If \(\frac{3 x-5}{6}+8 \geq 4+\frac{2 x}{3}\), then

The value of the integral \(I=\int_{-1}^1\left(x+x^3+x^5\right) d x\) is:

The area (sq. units) bounded by the curve \(y=\sin x, \pi \leq x \leq 2 \pi\) and the \(x\)-axis is

The solution of the differential equation \(\left(x^2+x y\right) d y=\left(x^2+y^2\right) d x\) is

If a person rides his motorbike \(x km\) at 30 km per hour, he has to spend ₹ 3 per kilometer on petrol; if he rides \(y km\) at a faster speed of 40 km per hour, the petrol cost increases to ₹ 4 per kilometer. If he has ₹ 100 to spend on petrol and wishes to find the maximum distance he can travel within one hour, then linear programming problem (LPP) formulation is:

In a meeting, 70% of the members favour and 30% oppose a certain proposal.
A member is selected at random and we take X = 0 if he opposed, and X = 1 if he is in favour.
Then, E(X) is:

The order and degree of the differential equation \(x\left(\frac{d y}{d x}\right)^3+\left(\frac{d^2 y}{d x^2}\right)^2-3 \frac{d y}{d x}+y=0\) respectively, are:

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, \(t_{2 g}\) 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 ( \(\Delta_{\circ}>P\) ) or high spin ( \(\Delta_{\circ}<P\) ) 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
What will be the crystal field splitting energy for tetrahedral \(\left[ CoCl _4\right]^{2-}\) if its value for octahedral \(\left[ CoCl _6\right]^{4-}\) is \(18,000 cm^{-1}\) ?

The corner points of feasible region determined by the following systems of linear inequalities :
\(2 x+y \leq 10, x+3 y \leq 15, x \geq 0, y 0\) are \((0,0),(5,0),(3,4)\) and \((0,5)\)
Let \(z=p x+q y\), when \(p, q>0\) then the relation between \(p\) and \(q\), so that maximum of \(z\) occurs at both points \((3,4)\) and \((0,5)\) is:

From the following, choose the one which are not secondary haloalkanes :
(A) 2-Bromopentane
(B) 1-Bromo-3-methylbutane
(C) 3-Bromopentane
(D) 2-Bromo-2- methylbutane
(E) 2-Bromo-3-methylbutane
Choose the correct answer from the options given below :

Read the passage given below and answer the question.
The transition elements which give the greatest number of oxidation states occur in or near the middle of the series. Manganese, for example, exhibits all the oxidation states from +2 to +7. The lesser number of oxidation states at extreme ends stems from either too few electrons to lose or share (Sc, Ti) on too many d-electrons (hence fewer orbitals available in which to share electrons with others) for higher valence (Cu, Zn). Thus early in the series scandium (II) is virtually unknown and titanium (IV) is more stable than Ti (III) or Ti (II). At the other end, the only oxidation state of zinc is +2 (no d electrons are involved). The maximum oxidation states of reasonable stability correspond in value to the sum of the s- and d-electrons upto manganese (\(TiO_2, VVO^+, CrVIO_4^{2-}, MnVIIO_4\)) followed by a rather abrupt decrease in stability of higher oxidation states, so that the typical species to follow are \(Fe^{II,III}, Co^{II,III}, Ni^{II}, Cu^{I,II}, Zn^{II}\).
The variability of oxidation states, a characteristic of transition elements, arises out of incomplete filling of d-orbitals in such a way that their oxidation states differ from each other by unity, e.g. \(V^{II}, V^{III}, V^{IV}, V^{V}\). This is in contrast with the variability of oxidation states of non transition elements where oxidation states normally differ by a unit of two.
An interesting feature in the variability of oxidation states of the d-block elements is noticed among the groups (groups 4 through 10). Although in the p-block, the lower oxidation states are favoured by the heavier members (due to inert pair effect), the opposite is true in the groups of d-block. For example, in group 6, Mo(VI) and W(VI) are found to be more stable than Cr(VI). The Cr(VI) in the form of dichromate in acidic medium is a strong oxidising agent, whereas \(MoO_3\) and \(WO_3\) are not.
Low oxidation states are found when a complex compound has ligands capable of \(\pi\)-acceptor character in addition to the \(\sigma\)-bonding. For example, in \(Ni(CO)_4\) and \(Fe(CO)_5\), the oxidation state of nickel and iron is zero.
The elements giving minimum number of oxidation states are in extreme left and right in 3d series.
This is because :

Arrange the given products formed during sewage treatment in correct sequence?
(A) Biogas
(B) Activated sludge
(C) Flocs
(D) Primary sludge
Choose the correct answer from the options given below:

The solution of the differential equation \((x+1) \frac{d y}{d x}+1-2 e^{-y}=0 ; y(0)=0\) is