Let \(P=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix and let \(Q=\left[b_{i j}\right]\) where \(b_{i j}=2^{i+j} a_{i j}, \forall 1 \leq i, j \leq 3\). If the determinant of \(P\) is 2 , then the determinant of \(Q\) is:

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

For a \(3 \times 3\) matrix \(A\), if \(A(\operatorname{adj} A)=\left[\begin{array}{ccc}99 & 0 & 0 \\ 0 & 99 & 0 \\ 0 & 0 & 99\end{array}\right]\) then \(\operatorname{det}(A)\) is equal to:

The value of the determinant \(\left|\begin{array}{ccc}a \cos \theta & b \sin \theta & 0 \\ -b \sin \theta & a \cos \theta & 0 \\ 0 & 0 & c\end{array}\right|\) is :

Irregular variations in a time series are not caused by:

Let \(A=\left[\begin{array}{ccc}0 & 2 & -5 \\ -2 & 0 & -7 \\ 5 & 7 & 0\end{array}\right]\) and \(B=\left[\begin{array}{ccc}1 & 5 & 8 \\ 2 & 7 & 4 \\ 3 & 9 & 10\end{array}\right]\) then |AB|=

The window of the house is in the form of a rectangle surmounted by a semi-circular portion, having a perimeter of 10 m as shown in figure.

If \(x\) and \(y\) represent the length and breadth of rectangular region, then the area A of the window in terms of \(x\) is :

match List - I with List - II

List - I	List - II
(A) Equations of line through (5, -4, 6) with direction ratios 3, 7, 2	(I) \(\frac{x+3}{5}=\frac{y+7}{-4}=\frac{z+2}{6}\)
(B) Equations of line through (3, 7, 2) with direction ratios 5, -4, 6	(II) \(\frac{x-3}{5}=\frac{y-7}{-4}=\frac{z-2}{6}\)
(C) Equations of line through (-5, 4, -6) with direction ratios 3, 7, 2	(III) \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\)
(D) Equations of line through (-3, -7, -2) with direction ratios 5, -4, 6	(IV) \(\frac{x+5}{3}=\frac{y-4}{7}=\frac{z+6}{2}\)

Choose the correct answert from the option given below :

Let \(y(x)=a(x+1) \log (x+1)+b x+5\) be the solution of the differential equation \(e^{\frac{d y}{d x}}=x+1, y(0)=5\), then the value of \((a+b)\) is :

Read the following passage carefully and answer the given questions.
Some strains of Bacillus thuringiensis produce proteins that kill certain insects such as lepidopterans (tobacco budworm, armyworm), coleopterans (beetles) and dipterans (flies, mosquitoes). B. thuringiensis forms protein crystals during a particular phase of their growth. Transgenic animals that produce useful biological products can be created by the introduction of the portion of DNA (or genes) which codes for a particular product such as human protein (a-1-antitrypsin) used to treat emphysema. In 1997, the first transgenic cow, Rosie, produced human protein-enriched milk (2.4 grams per litre). The milk contained the human alpha-lactalbumin and was nutritionally a more balanced product for human babies than natural cow-milk. A nematode Meloidegyne incognitia infects the roots of tobacco plants and causes a great reduction in yield. A novel strategy was adopted to prevent this infestation which was based on the process of RNA interference (RNAi). This method involves silencing of a specific mRNA due to a complementary dsRNA molecule that binds to and prevents translation of the mRNA (silencing). Biopiracy is the term used to refer to the use of bio-resources by multinational companies and other organisations without proper authorisation from the countries and people concerned without compensatory payment. ELISA is based on the principle of antigen-antibody interaction. Infection by pathogen can be detected by the presence of antigens (proteins, glycoproteins, etc.) or by detecting the antibodies synthesised against the pathogen.
mRNA silencing through RNA interference (RNAi) refers to the:

Ethyl bromide is converted to succinic acid by using the following reagents in steps:
 A. \(KCN\)
B. \(Br_2\)
C. \(KOH\ (alc.)\)
D. \(H_2O/H^+\)
Choose the correct sequence of conversion:

The value of the definite integral \(I=\int_{-1}^1 \frac{1}{1+\sqrt{e^x}} d x\) is :

The IUPAC name of the given compound is