Match (List I) with (List II)

(List I)	(List II)
A. \(\sin 2 x\)	\(I.\) decreases from 1 to 0 in \(\left(0, \frac{\pi}{2}\right)\)
B. \(\tan x\)	\(II.\) decreases if \(x \in\left(0, \frac{\pi}{6}\right)\)
C. \(\cos x\)	\(III.\) increases from 0 to 1 in \(\left(0, \frac{\pi}{4}\right)\)
D. \(\cos 3 x\)	\(IV.\) is an increasing function in each quadrant

Choose the correct answer from the options given below:

In the method of least squares, we find trend values using:

A random variable X has the following probability distribution

X	0	1	2	3	4	5
P(X)	0.1	k	0.2	2k	0.3	k

Then \(P(X<3)\) is

If \(A\) and \(B\) are square matrices of the same order, then \((A+B)(A-B)\) is equal to

If \(A=\left[\begin{array}{ll}1 & 0 \\ 3 & 1\end{array}\right]\) and \(A^4=\left[\begin{array}{ll}1 & 0 \\ k & 1\end{array}\right]\), then the value of \(k\) is :

The number of one - one functions from \(A=\{1,2,3,4\}\) to itself is:

Which of the following inequalities holds true?
(A) \(\sqrt{5}+\sqrt{3}>\sqrt{6}+\sqrt{2}\).
(B) If \(a>b\) and \(c<0\), then \(\frac{a}{c}<\frac{b}{c}\).
(C) \(\frac{1}{x^2}>\frac{1}{x}>1\), if \(0<x<1\).
(D) If \(a\) and \(b\) are positive integers and \(\frac{a-b}{6.25}=\frac{4}{2.5}\), then \(b>a\).
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A plane meets the coordinates axes at A, B and C such that centroid of \(\triangle A B C\) is at the point (2, 3, 1). If the equation of the plane is 3x + 2y + 6z = k, then k is:

Read the following passage carefully and answer the given questions. During anaerobic sludge digestion, bacteria produce a mixture of gases such as methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as source of energy as it is inflammable. The bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. You must have noticed that bottled fruit juices bought from the market are clearer as compared to those made at home. This is because the bottled juices are clarified by the use of pectinases and proteases. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. 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 bottled juices are clarified by the combination of -

Answer the question on the basis of passage given below :
Adsorption is the phenomenon of attracting and retaining the molecules of a substance on the surface of a solid resulting into a higher concentration on the surface than in the bulk. The substance adsorbed is known as adsorbate and the substance on which adsorption takes place is called adsorbent. In physisorption adsorbate is held to the adsorbent by weak van der Waals forces and in chemisorption by strong chemical bond.
Out of Milk, butter, ice cream, fruit juice and lemonade. Identify the one which is not a colloid.

Answer the question on the basis of passage given below :
All transition elements display typical metallic properties like high melting point, boiling point, tensile strength, high malleability and ductility, luster, etc. Their metallic character increases down the group and elements of 4d and 5d series exhibit higher metallic characters. They show variable oxidation states and forms a number of coloured compounds due to the presence of unpaired electrons which shows d-d transitions and makes them paramagnetic.
Which out of the following coordination compounds are colourless in nature?
A. \(\left[ Cr \left( NH _3\right)_6\right] Cl _3\)
B. \(\left[ Co \left( NH _3\right)_6\right] Br _3\)
C. \(K _3\left[ Fe ( CN )_6\right]\)
D. \(\left[ MnCl _2(e n)_2\right]^{+}\)
E. \(\left[ Ni ( CO )_4\right]\)
Choose the correct answer from the options given below :

A circular loop of area \( A = 0.01 \text{ m}^2 \) carrying a current of \( I = 10 \text{ A} \), is held in a magnetic field of \( B = 0.1 \text{ T} \) parallel to the plane of the loop. The torque acting on the loop 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 complez 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 \((3 / 5) \Delta_{\circ}\) 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.
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