The condition on \(a\) and \(b\), such that for \(y=\frac{a}{x}-\frac{b}{x^2}, \frac{d y}{d x}=0\) at \(x=1\) is :

Let \(X\) denotes the number of heads in a simultaneous toss of three coins, then \(P(0< X< 3)\) is

Which of the following are correct?
(A) A statistical hypothesis is a statement about the population parameter which may be true or false.
(B) In a statistical hypothesis testing procedure, a Type I error is made when the null hypothesis is accepted when it is false.
(C) In a statistical hypothesis testing procedure, a Type II error is made when the null hypothesis is rejected when it is true.
(D) The number of degrees of freedom of a statistic is the number of independent variates used to compute that statistic.
Choose the correct answer from the options given below :

\(\frac{d^2}{d x^2}\left(\operatorname{det}\left[\begin{array}{cc}x^3 & x \\ 2 & e^x\end{array}\right]\right)\) equals

The area (in sq. units) of the region enclosed by the curve \(9 x^2+4 y^2=36\) is

\(\int_0^{\pi / 2} \frac{\sin ^8 x}{\sin ^8 x+\cos ^8 x} d x\) is equal to

A machine has been producing steel tubes of mean inner diameter of 2 cm.
A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm.
Test the hypothesis that the machine is working in the proper order at 5% level of significance and answer, which of the following statements are correct?
[Given that \(t_{0.05}=2.262\)]
(A) The value of the test statistic is t = 0.476
(B) The null hypothesis is rejected at \(5 \%\) level of significance.
(C) The null hypothesis is accepted at \(5 \%\) level of significance.
(D) The degree of freedom is 10
Choose the correct answer from the options given below:

Which of the following reactions has a positive \(\Delta S^{\circ}\) value?

The refractive indices of glass and water are 3/2 and 4/3, respectively. If light passes from glass to water, the critical angle is

If \(0<2 x<1\), which of the following is smallest?

Read the passage carefully and answer the questions given along
Our immune system plays the major role in preventing diseases when we are exposed to disease-causing agents. Immunity is of two types: (i) Innate immunity and (ii) Acquired immunity. The human immune system consists of lymphoid organs, tissues, cells and soluble molecules like antibodies. The immune system also plays an important role in allergic reactions, auto-immune diseases and organ transplantation. The primary lymphoid organs are bone marrow and thymus. The secondary lymphoid organs are spleen, lymph nodes, tonsils, Peyer's patches of small intestine and appendix. There is lymphoid tissue also located within the lining of the major tracts (respiratory, digestive and urogenital tracts) called mucosa associated lymphoid tissue (MALT). It is important to note that HIV/AIDS is not spread by mere touch or physical contact; it spreads only through body fluids. AIDS has no cure, prevention is the best option. Adolescence is accompanied by several biological and behavioural changes and surveys and statistics show that use of drugs and alcohol has been on the rise especially among the youth. Factors associated with drug and alcohol abuse among adolescents are a child's natural curiosity, stress, the perception among youth that it is 'cool' or progressive to smoke, media exposure and an unstable or unsupportive family structures and peer pressure. It is best to identify the situations that may push an adolescent towards use of drugs or alcohol, and to take remedial measures well in time.
Which one of the following structures is not an example of a cellular barrier in innate immunity?

Assisted Reproductive Technology (ART) does NOT include:

If we double the number of turns in a moving coil galvanometer, the voltage sensitivity of the galvanometer