The value of the determinant \(\left|\begin{array}{cc}a \cos \theta & a \sin \theta \\ -a \sin \theta & a \cos \theta\end{array}\right|\) is:

Match List I with List II

LIST- I	LIST II
A. \(l x+m y+n z=d\) is	I. Equation of plane passing through a given point and normal to given vector
B. \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) is	II. Equation of plane in normal form
C. \((\vec{r}-\vec{a}) \cdot \vec{n}=0\)	III. Plane passing through the intersection of two planes
\(D .\left(a_1 x+b_1 y+c_1 z+d_1\right)+\lambda\left(a_2 x+b_2 y+c_2 z+d_2\right)=0\)	IV. Intercept form of plane

Choose the correct answer from the options given below :

\(\int \frac{f^{\prime}(x)}{f(x) \log _e[f(x)]} d x\) is equal to

\(y=x(x-3)^2\) decreases for the value of \(x\) given by

The probabilities of occurrence of two events \(E\) and \(F\) are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14 . The probability that neither \(E\) nor \(F\) occurs is

The mean of the number of heads in a simultaneous toss of three coins is :

If matrix \(A=\left[\begin{array}{ll}3 & x \\ y & 0\end{array}\right]\) and \(A^{\prime}=A\), then :

Azeotropes are binary mixtures having

For what value of \(k\), the following system of equations has infinitely many solutions?
\(x+2 y=5\), \(3 x+k y=15\)

Reaction of benzaldehyde with acetophenone in the presence of methanolic NaOH yields a major stable product as

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.
Which one of the following enzymes works as molecular glue?

Conditions for obtaining a interference pattern of light are
(A) Two waves should be coherent
(B) The distance of the screen from the slits should be small
(C) The separation between slits should be small
(D) Two coherent waves should have nearly the same amplitude
Choose the correct answer from the options given below:

The value of \(a+b+c+d\) if \(\left[\begin{array}{ll}2 a+b & a+2 b \\ 2 c+d & c+2 d\end{array}\right]=\left[\begin{array}{ll}8 & 4 \\ 3 & 3\end{array}\right]\) is :