If \(y=\sqrt{a x+b}\), then \(y \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2=\)

The maximum value of \(z=5 x+7 y\) subjected to constraints \(x+y \leq 5, x \geq 0, y \geq 0\) is :

Area lying in the first quadrant and bounded by the circle \(x^2+y^2=4\) and the lines x = 0 and x = 2 is:

Assuming the same rate of change continues for the following data :

Year(X)	2019	2020	2021	2022	2023
Profit in Percentage (y)	38	40	65	72	69

The equation of the straight line trend using the least square method is :

For what value of k, the following system have a unique solution? (where R is set of real numbers)
x + y + z = 1
2x + 3y + 4z = 3
x - y + kz = 5

A coin is tossed and a die is thrown. The probability that the outcome will be a tail on the coin or a number greater than 3 on the die is

Consider the following hypothesis test:
\(H_0: \mu \leq 16\)
\(H_a: \mu>16\)
A sample of 25 provided a sample mean \(\bar{x}\) = 17 and a sample standard deviation S = 3.5, then the value of the test statistic in t-test 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 comples to yield three orbitals of lower energy, \(t_{2 g}\) 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 \(\Delta_{\circ}\) 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 order of the ligands in their increasing crystal field splitting:

The value of \(\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)+\sin ^{-1}\left(\frac{1}{2}\right)\) is:

The equation of motion of a missile is x = 3t, y = -4t, z = t where time 't' is in seconds and the distance is measured in kilometres.
If the position of rocket at a certain instant of time is (5, -8, 10), then what will be the height of the rocket from the ground? (Ground is considered as xy plane)

Two lenses of power 3D and -1D are kept in contact. The focal length of the combined lens would be :

In a time series, the variations which occur due to general tendency of the data to increase or decrease over a long term are known as :

Match List-l with List-ll

List-l (Enzymes)	List-II (Functions)
(A). DNA polymerase	(I). Transcribes precursor of mRNA (hnRNA)
(B). DNA ligase	(II). Transcribes rRNAs
(C). RNA polymerase	(III). Joins the DNA fragments
(D). RNA polymerase II	(IV). Catalyzes the polymerization of deoxynucleotides

the correct answer from the options given below: