The rate of change of the area of an equilateral triangle with respect to its side when its side \(=2\) is :

Two percent of the bolts manufactured in a factory are found to be defective. Using the Poisson distribution, the probability that in a sample of 100 bolts chosen at random, exactly two will be defective, is : [Given \(e^{-2}=0.135\)]

The value/s of \(\lambda\) for which the given system of equations will have infinitely many solutions \(x+y-6 z=0 ; \lambda x-y+2 z=0 ;-3 x+\lambda y+2 z=0\), is/are:

The solution of the differential equation \(\cos x \sin y dx + \sin x \cos y dy = 0\) is:

Let \(y=\cos \left(\sin x^2\right)\), then the value of \(\frac{d y}{d x}\) at \(x=\frac{\sqrt{\pi}}{2}\) is equal to

The function \(f(x)=\left\{\begin{array}{cl}\frac{x^2+2 x-3}{x-1} & , x \neq 1 \\ 2 & , x=1\end{array}\right.\) is :

If \(r_{e f f}=\) effective rate of interest, \(r=\) nominal rate of interest and \(m=\) number of conversion periods per year, the relationship between nominal rate and effective rate of interest is :

Mr. 'X' wishes to purchase a house for ₹ \(49,65,000\) with a down payment of ₹ \(15,00,000\) and balance amount in EMI for 25 years. If bank charges \(6 \%\) per annum compounded monthly. Then the EMI is: [Given that \((1.005)^{300}=4.4650\) ]

Read the following paragraph and answer the given questions.
RNAi takes place in all eukaryotic organisms as a method of cellular defense. The source of this complementary RNA could be from an infection by viruses having RNA genomes or mobile genetic elements (transposons) that replicate via an RNA intermediate. Insulin from an animal source, though caused some patients to develop allergy or other types of reactions to the foreign protein. Insulin consists of two short polypeptide chains: chain A and chain B, that are linked together by disulphide bridges. Transgenic mice are being developed for use in testing the safety of vaccines before they are used on humans. Transgenic mice are being used to test the safety of the polio vaccine. If successful and found to be reliable, they could replace the use of monkeys to test the safety of batches of the vaccine. Specific Bt toxin genes were isolated from Bacillus thuringiensis and incorporated into the several crop plants such as cotton. The choice of genes depends upon the crop and the targeted pest, as most Bt toxins are insect-group specific. The toxin is coded by a gene cryIAc named cry. There are a number of them, for example, the proteins encoded by the genes cryIAc and cryIIAb control the cotton bollworms, that of cryIAb controls corn borer.

The Bt toxin gene cryIAb controls the:

Identify the correct statement(s).
(A) In any lens, the optical centre is always at the centre of the lens.
(B) In a lens, if the curvature of its two surfaces is equal, the optical centre is at the centre of the lens.
(C) A light ray passing through the optical centre of a lens never shifts from the direction of the incident ray.
(D) A ray of light incident perpendicularly on a lens passes through its focus after refraction.
Choose the correct answer from the following given options.

Darwin's finches and Australian marsupials are an example of:

The function \(f(x)=\left\{\begin{array}{ll}\frac{|x|}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.\) is:

The amine which will not undergo benzoylation in alkaline medium is