkenes.
Due to the steric hinderance, secondary and tertiary alcohols undergo elimination reaction rather than substitution reaction yielding alkene as major product. Ethers can also be obtained by heating alkyl halides with sodium or potassium alkoxides.
Substituted Ethers (secondary & tertiary) can also be prepared using this method.
If primary halides are replaced with secondary and tertiary halides, alkenes are the major products. The compound with the lowest boiling point is

If \(A=\left[\begin{array}{ll}3 & -5 \\ 4 & -7\end{array}\right]\), then \(A^{-1}\) is:

In the hydrogen atom, an electron revolves around the nucleus in a circular orbit of radius \( 5.11 \times 10^{-11} \text{ m} \) with a frequency of \( 6.8 \times 10^{15} \text{ Hz} \). The magnetic field at the centre of the orbit is :