Maximum value of the objective function Z = 4x + 3y subject to the constraints \(x+y \geq 4, x \geq 4, y \geq 0\) exists at

\(\int\left(x+\frac{1}{x}\right)^2 d x\) equals :

Pipe A can independently empty a half-filled tank in 3 hours whereas pipe B can alone fill the same tank in 4 hours, when the tank is completely empty. In how much time.

The total cost \(C(x)\) associated with the production of \(x\) units of an item is given by \( C(x)=0.001 x^3+0.06 x^2+20 x+500\)
The marginal cost when 10 units are produced is:

Match List - I with List - II

List - I (Example)	List - II (Component of Time Series)
(A) Lockouts and strikes	(I) Secular trend
(B) Rise and fall of share market	(II) Seasonal trend
(C) Continuous decline in death rate	(III) Irregular trend
(D) The rise in prices before Diwali	(IV) Cyclical trend

Choose the correct answer from the options given below:

For the function \(f(x)=x^{1 / x}, x>0\), which of the following are correct?
(A) \(x=0\) is the only point where extremum may occur.
(B) The given function is maximum at \(x=e\).
(C) The function has no extreme value for \(x>0\).
(D) The maximum value of the function \(f(x)\) is \(e^{1 / e}\)
Choose the correct answer from the options given below :

If lines \(\frac{x+5}{5 \lambda+2}=\frac{4-2 y}{10}=\frac{1-3 z}{-3}\) and \(\frac{x-2}{1}=\frac{1+2 y}{4 \lambda}=\frac{2+z}{3}\) are perpendicular, then value of ' \(\lambda\) ' is

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.
Malignant malaria is caused by _____ is the most serious one and can even be fatal.

The total energy of the electron in a hydrogen atom is \( -1.51 \text{ eV} \).
A. Its Kinetic energy is \( -1.51 \text{ eV} \)
B. Its potential energy is \( -3.02 \text{ eV} \)
C. electron is in second excited state
D. The ground state energy of hydrogen atom is 13.6 eV
Choose the correct answer from the options given below:

The probability that a plant \(A\) survives is \(\frac{3}{4}\) and the probability that another plant \(B\) survives is \(\frac{1}{3}\).
The probability that only one of them survives is :

The first isolated restriction endonuclease was:

If \(A=\left[\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0\end{array}\right]\), then the value of \(\operatorname{det}(\operatorname{adj}(2 A))\) is :

Verhulst-Pearl logistic growth equation can be represented by -