In an LPP, if the objective function Z = ax + by has the same maximum value at two corner points of the feasible region, then the number of points of which \(Z_{\max }\) Occurs is:

The value of \(\tan ^{-1}(2)+\tan ^{-1}(3)\) is equal to

The present value of a perpetuity of ₹2,500 payable at the end of each year, if money is worth 10% compounded annually, is:

The corner points of the feasible region determined by the system of linear inequalities are \((0,0)\), \((2,4),(7,3)\) and \((5,0)\). If the maximum value of \(z=p x+q y\), where \(p, q>0\) occurs at both \((2,4)\) and \((7,3)\), then :

The value of \(\sin ^{-1} \frac{3}{5}-\sin ^{-1} \frac{8}{17}\) is :

Match List-l with List-ll

List-l	List-ll
(A) Angle between \(\hat{i}\) and \(-\hat{j}\) is	(I) \(\frac{\pi}{6}\)
(B) Angle between \(2 \hat{i}+\hat{k}\) and \(10 \hat{i}+5 \hat{k}\) is	(II) \(\frac{\pi}{4}\)
(C) Angle between \(\hat{i}\) and \(\hat{i}+\hat{j}\) is	(III) \(2 \pi\)
(D) Angle between \(\sqrt{3} \hat{j}-\hat{k}\) and \(\hat{j}\) is	(IV) \(\frac{\pi}{2}\)

Choose the correct answer from the options given below :

If \(x=a \cos ^3 \theta, y=a \sin ^3 \theta\). The value of \(\frac{d^2 y}{d x^2}\) at \(\theta=\frac{\pi}{6}\) is :

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The DNA fragments can be separated by a technique known as gel electrophoresis. Since DNA fragments are negatively charged molecules, they can be separated by forcing them to move towards the anode under an electric field through a medium/matrix. The separated DNA fragments can be visualised only after staining the DNA with a compound known as ethidium bromide followed by exposure to UV radiation. The repeated amplification of DNA is achieved by the use of a thermostable DNA polymerase (isolated from a bacterium, Thermus aquaticus). After completion of the biosynthetic stage, the product has to be subjected through a series of processes before it is ready for marketing as a finished product. The downstream processing and quality control testing vary from product to product. The convention for naming restriction enzymes is the first letter of the name comes from the genus and the second two letters come from the species of the prokaryotic cell from which they were isolated.
The downstream processing does not include -

At sunrise and sunset the sun appears reddish due to

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According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1\) )d, ns, np 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.
Which of the following is not true about the limitations of VBT

Which of the following is a free living, effective biocontrol agent, found very often in the root ecosystem?

You must have seen the weed Calotropis growing in abandoned fields. The plant produces highly poisonous _______ and that is why you never see any cattle or goats browsing on this plant.

A 200-turn coil of self-inductance 20 mH carries a current of 4 mA. The magnetic flux linked with each turn of the coil is: