The tangent to the curve \(y=x^3-11 x+5\) at \(x=2\) is parallel to :
(A) \( y=x+1\)
(B) \(y=-x+2\)
(C) \(2 y=x+3\)
(D) \(2 y=2 x+4\)
Choose the correct answer from the options given below :

A man rows 24 km upstream in 6 hours and 36 km downstream in 6 hours each time, then the speed of the boat in the still water is

A boat covers 32 km upstream and 36 km downstream in 7 hours.
Also, it covers 40 km upstream and 48 km downstream in 9 hours.
The speed of the boat in still water is :

Consider the function \(f: N \rightarrow N\) given by \(f(x)=\left\{\begin{array}{ll}x+1, & \text { if } x \text { is odd } \\ x-1, & \text { if } x \text { is even }\end{array}\right.\)

Which of the following statements are true?
(A) Degree of the differential equation \(\frac{d^2 y}{d x^2}+y=1\) is 2.
(B) Number of arbitrary constants in the particular solution the differential equation of order 4 are 4.
(C) \(y^2=a\left(b^2-x^2\right)\) will be the general solution of a differential equation of order 2; where a and b are two arbitrary constants.
(D) \(f(x, y)=\frac{2 x-y}{x+3 y}\) is a homogeneous function of degree 1.
(E) Integrating factor of \(x \frac{d y}{d x}-y=2 x^2\) is \(\frac{1}{x}\).
Choose the correct answer from the options given below:

If corner points of a feasible region are \((0,0),(2,0),\left(\frac{20}{19}, \frac{45}{19}\right)\) and \((0,3)\), then
(A) Maximum value of \(z=5 x+3 y\) is 10
(B) Minimum value of \(z=5 x+3 y\) is 0
(C) Maximum value of \(z=5 x+3 y\) is \(\frac{235}{19}\) and minimum value is 0
(D) Maximum value of \(z=5 x+3 y\) is 10 and minimum value is 0
Choose the correct answer from the options given below :

In a box containing 100 bulbs, 10 are defective.
The probability that out of a sample of 5 bulbs none is defective is:

If \(\hat{a}, \hat{b}\) and \(\hat{c}\) are three units vectors and \(\hat{a}+\hat{b}+\hat{c}=\overrightarrow{0}\), then the angle between \(\hat{a}\) and ( \(-\hat{b}\) ) is

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 of the following technique is not very useful to detect the cancers of internal organs?

Electric field at a point due to a uniformly charged infinite plane sheet is

Identify the basic features of the photon picture of electromagnetic radiation from the following:
(A) Each photon has energy \(E = h\nu\) in vacuum.
(B) Photons are electrically neutral and are not deflected by electric and magnetic fields.
(C) All photons of light of a particular frequency \(\nu\) or wavelength \(\lambda\) have the same energy and momentum.
(D) If the intensity of light of a given wavelength is increased, there is an increase in the number of photons incident on a given area in a given time.
Choose the correct answer from the options given below:

When \(Br_2\) is treated with aqueous solution of \(NaF, NaCl, NaI\) separately:

Which of the following equation correctly describes 'Net primary productivity'?