If the equation of a floor of a room is given by \(x+y-z+4=0\) and the equation of roof is given by \(x+y-z+5=0\). Then, the height of the room is :

The probability that in a year of the 22nd century chosen at random, there will be 53 Sundays is:

If \(\hat{i}, \hat{j}, \hat{k}\) are unit vectors along X, Y, Z axes then,
(A) \(\hat{i} \times \hat{i}=1\)
(B) \(\hat{j} \cdot \hat{j}=0\)
(C) \(\hat{i} \times \hat{j}=\hat{k}\)
(D) \(\hat{k} \times \hat{i}=\hat{j}\)
(E) \(\hat{i} \cdot \hat{j}=0\)
Choose the correct answer from the options given below:

If A is a non-identity invertible symmetric matrix, then \(A^{-1}\) is:

The relation \(R\) on the set \(A\) of points in a plane, given by \(R=\{(P, Q)\) : Distance of the point \(P\) from the origin is same as the distance of

If \(A\) is a square matrix and \(I\) is an identity matrix of same order such that \(A^2=A\), then \((2 I+A)^2-5 A\) is :

For finding absolute maximum and minimum values of a function \(f\) given by
\(f(x)=2 x^3-15 x^2+36 x+1 \text { on }[1,5]\)
A. the critical points in \((1,5)\) are 2 and 3 .
B. the absolute maximum value of \(f\) on \([1,5]\) is 29 .
C. the absolute minimum value of \(f\) on \([1,5]\) is 24 .
D. the absolute minimum value of \(f\) on \([1,5]\) is 20 .
E. we evaluate the value of \(f\) at critical points and at the end points of the interval \([1,5]\).
Choose the correct answer from the options given below:

The work done by the field of a nucleus in a complete circular orbit of the electron is

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).
In anaerobic sludge digesters, the biogas produced by the bacteria, is composed of -

Which of the following is only composed of exons?

Reduction of nitrobenzene by which of the following reagent does not give aniline?

The common name and IUPAC name of the following compound, respectively, are

The direction ratios of the line perpendicular to the lines \(\frac{x-7}{-6}=\frac{y+17}{4}=\frac{z-6}{2}\) and \(\frac{x+5}{6}=\frac{y+3}{3}=\frac{z-4}{-6}\) are proportional to: