If the matrix \(A=\left[\begin{array}{lll}0 & b & 1 \\ 2 & 0 & c \\ 3 & 4 & 0\end{array}\right]\) is a symmetric matrix, then the values of \(b\) and \(c\) are:

The planes 2x - y + 4z = 5 and \(5 x-\frac{5}{2} y+10 z=6\) are

If the product of matrices \(\left[\begin{array}{cc}\cos ^2 \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^2 \theta\end{array}\right]\) and \(\left[\begin{array}{cc}\cos ^2 \beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & \sin ^2 \beta\end{array}\right]\) is a null matrix, then \(\theta\) and \(\beta\) differ by:

In \(\triangle A B C\), which of the following is not true is:
(A) \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}=\overrightarrow{0}\)
(B) \(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{A C}=\overrightarrow{0}\)
(C) \(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{C A}=\overrightarrow{0}\)
(D) \(\overrightarrow{A B}-\overrightarrow{C B}+\overrightarrow{C A}=\overrightarrow{0}\)
(E) \(\overrightarrow{B A}-\overrightarrow{B C}-\overrightarrow{C A} \neq \overrightarrow{0}\)

Consider the function \(f: N \rightarrow N\) given by \(f(x)=\left\{\begin{array}{ll}x+1, & \text { if } x \text { is odd } \\ x-1, & \text { if } x \text { is even }\end{array}\right.\)

If \(A=\left[\begin{array}{cc}2-x & 1 \\ 3 & y+2\end{array}\right]\) and \(B=\left[\begin{array}{cc}y & 1 \\ 3 & 2 x+1\end{array}\right]\) are such that A = B, then values of z and y are respectively :

The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin |x|-\cos |x|) d x\) is :

Find the equivalent capacitance between the points A and B.

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).
Which statement is true about LAB ?

The type of pollination in which genetically different pollen grains fall on the stigma is known as:

If A is a square matrix of order 3 and |A| = 5, then | adj A| is:

\(\int \frac{(x-3) e^x}{(x-1)^3} d x\) is equal to

Homologous organs represent: