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:

Which of the following are correct?
(A) For a square matrix \(A\), if \(A^3=I\), then \(A^{-1}=A^2\).
(B) The determinant of only a square matrix can be defined.
(C) If \(A\) is a square matrix of order 3 , then the number of minors of the matrix \(A\) is 3 .
(D) If \(A\) and \(B\) are two non-singular matrices of the same order, then \((A B)^{-1}=B^{-1} A^{-1}\).
Choose the correct answer from the options given below :

The maximum value of \(Z\) for the linear programing problem maximize \(Z=x+y\) subject to the constraints \(x+4 y \leq 8,2 x+3 y \leq 12,3 x+y \leq 9, x \geq 0, y \geq 0\) is :

The plane \(2 x-(1+\lambda) y+3 \lambda z=0, \lambda \in R\) passes through the intersection of planes :

Let \(A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\) and \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\). If \(A^T+A=I\), then:

The value of the determinant \(\left|\begin{array}{lll}a-b & b+c & a \\b-c & c+a & b \\c-a & a+b & c\end{array}\right| \text { is : }\)

If \(\left|\begin{array}{ccc}x-2 & 1 & 2 \\ 1 & 2 x & 1 \\ -1 & 1 & 1\end{array}\right|=3\) then the values of x are:

The longest wavelength in Balmer Series of hydrogen spectrum will be:

We use microbes and microbially derived products almost every day. Bacteria called lactic acid bacteria (LAB) grow in milk to convert it into curd. The dough, which is used to make bread, is fermented by yeast called Saccharomyces cerevisiae. Microbes are used to produce industrial products like lactic acid, acetic acid and alcohol, which are used in a variety of processes in the industry. However, certain bacteria, which grow anaerobically on cellulosic material, produce large amount of methane along with CO2 and H2. These bacteria are collectively called methanogens, and one such common bacterium is Methanobacterium. These bacteria are commonly found in the anaerobic sludge during sewage treatment. Microbes are also used for commercial and industrial production of certain chemicals like organic acids, alcohols and enzymes.
Examples of acid producers are Aspergillus niger (a fungus) of citric acid. You are familiar with the commonly used antibiotic Penicillin. Do you know that Penicillin was the first antibiotic to be discovered, and it was a chance discovery? 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.
Which one of the following statements is not true with respect to microbes?

The dark purple crystals of potassium permanganate are isostructural with:

\(\frac{d}{d x}\left(e^{2 \log _e x^3}\right)\) equals

Read the following passage carefully and answer the given questions
The earthworm is referred to as the farmer's friend. This is because they help in the breakdown of complex organic matter as well as in loosening of soil. Similarly, decomposers break down complex organic matter into inorganic substances like carbon dioxide, water and nutrients and the process is called decomposition. Dead plant remains such as leaves, bark, flowers and dead remains of animals, including fecal matter, constitute detritus, which is the raw material for decomposition. Some of the steps in the process of decomposition are fragmentation, catabolism, leaching, humification etc.
The degradation of detritus into simpler inorganic substances by the bacterial and fungal enzymes is called as-

Consider the differential equation \(x d y=(x+y) d x\). Which of the following are true?
(A) It is a homogenous differential equation
(B) It is a differential equation of order 2
(C) The general solution of the differential equation contains 2 arbitrary constants
(D) Integrating factor of differential equation is \(\frac{1}{x}\)
(E) Degree of the differential equation is not defined
Choose the correct answer from the options given below: