If the function \(f(x)=\left\{\begin{array}{ll}\frac{\sin 5 x}{3 x}, & x \neq 0 \\ \frac{k}{3}, & x=0\end{array}\right.\) is continuous at \(x=0\), then \(k^2-2 k+10\) is equal to:

The minimum value of the function \(f(x)=x^3+(10-x)^3\) occurs at :

\(\int \frac{1}{x\left(x^5-1\right)} d x\) is equal to

The general solution of the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\)

If the matrix \(M=\left[\begin{array}{ccc}0 & -1 & 3 \alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0\end{array}\right]\) is skew-symmetric, then

The probability distribution of a random variable \(x\) is, \(P(x)=\frac{k}{2^x}, x=0,1,2,3\). Then
Match List-I with List-II

List-I	List-II
(A) k	(I) \(\frac{2}{15}\)
(B) \(P ( x =1)\)	(II) \(\frac{1}{5}\)
(C) \(P (1<x<3)\)	(III) \(\frac{8}{15}\)
(D) \(P(x \geq 2)\)	(IV) \(\frac{4}{15}\)

Choose the correct answer from the options given below :

The area of the region bounded by the curve \(y^2=x\) and the lines \(x=1, x=4\) is:

Read the passage below and answer the question:
The accumulation of molecular species at the surface rather than in the bulk of a solid or liquid is termed as adsorption. There are two types of adsorption. In physisorption, attractive forces are mainly Van der Waals while in chemical adsorption adsorbate is held with chemical bonds adsorbent. Adsorption increases with increase in pressure and decreases as temperature is increased.
Which of the following is not characteristic of chemical adsorption?

Arrange the following in increasing order of their basic strength in aqueous phase :
(A) \(CH_{3}NH_{2}\)
(B) \(C_{6}H_{5}CH_{2}NH_{2}\)
(C) \(C_{6}H_{5}NH_{2}\)
(D) \((CH_{3})_{3}N\)
Choose the correct answer from the options given below :

Electromagnetic wave travels with speed of \(2.0 \times 10^8 m / s\) in any medium of relative permeability ( \(\mu_r\) ) 1.5. Find the relative permittivity of the medium ( \(\epsilon_r\) )

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in 'Swiss cheese' are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
Propionibacterium sharmanii is used for the production of -

The correct order for increasing the boiling point of hydrides (given below) of group 15 is:
(A) \(NH _3\)
(B) \(PH _3\)
(C) \(AsH _3\)
(D) \(SbH _3\)
(E) \(BiH _3\)
Choose the correct answer from the options given below :

A cylindrical drum of radius 7 cm and height 2 m is being kept in a vertical position filled with milk. If the milk is leaking at \(14 cm^3 / sec\) from its lower base, then the rate of decrease in the level of milk is: [Take \(\pi=\frac{22}{7}\) ]