The direction cosines of z-axis are:

The variance of number of heads in three tosses of a coin is:

The particular solution of the differential equation \(\frac{d y}{d x}+\frac{3 y}{x}=0, y(1)=1\) is

The area of a triangle having the points \(A (1,1,1), B (1,2,3)\) and \(C (2,3,1)\) as its vertices is :

The matrix \(A=\left[\begin{array}{lll}0 & 0 & 5 \\ 0 & 5 & 0 \\ 0 & 0 & 5\end{array}\right]\) is a:

One root of the equation \(\left|\begin{array}{ccc}3 x-8 & 3 & 3 \\ 3 & 3 x-8 & 3 \\ 3 & 3 & 3 x-8\end{array}\right|=0\) is:

Area of region bounded by the curve \(y^2=4 x\) y-axis and the line y = 3 is:

Read the passage carefully and answer the questions:
The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the octahedral complex to yield three orbitals of lower energy, \(t_2\), set and two orbitals of higher energy, \(e_g\) set. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting and the energy separation is denoted by \(\Delta_0\). Thus energy of the two \(e_g\), orbitals will increase by \((3 / 5) \Delta_0\) and that of three \(t_{2 g}\) will decrease by \((2 / 5) \Delta_0\). The crystal field splitting \(\Delta_0\) depends upon the field produced by the ligand and charge on the metal ion. Some ligands are able to produce strong fields, in which case the splitting will be large, whereas others produce weak fields and consequently result in small splitting of \(d\) orbitals. Relative magnitude of crystal field splitting energy \(\Delta_0\) and pairing energy, \(P\) (energy required for electron pairing in a single orbital) determine the formation of low spin ( \(\Delta_0>P\) ) or high spin ( \(\Delta_0<P\) ) complex. In tetrahedral coordination entity formation, the \(d\)-orbital splitting is inverted and is smaller as compared to the octahedral field splitting. The crystal field theory attributes the colour of the complex to \(d-d\) transition of the electron. The colour of the coordination compounds depends on the crystal field splitting. In the absence of ligand, crystal field splitting does not occur and hence the substance is colourless.
Choose the correct statement from the following:

Two long parallel wires separated by a distance of 2.50 cm, repel each other with a force of \( 4 \times 10^{-5} \text{ N/m} \). The current in one wire is 0.5 A. The current in the second wire will be

Arrange the following in the increasing order of oxidation state of nitrogen :
(A) \(NO\)
(B) \(NO_2\)
(C) \(N_2O_5\)
(D) \(N_2O\)
(E) \(N_2O_3\)
Choose the correct answer from the options given below :

About ______ the first cellular forms of life appeared on earth.

Read the following passage given below and answer the question.
Production of electricity by thermal plants is not very efficient method. It is now possible to make cells in which reactants are fed continuously to the electrodes and products are also removed continuously. One of the most successful fuel cells is hydrogen fuel cell.
Which is not true for the fuel cell?

Which one of the following infections is completely curable?