The relation R in the set \(\mathbb{R}\) of real numbers defined as \(R = \{(a, b) : a \le b^2\}\) is:

Which of the following are examples of irregular trends in a time series?
(A) Decrease in production due to a sudden strike.
(B) The rise in prices before festivals.
(C) Unusual rise in income of the printing press due to the announcement of an election.
(D) Fall in crop yield due to floods.
Choose the correct answer from the options given below :

The area (in sq. units) of the region bounded by the curve \(y=\sqrt{16-x^2}\) and x-axis is

Which of the following are linear first order differential equations?
(A) \(\frac{d y}{d x}+P(x) y=Q(x)\)
(B) \(\frac{d x}{d y}+P(y) x=Q(y)\)
(C) \((x-y) \frac{d y}{d x}=x+2 y\)
(D) \(\left(1+x^2\right) \frac{d y}{d x}+2 x y=2\)
Choose the correct answer from the options given below:

If it is 7:00 pm currently in the clock, what will the clock show (in am or pm) after 674 hours?

If \(\vec{a}\) is any vector, then \(|\vec{a} \times \hat{i}|^2+|\vec{a} \times \hat{j}|^2+|\vec{a} \times \hat{k}|^2\) is equal to

500 g of jaggery syrup has 30% jaggery in it.
The quantity of jaggery that should be added to make it 50% of syrup is:

Two pipes A and B can fill a tank respectively in 30 min and 45 min.
Both A and B are opened together for some time and then pipe B is turned off.
If the tank is completely filled in 20 min, find after how many minutes pipe B is turned off.

The shortest distance between the lines \(\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})\) and \(\vec{r}=(2 \hat{i}+4 \hat{j}+5 \hat{k})+\mu(3 \hat{i}+4 \hat{j}+5 \hat{k})\) is equal to :

It is given that mean and variance of a binomial variate X are 2 and 1 respectively. Find the probability that X takes value greater than or equal to 1.

Read the following passage carefully and answer the given questions.
Plasmodium, a tiny protozoan is responsible for the disease malaria. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal. Innate immunity consist of four types of barriers. These are – (i) Physical barriers- skin on our body is the main barrier which prevents entry of the micro-organisms. (ii) Physiological barriers- Acid in the stomach, saliva in the mouth, tears from eyes-all prevent microbial growth. (iii) Cellular barriers- Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. (iv) Cytokine barriers- virus-infected cells secrete proteins called interferons which protect non-infected cells from further viral infection. Allergy is due to the release of chemicals like histamine and serotonin from the mast cells. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Drugs like barbiturates, amphetamines, benzodiazepines, and other similar drugs, that are normally used as medicines to help patients cope with mental illnesses like depression and insomnia, are often abused.
Which one of the following drugs is not used as medicine to help patients cope with mental illnesses like depression and insomnia?

Read the passage carefully and answer
Carbohydrates are essential for both plants and animals. Some common examples of carbohydrates are glucose, fructose, sucrose, glycogen, cellulose and starch. They form a major portion of our food. Honey has been used for a long time as an instant source of energy in Ayurvedic medicine. Carbohydrates are used as storage molecules in both plants and animals. The cell wall of bacteria and plants is made of cellulose. We use cellulose in the form of wood for making furniture and as cotton fiber for clothing. They also provide raw material for textiles, paper, lacquers, and breweries. Carbohydrates are classified as Monosaccharides, Oligosaccharides and Polysaccharides on the basis of their behavior towards hydrolysis.
Monosaccharides cannot be hydrolyzed further to simpler units whereas oligosaccharides can yield 2-10 units of monosaccharides on hydrolysis. Polysaccharides, which are non-sugars yield a large number of monosaccharide units on hydrolysis.
Answer the following questions about carbohydrates:
Identify a component of starch from among the following.

Read the following paragraph carefully and answer the given questions.
Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. For determining the cause of allergy, the patient is exposed to or injected with very small doses of possible allergens, and the reactions studied. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. This is obtained by acetylation of morphine. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal.
Which one of the following is not the cellular barrier of innate immunity?