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correct option:

(A) 5/√6

Answer is (c) (1,3)

Answer is (b) A^-1 exists

Answer is (c) equivalence

Answer is (c) (x + 3)/2

Answer is (a) x(dy/dx) = y

(A) y = 4sin3x 

Answer is (a) 2

Correct option:

(B) 1/(x log log log x)

Correct option:

(A) 2 

Correct option:

(B) (x - 3)/2

Correct option is (D) Mean of observations

In the formula \(\overline X=\frac{\sum x_i}{n},\) the letter \(\overline X\) represents mean of the observation.

D) Mean of observations

Correct option:

(D) -π/4

Option : (b)

y = mx + 1 … . . (1) and 

y² = 4x … … (2) 

Substituting (1) in (2) : 

(mx + 1)² = 4x 

⇒ m²x² + (2m − 4)x + 1 = 0 … … . . (3) 

As line is tangent to the curve 

⇒ line touches the curve at only one point 

⇒ (2m − 4)² − 4m² = 0 

⇒ m = 1

Option : (c)

Let f(x) = [x(x − 1) + 1]^1/3, 0 ≤ x ≤ 1

f'(x) = \(\frac{2x-1}{3(x^2-x+1)^{\frac{2}{3}}}\)

Let f'(x) = 0

⇒ x = \(\frac{1}{2}\)∈ [0,1]

f(0) = 1, f(\(\frac{1}{2}\))

= \((\frac{3}{4})^\frac{1}{3}\) and f(1) = 1

∴ Maximum value of f(x) is 1.

(D) (x – y)(y – z) (z – x)

Correct option:

(A) 2 

Ascospores type of spores are produced sexually.

Enzymes responsible for alcoholic fermentation Zymase.

There are two equations provided here.

3x – 5y – 4 = 0

3x -5y = 4 ……. (1).

9x = 2y + 7

9x -2y = 7 ……(2).

We have to solve for x and y in the given equations by the elimination method

Now only considering equations (1) and (2)

3x – 5y = 4………(1)

9x – 2y = 7 ……(2)

On multiplying eq. (1) by 3, we get….

3 (3x – 5y = 4)

9x – 15y = 12…… (3).

Now we can easily subtract (2) and (3) to get…..

9x – 2y = 7

– 9x – 15y = 12.

– +13 y = -5

Or we get 13 y = -5 or y = -5/13.

Now we substitute the value of Y = -5/13 in equation (1) we get the value of x.

3x – 5(-5/13) = 4

3x +25/13 = 4

39x + 25 = 52

Or 39x = 52 – 25

39x = 27

Or x = 27/39 = 9 / 13

So x = 9/13 and y = -5/13.

x = 9/13 and y = -5/13 

Answer: False

Answer is (d) cot^-1 x + k

Test statistic is the statistic whose distribution test being conducted.

When expected frequencies are less than 5.

• Uniform demand 

• Production is Instantaneous 

• Lead time is Zero 

• Shortages are allowed.

Answer is (a) (1/a) tan^-1 (x/a) + k

When expected frequencies are below 5(E_i < 5). 

An Defect is quality characteristic which does not conform to specifications.

‘Interpolation’ is the technique of estimating a value of the dependent variable (y) for any intermediate value of the independent variables (x) ‘Extrapolation’ is the technique of estimating the value of dependent variable (y) for any value of the independent variable (x) which is outside the range of the given series.

1. Total number of frequency N should be large. 

2. Theoretical/Expected frequency each E_i should be 5 or more (E_i ≥ 5) if any one is below it should be pooled with the adjacent frequencies (untill together 5 or more) 

3. If any one of the parameter is estimated from the data for each such estimation one degrees of freedom should be lessened.

Because of non-negative restrictions both x and y are positive on x, y – plane.

Geometric mean.

Under small sample tests t-distribution is used 

1. to test whether the population has the given mean, 

2. to test whether two population means are equal (=/>/<)

Number of competitors should be finite.

The direction cosines of the normal to YZ plane are 1,0,0

Geometric mean (G. M).

(C) 2x + 5y – 6z + k = 0 

Correct option :

(A) 0

(A) x/√(1 + x²)

Answer is (d) 2tan x sec² x

Correct option :

(D) 0

Correct option:

(B) -(1/5)

Answer is (c) 3x² cos (x³)

Correct option:

(B) 1/6

Answer is (a) 0

Statistics is a subject that provides a body of principles and methodology for designing the methods of collecting, summarizing, analyzing, and interpreting data, drawing valid conclusions, and reaching a decision.

The use of statistics in any scientific investigation is indispensable. The detailed exposition of the subject in terms of its methods and importance can be found in many texts.

Our aim in this text is to make a brief overview of some statistical tools that will guide a researcher to statistically analyzing his/her data, interpreting and generalizing his/her results, and then assessing the extent of uncertainty underlying these generalizations.

Two broad classifications of the subject of statistics have been made in our endeavor to present statistical methods for analyzing and interpreting the results: descriptive statistics and inferential statistics.

We introduce the concepts of these two terms in turn.

The most common forms of descriptive statistics in use are measures of central tendency and variability of data.

Descriptive statistics are the tools that can enable us to describe a large volume of data in a summarized fashion, making it easy to comprehend.

When your findings are from a probability sample, summary descriptions, or statistics derived from this sample may be used to infer about the corresponding population parameters under certain assumptions about the distribution of the underlying population.

This falls under inferential statistics.

Statistical procedures that allow you to infer from what you found in a representative sample to the whole population are called inferential statistics.

Such statistics may be used to test hypotheses about the relationships that may exist within a population under study.

Put differently, and this is done by asking whether the patterns found in the sample data would differ from those in the population from which the sample data were drawn.

Answer is (a) 1/√(1 - x²)

• Controlling the quality of the product during the manufacturing process itself is called process control. 

• Controlling the quality of the finished products/manufactured products is called product control.

S = {(1, 1), (2, 2), ... , (6, 6)}

n(S) = 36

E = getting 7 or 11

= {(1, 6), (2, 5), (3, 4), (4, 3), (5,2), (6, 1), (5, 6), (6, 5)}

n(E) = 8

p(E) = n(E)/n(S) = 8/36 = 2/9

\(=x^2_{cal} =\frac {ns^2}{σ^2} = \frac{10 \times20}{25} =8\).