If \(x^2-y^2=t-\frac{1}{t}\), and \(x^4+y^4=t^2+\frac{1}{t^2}\), then which of the following is correct?

\(\int_0^2\left(x^2+1\right) d x\) is equal to

Which of the following is the range of the principal value of \(\sec ^{-1} x\) ?

\(\int_0^1 \tan ^{-1}\left(\frac{2 x-1}{1+x-x^2}\right) d x\) is equal to :

Difference between the maximum and minimum value of the objective function \(Z=5 x+3 y\), subject to the constraints: \(3 x+5 y \leq 15 ; 5 x+2 y \leq 10 ; x, y \geq 0\) is:

A die is thrown 4 times and getting 3 is considered a success. The probability of 2 successes is :

If \(x=4 t\) and \(y=\frac{4}{t}\), then \(\frac{d^2 y}{d x^2}\) is

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According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1) d, n s, n p\) 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.
Which of the following represents the correct pair?

Area of the region bounded by \(y=x^2\) and the line \(y=16\) is

Match List-I with List-II

List-I (Terms)	List-II (Features)
(A) Humus	(I) Mass of living organisms
(B) Detritus	(II) Dark-colored amorphous substance
(C) Producers	(III) Dead remains of plants and animals
(D) Biomass	(IV) Phytoplankton

Choose the correct answer from the options given below:

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Haloalkanes undergo nucelophilic substitution reactions owing to the polarity of CX bond. The nucleophile reacts with the haloalkane on the carbon possessing a partial positive charge holding the halogen atom. The halogen atom \(X\) is replaced by a nucleophile. Depending on the kinetics and mode of bond breaking, the mechanism can be either SN1 or SN2 reaction. The rate of SN1 reaction is governed by the stability of carbocation and in SN2 reaction, the rate of reaction is governed by steric factor. Chirality is the main factor in both SN1and SN2. In SN1 reaction, the chirality of alkyl halide is accompanied by racemization of the product, while in SN2 reaction, the product is characterized by inversion of configuration. The structure of alkyl halide and the nature of the solvent also governs the mechanism of the substitution.
Which of the following bromo isomer of butane will react at a faster rate towards SN2 reaction?

The value of a washing machine purchased two years ago depreciates at the annual rate of 10%. If its present value is ₹ 50,000, then its value after two years will be :

C and Z in the species-area relationship log S = log C + Z log A represents: