The maximum slope of the curve \(y=-x^3+3 x^2+9 x-27\) is:

The slope of the normal to the curve \(y=x^3-4 \sin x\) at \(x=0\) is:

Let \(\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\). Then \(y\) is :

Ajesh has set up a sinking fund in order to have ₹ 10,00,000 after 10 years for his son's education. The amount should be set aside at the end of every 6 months into an account paying \(5\%\) per annum compounded half yearly is : [given \((1.025)^{20}=1.6386\) ]

\(y=e^{x+e^{\infty+e^{x+\cdots \infty}}}\) then \(\frac{d y}{d x}=\) ______________

The number of all possible matrices of order \(3 \times 4\) with each entry 1, 2 or 3 are:

Match List - I with List - II. Given \(P(A)=\frac{1}{3}\) and \(P(B)=\frac{1}{5}\) where \(A\) and \(B\) are independent.

LIST I	LIST II
A. \(P(A \cap B)\)	I. \(\frac{7}{15}\)
B. \(P \left( A ^{\prime} \cap B ^{\prime}\right)\)	II. \(\frac{2}{5}\)
C. \(P\) (at least one of the two events takes place)	III. \(\frac{1}{15}\)
D. \(P\) (only one event takes place)	IV. \(\frac{8}{15}\)

Choose the correct answer from the options given below:

Read the passage carefully and give the answers to the question :
Does the number of species in a community really matter to the functioning of the ecosystem? This is a question for which ecologists have not been able to give a definitive answer. For many decades, ecologists believed that communities with more species, generally, tend to be more stable than those with less species. According to the International Union for Conservation of Nature and Natural Resources (IUCN) (2004), the total number of plant and animal species described so far is more than 1.5 million.
The following are the various hypotheses proposed in explaining the greatest biological diversity in tropics except :

When data of the variable is collected at distinct time intervals for a specified period of time, it is called

Arrange the following historical events in the history of life in chronological order?
(A) Natural selection
(B) Self-replicating molecules
(C) Eukaryotic cell
(D) Prokaryotic cell
Choose the correct answer from the options given below:

\(H_3 C-CH_2-CH_2-CH_2-CH=CH_2 \xrightarrow{HBr} A\)
A will be

Radiation of frequency \(2 v _0\) incident on a metal with threshold frequency \(v _0\) results in maximum kinetic energy of photoelectrons as :

An insulating rod of length \( L \) carries charge \( q \) distributed uniformly on it. The rod is pivoted at one end and is rotated at a frequency \( f \) about a fixed perpendicular axis. Then the magnetic dipole moment of the system is :