Which of the following functions \(f(x)\) are differentiable at \(x=0\) ?
(A) \(|x|\)
(B) \(|x-1|\)
(C) \(|\sin x|\)
(D) \(|\cos x|\)
(E) \(x^2\)
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If \(y=\log \left[\frac{x^2}{e^2}\right]\) then value of \(\frac{d^2 y}{d x^2}\) is :

A vector of magnitude 8 units in the direction perpendicular to both the vectors \(\hat{i}+\hat{j}+\hat{k}\) and \(2 \hat{i}+\hat{k}\) is

The EMI (in) under the flat rate on a loan of 6,00,000 rupees with 20% annual interest for 5 years is :

For the differential equation \((x+y) d y+(x-y) d x=0\), which of the following is/are correct?
(A) Differential equation is homogeneous
(B) Order of differential equation is 1
(C) Integrating factor of differential equation is \(e ^{ x }\)
(D) Degree of the equation is not defined
Choose the correct answer from the options given below:

The function \(f(x)=\frac{x}{3}+\frac{3}{x}\) is increasing in the interval:

Match (List I) with (List II)

(List I)	(List II)
A. For any vector \(\vec{a}, \vec{a} \times \vec{a}\)	\(I\). \(|\vec{a}|^2\)
B. For any vector \(\vec{a}, \vec{a} \cdot \vec{a}\)	\(II.\) \(0\)
C. If either \(\vec{a}=\overrightarrow{0}\) or \(\vec{b}=\overrightarrow{0}\), then \(\vec{a} \cdot \vec{b}\) is	\(III.\) \(|\vec{a}||\vec{b}|\)
D. If the angle between \(\vec{a}\) and \(\vec{b}\) is \(\theta=0\), then \(\vec{a} \cdot \vec{b}\) is	\(IV.\) \(\overrightarrow{0}\)

Choose the correct answer from the options given below:

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 micro-organisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
BOD is a measure of-

Correct order of intermolecular forces of attractions in the following solids is :
(A) NaCl
(B) \(I_2\)
(C) Ice
(D) Iron
(E) Rubber
Choose the correct answer from the options given below :

Total internal reflection can take place only, if

Read the following passage carefully and answer the given questions.
Productivity in the Ecosystem
The flow of energy through a terrestrial ecosystem starts with the harnessing of sunlight by autotrophs. The radiant energy is converted by photosynthesis to organic compounds is known as productivity. The productivity is expressed in units of energy per unit of area per unit time. The amount of organic matter present at any given time is biomass. The productivity of any ecosystem is affected by many parameters like weather conditions, nutrient availability etc. The productivity of any ecosystem does vary seasonally and yearly.
The amount of organic matter accumulated in an ecosystem at a given time is known as-

Which one of the followings is not a feature of insecticidal protein crystals formed by Bacillus thuringiensis: -

A capacitor of capacitance C is first charged by a battery of V volts. It is disconnected from the battery and is filled with a dielectric of dielectric constant K. The ratio of energy stored in the two cases, respectively will be