Match List 1 with List 2.

List - 1 (Matrix)	List - 2 (Determinants)
(A) \(\operatorname{Matrix}(A)=\left[\begin{array}{cc}1 & 7 \\ -3 & 5\end{array}\right]\)	(I) 24
(B) \(\operatorname{Matrix}(B)=\left[\begin{array}{cc}-2 & 5 \\ -3 & -3\end{array}\right]\)	(II) 32
(C) \(\operatorname{Matrix}(C)=\left[\begin{array}{ll}-12 & 8 \\ -16 & 8\end{array}\right]\)	(III) 21
(D) \(\operatorname{Matrix}(C)=\left[\begin{array}{ll}-12 & 8 \\ -16 & 8\end{array}\right]\)	(IV) 26

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Match List-I with List-II

List-I	List-II
(A) \(\int \frac{d x}{x^2-16}\)	(I) \(\frac{1}{8} \log \left|\frac{4+x}{4-x}\right|+c\)
(B) \(\int \frac{d x}{x^2+16}\)	(II) \(\log \left|x+\sqrt{x^2-16}\right|+c\)
(C) \(\int \frac{d x}{16-x^2}\)	(III) \(\frac{1}{8} \log \left|\frac{x-4}{x+4}\right|+c\)
(D) \(\int \frac{d x}{\sqrt{x^2-16}}\)	(IV) \(\frac{1}{4} \tan ^{-1}\left(\frac{x}{4}\right)+c\)

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If \(f(x)=\left\{\begin{array}{cl}\frac{k \cos x}{\pi-2 x}, & x \neq \frac{\pi}{2} \\ 3, & x=\frac{\pi}{2}\end{array}\right.\) is continuous at \(x=\frac{\pi}{2}\), then \(k\) is :

If \(A\) is a square matrix such that \(A^2=A\) and \(I\) is the identity matrix of same order as \(A\), then the matrix \((2 I+A)^3-19 A-3 I\) is equal to

The value of \(x\), if \(\cos^{-1} \sqrt{3}x + \cos^{-1} x = \frac{\pi}{2}\) is:

If \(A\) is a \(3 \times 3\) matrix such that \(|\operatorname{adj} A|=9\) and \(\left|k A^{-1}\right|=9\), then the value of \(k\) are :

If \(A = \begin{bmatrix} 1 & 2 & -1 \\ 2 & 3 & 0 \\ 1 & -1 & 2 \end{bmatrix}\), then Adj(A) 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).
BOD refers to the -

The quantity of charge required to obtain \(2\text{ mol}\) of \(\text{Mn}^{2+}\) from \(\text{MnO}_4^{-}\) is:

Match List-l with List-II:

List-I (Interspecies Relationships)	List-II (Features)
(A) Commensalism	(I) One species is benefitted at the expense of the other
(B) Mutualism	(II) One species is harmed and the other is unaffected
(C) Amensalism	(III) Both the species are benefitted
(D) Parasitism	(IV) One species benefits and other remains unaffected

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Self Inductance of coil causes :

Which of the statements are true, If a convex lens is placed in water:
a. Its focal length increases.
b. Its focal length decreases.
c. Rays bend less when they move through lens.
d. There is no change in radius of curvature.
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In a pure crystal, the lattice point cannot be occupied by: