Arrange the following as per statistical inferences :
(A) Data Analysis
(B) Population
(C) Making Inferences
(D) Data Collection
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The angle between the vectors \(\hat{i}-2 \hat{j}+\hat{k}\) and \(3 \hat{i}+2 \hat{j}+\hat{k}\) is:

The value of the integral \(\int_0^\pi 2 x \sin ^3 x\ d x\) is:

In binomial distribution with \(n =10\) and \(P =1\), the probability of the event that unequal number 3 of failures and successes occur is:

\(\int \frac{1}{(x+1)(x+2)} d x\) is equal to

If X is a normal variate with mean 16 and standard deviation 4, then the value of the standard normal variate Z corresponding to X = 17 is :

A machine costing ₹ \(36000\) has an effective life of 5 years with scrap value of ₹ \( 5000\) following a linear method of depreciation. Which are correct?
(A) The value of the machine after 1 year is ₹ \(31000\)
(B) The value of the machine after 2 years is ₹ \(23600\)
(C) The value of the machine after 3 years is ₹ \(18400\)
(D) The value of the machine after 4 years is ₹ \(11200\)
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Microbes in human welfare Microbes are a very important component of life on earth. we use microbes and microbially derived products almost every day, like LAB to make curd, yeast to make bread, adding flavor and texture to cheese, producing antibiotics etc. The addition of LAB, improves the milk quality by increasing vit. B12. Cheese is one of the oldest food items in which microbes are used. For example, large holes in Swiss cheese are due to CO2 produced by the bacterium Propionibacterium sharmanii. The BOD test measures the rate of uptake of oxygen by microorganisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water. The greater the BOD of waste water, the more is its polluting potential. In anaerobic sludge digesters, bacteria produce a mixture of gases like methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as a source of energy. Biofertilisers are organisms that enrich the nutrient quality of the soil. The main sources of biofertilisers are bacteria, fungi and cyanobacteria. You have studied about the nodules on the roots of leguminous plants formed by the symbiotic association of Rhizobium. These bacteria fix atmospheric nitrogen into organic forms, which is used by the plant as a nutrient. Other bacteria can fix atmospheric nitrogen while free-living in the soil (examples Azospirillum and Azotobacter).
Identify, the free living bacteria in soil, which can fix the atmospheric nitrogen.

Of the incident solar radiation, what is the percentage of photosynthetically active radiation (PAR)?

A wheel with 8 metallic spokes each 50 cm long is rotated with a speed of 120 rev/min in a plane normal to the horizontal component of the earth's magnetic field. The earth's magnetic field is \( 0.4 \text{ G} \). Calculate the emf induced between the axle and the rim.

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 ( \(Ti ^{I V} O _2, V^V O _2^{+}, Cr ^{V I} O _4^{2-}\), \(Mn ^{\text {VII }} O _4\) ) followed by a rather abrupt decrease in stability of higher oxidation states, so that the typical species to follow are \(Fe ^{I I, I I I}, Co ^{I I, I I I}, Ni ^{I I}, Cu ^{I, I I}, Zn ^{I I}\).
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. \(V ^{I I}, V^{I I I}, V^{I V}, V^V\). 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 dblock. 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.
Which amongst the following is a better oxidising agent than other ?

Two pipes A and B can fill a tank in 20 minutes and 30 minutes respectively.
Both pipes A and B are opened together for some time and then pipe B is turned off.
If the tank is filled in 15 minutes, then after how many minutes is pipe B turned off?

Match List I with List II

LIST I	LIST II
A. \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)	I. Linear Differential Equation of type \(\frac{d y}{d x}+P y=Q\)
B. \(\frac{d y}{d x}-2 y=\cos 3 x\)	II. Homogeneous Differential Equation
C. \(x^2 \frac{d y}{d x}=x^2-2 y^2+x y\)	III. Linear Differential Equation of type \(\frac{d x}{d y}+P x=Q\)
D. \(\left(x+2 y^3\right) \frac{d y}{d x}=y\)	IV. Variable separable

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