Which one of the following set of constraints does the given shaded region represent?

Consider the following hypothesis test:
\(\begin{aligned} & H_0: \mu=20 \\ & H_a: \mu \neq 20\end{aligned}\)
A sample of 40 provided a sample mean of 19. The standard deviation is 3. Then the value of the t-test statistic is:

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

A fair coin is tossed 100 times
Find the probability of getting head an odd number of times

Consider the following statements with respect to Probability distribution:
(A) In a given sample space not more than four random variables can be defined
(B) Mean expectation value is a parameter and not a statistic
(C) In a Gaussain distribution if only mean and mode are same it is said to be normally distributed
(D) Probability of 'r' success in 'n' Bernoulli is P(r) = \({ }^n C_r q^r p^{n-r}\)
(E) Variance enables us to study the variability of a random variable from the mean expectation

Choose the correct answer from the options given below:

For the objective function \(Z = 3x + 5y\) subject to constraints \(x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0\):

\(\int_1^4|x-2| d x\) is equal to:

Read the passage carefully and answer the Questions
Haloalkanes undergo nucelophilic substitution reactions owing to the polarity of CX bond. The nucleophile reacts with the haloalkane on the carbon possessing a partial positive charge holding the halogen atom. The halogen atom X is replaced by a nucleophile. Depending on the kinetics and mode of bond breaking, the mechanism can be either SN1 or SN2 reaction. The rate of SN1 reaction is governed by the stability of carbocation and in SN2 reaction, the rate of reaction is governed by steric factor. Chirality is the main factor in both SN1and SN2. In SN1 reaction, the chirality of alkyl halide is accompanied by racemization of the product, while in SN2 reaction, the product is characterized by inversion of configuration. The structure of alkyl halide and the nature of the solvent also governs the mechanism of the substitution.
What is meant by the term racemic mixture?

\(\int \frac{\log _e x}{\left(1+\log _e x\right)^2} d x\)2 is equal to

The correct statements regarding order and molecularity of the reaction are:
(A) The order of a reaction can be zero or an integer.
(B) The molecularity of a reaction can be zero or an integer.
(C) The molecularity cannot be zero or fraction.
(D) The order of a reaction is decided by the slowest step of the complex reaction.
Choose the correct answer from the options given below:

If \(x\) is real, the minimum value of \(f(x)=x^2-8 x+20\) is :

In a double slit experiment, if the distance between the slits be reduced to half, then the new fringe width will be

The area bounded by \(x = 1, x = 2, xy = 1\) and x-axis is :