A boat covers a distance of 24 km upstream and returns to the same point in a total time of 4 hours. If the speed of the boat in downstream is twice its speed in upstream, then the speed of boat in upstream is:

The total cost of producing \(x\) generators is given by
\(TC =x^3-60 x^2+1500 x+2000\)
The Marginal Cost \(( MC )\), when \(x=10\) units is:

The principle value of \(\cos ^{-1}\left(-\frac{1}{2}\right)\) is :

If \(x=-1\) and \(x=-2\) are the extreme points of \(f(x)=\alpha \log |x|+\beta x^2+x\) then

For finding absolute maximum and minimum values of a function \(f\) given by
\(f(x)=2 x^3-15 x^2+36 x+1 \text { on }[1,5]\)
A. the critical points in \((1,5)\) are 2 and 3 .
B. the absolute maximum value of \(f\) on \([1,5]\) is 29 .
C. the absolute minimum value of \(f\) on \([1,5]\) is 24 .
D. the absolute minimum value of \(f\) on \([1,5]\) is 20 .
E. we evaluate the value of \(f\) at critical points and at the end points of the interval \([1,5]\).
Choose the correct answer from the options given below:

The value of \(\tan ^{-1} 1+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)\) is

\(\int\left(\frac{1}{\log _e t}-\frac{1}{\left(\log _e t\right)^2}\right) d t\) is equal to

Match List - I with List - II.

List - I (lon)	List - II (Magnetic Moment (BM))
(A) \(V^{2+}\)	(1) 1.73
(A) \(Ni ^{2+}\)	(II) 3.87
(C) \(Cu ^{2+}\)	(III) 4.90
(D) \(Cr ^{2+}\)	(IV) 2.84

Choose the correct answer from the options given below :

Read the passage and answer the question : There are thousands of varieties of rice in India alone. The diversity of rice in India is one of the richest in the world. Basmati rice is distinct for its unique aroma and flavour and 27 documented varieties of Basmati are grown in India. There is reference to Basmati in ancient texts, folklore and poetry, as it has been grown for centuries. Several attempts have also been made to patent uses, products and processes based on Indian traditional herbal medicine, e.g. turmeric, neem. There has been growing realisation of the injustice, inadequate compensation and benefit sharing between developed and developing countries. The Indian Parliament has recently cleared a Bill that takes such issues into consideration, including patent terms emergency provisions and research and development initiative.
How many varieties of rice India has?

Answer the question on the basis of passage given below :
Nitrogen differs from rest of the members of group 15 due to its small size, high electronegativity, high ionization enthalpy and non-availability of d orbitals. Nitrogen has unique ability to form pa - pa multiple bonds with itself and with other elements having small size and high electronegativity. Heavier elements of this group do not form pa - pr bonds as their atomic orbitals are so large and diffuse that they cannot have effective overlapping.
Group 15 elements forms oxides of type \(E _2 O _3\). Acidic oxides are formed by
A. Bi
B. Sb
C. P
D. As
E. N
Choose the correct answer from the options given below:

Two point charges + 200 \( \mu \)C are placed at a distance of 2 mm from each other in a uniform electric field of \( 10^5 \text{ N/C} \), such that the line joining the negative to positive charge makes an angle 30° with the field. The torque acting on the system is:

Which one of the following is the cathodic reaction in a fuel cell?

If \(f(x)=\left\{\begin{array}{ll}a x-1 & \text { if } x \geq 1 \\ 2 x+1 & \text { if } x<1\end{array}\right.\) is continuous at x = 1, then a equals