Pipe A can fill a tank three times faster than pipe B.
If both pipes A and B are running together, they can fill the tank in 12 minutes.
Then the time taken by pipe B alone to fill the tank is :

For \(x \in(0,1)\), the derivative of \(\sin ^{-1} x\) with respect to \(\cos ^{-1} \sqrt{1-x^2}\) is:

The curve \(x=y^2\) and \(x y=k\) cut orthogonally, then \(k^2\) is equal to :

Corner points of the feasible region for an LPP are: \((0,2),(3,0),(6,0),(6,8)\) and \((0,5)\). Let \(z=4 x+6 y\) be the objective function. Then, Max \(z\) - Min \(z\) is equal to:

If we take \(8\) identical slips of paper and write the number \(0\) on one of them, the number \(1\) on three of the slips, the number \(2\) on three of the slips and the number \(3\) on one of the slips. These slips are folded, put in a box and roughly mixed. One slip is drawn at random from the box. If \(X\) is the random variable denoting the number written on the drawn slip, the variance of \(X\) is:

The value of \(\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)+2 \cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)\) is:

The probabilities of occurrence of two events A and B are 0.45 and 0.20 respectively.
The probability of their simultaneous occurrence is 0.06.
The probability that neither A nor B occurs is

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

The correct expression and unit of molar conductivity (\(\Lambda_{m}\)) are:

When a species become extinct, the plant and animal species associated with it in an obligatory way also become extinct. This is known as-

Read the passage carefully and answer the Questions:
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1\) )d, \(n s, n p\) or \(n s, n p, n d\) orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on. These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding. While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. The Crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. The five \(d\) orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands in a complex, it becomes asymmetrical and the degeneracy of the \(d\) orbitals is lifted. It results in splitting of the \(d\) orbitals. The pattern of splitting depends upon the nature of the crystal field.
Select the incorrect statement about the Crystal field splitting in octahedral complexes.

The magnetic field in a travelling electromagnetic wave has a peak value of 30 nT. The peak value of electric field strength is : (speed of light in free space is \( 3 \times 10^8 \text{ m/s} \)).

Identify the equation representing logistic growth of a population.