If \(A\) is any event associated with sample space and If \(E_1, E_2, E_3\) are mutually exclusive and exhaustive events. Then which of the following are true?
(A) \(P(A)=P\left(E_1\right) P\left(E_1 \mid A\right)+P\left(E_2\right) P\left(E_2 \mid A\right)+P\left(E_3\right) P\left(E_3 \mid A\right)\)
(B) \(P(A)=P\left(A \mid E_1\right) P\left(E_1\right)+P\left(A \mid E_2\right) P\left(E_2\right)+P\left(A \mid E_3\right) P\left(E_3\right)\)
(C) \(P\left(E_i \mid A\right)=\frac{P\left(A \mid E_i\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(A \mid E_i\right) P\left(E_i\right)}, i=1,2,3\)
(D) \(P\left(A \mid E_i\right)=\frac{P\left(E_i \mid A\right) P\left(E_i\right)}{\sum_{i=1}^3 P\left(E_i \mid A\right) P\left(E_i\right)}, i=1,2,3\)
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The sum of order and degree of the differential equation \(y=x \frac{d y}{d x}+2 \sqrt{1+\left(\frac{d y}{d x}\right)^2}\) is

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

The real valued function \(f(x)=x^{15}+5 x^9+10\) is increasing for __________ .

In a survey question for a sample of 250 individuals, 120 persons gave response 'yes', 80 persons gave response 'no' and 50 gave 'no response'. The point estimate of the proportion in the population who responded 'yes' is:

If \(y=5 e^{2 x}+4 e^{3 x}\), then \(\frac{d^2 y}{d x^2}\) equals:

On differentiating \(e^{x^2}\) with respect to x, we get

When a ray of light enters from denser medium to rarer medium, its velocity is doubled. The critical angle for total internal reflection will be :

Answer the question on basis of passage given below :
Alcohols and phenol are formed when a hydrogen atom in a hydrocarbon, aliphatic and aromatic respectively, is replaced by -OH group. The substitution of a hydrogen atom in a hydrocarbon by an alkoxy or aryloxy group forms ethers.
Alcohols and phenol consist of two part, an alkyl/aryl group and a hydroxyl group. The properties of alcohols and phenols are primarily due to the hydroxyl group. The nature of alkyl and aryl groups simply modify these properties. Alcohol react both as nucleophiles and electrophiles whereas in phenols, the reactions that take place on the aromatic ring are electrophilic subtitution reactions.
Few compounds are given below (A to D). Arrange them in the increasing
order of their boiling points.
(A) Pentan-1-ol
(B) n-Butane
(C) Pentan-1-al
(D) Ethoxyethane
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If \(f(x)=x+\frac{1}{x}\). Which of the following is not a correct option?

Mercury is the only liquid metal at room temperature due to:

Identify the correct statements from the following in the context of electromagnetic waves.
(A) X-rays are not electromagnetic waves.
(B) Electromagnetic waves are transverse in nature.
(C) Electromagnetic waves can not be polarised.
(D) Electromagnetic waves travel with the speed of light in vacuum.
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Which of the following statements are correct?
(A) A nucleus of mass number A has a radius R given by the expression \( R = R_0 A^{1/3} \)
(B) Volume of nucleus is proportional to mass number A
(C) The density of nucleus increases with the radius of nucleus.
(D) Density of nuclear matter does not depend on its mass number A
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