The area (in square units) of the region bounded by the curve \(x^2=y\) and the straight line \(y=4\) in the first quadrant is equal to

\(\int \frac{x \sin ^{-1} x}{\sqrt{1-x^2}} d x=\)

The function \(f(x)=x^3, x \in R\), has:

If \(A=\left[\begin{array}{cc}x+4 & 2 \\ 3 & 1\end{array}\right] B =\left[\begin{array}{cc}x & 2 x-1 \\ 2 & 5\end{array}\right]\) and \(|A B|=12\), then the value(s) of \(x\) is/are :

Interval in which the function \(f\) given by \(f(x)=\tan x-4 x, x \in\left(0, \frac{\pi}{2}\right)\) is strictly decreasing is

A line passes through two points A(2, -3, -1) and B(8, -1, 2). The co-ordinates of a point on this line at a distance of 14 units from A are:
Α. (14, 1, 5)
Β. (86, 25, 41)
C. (-10, -7, -7)
D. (-82, -31, -43)
E. (10, 7, 7)
Choose the correct answer from the options given below:

The general solution of the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\) is :
(Where \(C\) is constant of integration)

Three capacitors of capacitances 2 \(\mu F\), 3 \(\mu F\) and 6 \(\mu F\) are connected in series. The combination is connected to a 12 V supply. The respective potential differences across them will be

DNA recombinant technology involves the use of restriction endonucleases. After having cloned the gene of interest and having optimised the conditions to induce the expression of the target protein, one has to consider producing it on a large scale. The cells harbouring cloned genes of interest may be grown on a small scale in the laboratory. In order to link the alien DNA, the vector needs to have very few, preferably single, recognition sites for the commonly used restriction enzymes. Presence of more than one recognition sites within the vector will generate several fragments, which will complicate the gene cloning. In Biolistics or gene gun method cells are bombarded with high velocity micro particles of gold or tungsten coated with DNA. The ligation of alien DNA is carried out at a restriction site present in one of the two antibiotic resistance genes. For example, you can ligate a foreign DNA at the BamH I site of tetracycline resistance gene in the vector pBR322. Since the DNA is enclosed within the membranes, we have to break the cell open to release DNA along with other macromolecules such as RNA, proteins, polysaccharides and also lipids. The RNA can be removed by treatment with ribonuclease whereas proteins can be removed by treatment with protease. Other molecules can be removed by appropriate treatments and purified DNA ultimately precipitates out after the addition of chilled ethanol.
If any protein encoding gene is expressed in a heterologous host, it is called as

Aqueous solutions from the following with highest boiling point will be : (Assume no dissociation or association)

The order and degree of the differential equation
\(x^2 \frac{d^2y}{dx^2} - 3 \left(\frac{dy}{dx}\right)^2 + y = 0\) are:

The value of the determinant \(\left|\begin{array}{ccc}a+b & b+c & c+a \\ c & a & b \\ 1 & 1 & 1\end{array}\right|\) is :

Consider the figure as shown below. If work done in bringing a point charge from infinity to points A, B and C are \( W_A, W_B \) and \( W_C \) respectively, then, which of the following relation is correct?