The absolute maximum value of the function \(f(x)=4 x-\frac{1}{2} x^2\) in the interval \(\left[-2, \frac{9}{2}\right]\) is

In a LPP, let \(R\) be the feasible region.
A. If R is unbounded then a max./min. value of objective function may not exist.
B. If R is bounded then a max. and min. value of objective function will always exist.
C. If a solution exists, it must occur at a corner point.
D. If R is bounded then max. will exist but min. may or may not exist for an objective function.
Choose the correct answer from the options given below:

If \(P, Q\) and \(R\) are three singular matrices given by
\(P=\left[\begin{array}{cc}2 & 3 a \\4 & 3\end{array}\right], Q=\left[\begin{array}{cc}b & 5 \\2 a & 6\end{array}\right]\)
and \(R=\left[\begin{array}{cc}a^2+b^2-c & 1-c \\ c+1 & c\end{array}\right]\), then the value of \((2 a+6 b+17 c)\) is:

Let \(A=\left[a_{i j}\right]_{3 \times 3}\) be a matrix, defined by \(a_{i j}=\left\{\begin{array}{ll}2 i+3 j & , i < j \\ 6 & , i=j \\ 3 i-2 j & , i > j\end{array}\right.\). The number of elements in A. which are greater than 6, is

If \(f(x)=\frac{1}{2 x+1}, x \neq-\frac{1}{2}\), then \(f[f(x)]\) is :

The integrating factor of differential equation \(\left[y(1-x \tan x)+x^2 \cos x\right] d x-x d y=0\) is :

If \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ -4 & 2 & 4\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & -2 \\ 3 & -4 \\ 2 & 4\end{array}\right]\) then product \(A B\) is :

\(\text{If} A \text{ and } B \text{ are independent events with }P(A)=0.8 \text{ and } P(B)=0.4,\text{ then the value of } P(A\cup B) \text{ is: \)

Arrange the following series of emission lines in the spectrum of hydrogen atom in sequence according to their increasing wavelengths:
(A) Paschen series
(B) Lyman series
(C) Balmer series
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In our body, cell growth and differentiation is highly controlled and regulated. In cancer cells, there is breakdown of these regulatory mechanisms. Cancerous cells just continue to divide giving rise to masses of cells called tumors. Benign tumors normally remain confined to their original location. The malignant tumors, on the other hand are a mass of proliferating cells. These cells actively divide and starve the normal cells by competing for vital nutrients. Cells sloughed from such tumors reach distant sites through blood, and wherever they get lodged in the body, they start a new tumor there. This property is called metastasis. Transformation of normal cells into cancerous cells may be induced by physical, chemical or biological agents called carcinogens.
Cancer is caused by:
A. Carcinogens
B. Protein rich foods
C. Primary lymphocytes
D. Oncogenes
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Carbohydrates such as cane sugar, glucose, starch etc. belong to a class of naturally occurring organic compounds. Structurally these are polyhydroxy aldehydes or ketones or the compounds which produce such units on hydrolysis. Carbohydrates which are sweet in taste are called sugars. Theses sugars can be reducing or non-reducing. Monosaccharides cannot be hydrolyzed to simpler units whereas Oligosaccharides yield two to ten monosaccharide units on hydrolysis. Polysaccharides yield large number of monosaccharide units on hydrolysis. This class of biomolecules is essential for living systems as they serve as storage molecules in plants and animals. In addition, they provide raw materials for important industries like furniture, textiles, paper, lacquers and breweries. Two aldopentose form important constituents of nucleic acids. Carbohydrates are also called saccharides (Greek: sakcharon which means sugar). Sugar which is used in household and in milk also comes under this class of biomolecules.
Reducing sugars respond to

The linear magnifications of the object in the conjugate positions of a convex lens are 4 and 2, respectively. The distance between the conjugate positions is 40 cm . This convex lens of glass \((u=1.5)\) is immersed in water \((u=1.33)\). The new focal length of convex lens in water is :

The inverse of the matrix \(\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6\end{array}\right]\) is