For two events A and B. Match List I with List II

LIST-I	LIST-II
A. \(P(\bar{A} \cap \bar{B})=P(\bar{A}) \cdot P(\bar{B})\)	I. P(A/B) ≥ P(A)
B. \(A \subset B\) and \(P(B) \neq 0\)	II. P(A) = P(B)
C. P(AUB) = P(A) + P(B)	III. A and B are independent events
D. P(A/B) = P(B|A)	IV. A and B are mutually exclusive events

Choose the correct answer from the options given below:

A pair of dice is thrown 7 times. If getting a total of 7 is considered a success, the probability of exactly 6 successe, is:

For independent events \( A_{1} , A_{2}, A_{3} , \ldots, A_{n}\) , if \(P(A_{i}\) \(= \frac{1}{i + 1}, \; i = 1, 2, 3 \ldots, n\), then the probability that none of the events occur is :

The general solution of the differential equation \(\log _e\left(\frac{d y}{d x}\right)=a x+b y\) is

If the sum and difference of squares of mean and variance of a Binomial distribution is \(\frac{225}{256}\) and \(\frac{63}{256}\) respectively, the \(P(X \geq 2)\) is :

Match List-l with List-ll

List-l (Mathematical Statement)	List-ll (Value)
(A) \(\hat{i} \cdot(\hat{j} \times \hat{k})\)	(I) \(-\hat{k}\)
(B) \(\hat{j} \cdot(\hat{i} \times \hat{k})\)	(II) 1
(C) \(\hat{i} \times(\hat{j} \times \hat{k})\)	(III) \(-1\)
(D) \(\hat{j} \times \hat{i}\)	(IV) \(\overrightarrow{0}\)

Choose the correct answer from the options given below :

The function \(f(x)=-\frac{3}{4} x^4-8 x^3-\frac{45}{2} x^2+163\) has a local maxima at

The area enclosed by the curve \(x=8 \cos t, y=4 \sin t\) between the lines \(x=0\) and \(x=4 \sqrt{3}\) is:

Match List-I with List-II

List-I (Contraceptive methods)	List-II (Examples)
(A) Non-medicated IUD	(I) LNG-20
(B) Hormone releasing IUD	(II) Multiload 375
(C) Copper releasing IUD	(III) Vaults
(D) Barrier method	(IV) Lippes loop

Choose the correct answer from the options given below:

In Rutherford's scattering experiment, if impact parameter \( b = 0 \), then the value of the scattering angle will be :

Two charges \( q_1 = 4 \times 10^{-7} \text{ C} \) and \( q_2 = -6 \times 10^{-7} \text{ C} \) are located 3 m apart. At what point on the line joining the two charges is the electric potential zero? (Consider potential zero at infinity.)

Which one of the following statements is correct with reference to the solubility of gases in liquid?

Read the passage carefully and answer the questions
Carbohydrates are naturally formed organic compounds from various plants. The common chemical formula, \(C _x\left( H _2 O \right)_y\), was considered as hydrates of carbon to derive the various carbohydrates. For instance, the general formula for glucose ( \(C _6 H _{12} O _6\) ) easily fits into the above formula. Carbohydrates also called saccharides are classified on the basis of their behavior towards hydrolysis. Monosaccharides cannot be hydrolyzed further to simpler units of polyhydroxy aldehyde or ketones. About 20 monosaccharides are known to occur in nature. Carbohydrates that yield large number of monosaccharide units are called polysaccharides. Amino acids contain ( \(- NH _2\) ) and ( - COOH ), which can be classified in various forms. Proteins are polymers of alpha amino acids which are connected to each other by a peptide bond or peptide linkage. When a native protein is subjected to physical change, its globules unfold and helix get uncoiled and the protein loses its biological activity. The protein is said to be denaturated.
The peptide linkage formed between the reaction of glycine and alanine is represented as