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Let us assume that, y = cos^-1( -1/2)

We can write this as:

cos y =- 1/2

cos y = cos (2π/3).

Thus, the Range of the principal value of cos^-1 is [0, π ].

Therefore, the principal value of  cos^-1( -1/2) is 2π /3.

Correct Answer is (C) \(\sqrt{\frac{1}{10}}\)

Let x =\(cos^{-1}\frac{4}{5}\)

⇒ cos x =\(\frac{4}{5}\)

Therefore \(sin\left(\frac{1}{2}cos^{-1}\frac{4}{5}\right)\) becomes sin(\(\frac{1}{2}\)x),i.e sin (\(\frac{x}{2}\))

We know that sin (\(\frac{x}{2}\)) = \(\sqrt{\frac{1-cosx}{2}}\)

=\(\sqrt{\frac{1-\frac{4}{5}}{2}}\)

=\(\sqrt{\frac{\frac{2}{5}}{2}}\)

sin(\(\frac{x}{2}\))=\(\sqrt{\frac{1}{10}}\)

Correct Answer is (B) \(\frac{\pi}{4}\)

Let , x = \(tan^{-1}\begin{Bmatrix}2cos\left(2sin^{-1}\frac{1}{2}\right)\end{Bmatrix}\)

⇒ x= \(tan^{-1}\begin{Bmatrix}2cos\left(2cos(\frac{\pi}{6})\right)\end{Bmatrix}\)

\((\because sin(\frac{\pi}{6})=\frac{1}{2})\)

⇒ x = tan^{-1\((2cos\frac{\pi}{3})\)}

⇒ x = tan^-1\((2(\frac{1}{2}))\)\(tan^{-1}1=\frac{\pi}{4}\)

\((\because cos(\frac{\pi}{3})=\frac{1}{2} and \tan(\frac{\pi}{4}=1)\)

Correct Answer is (B) \(\frac{5}{\sqrt{26}}\)

Given : \(cos^{-1}\frac{-1}{5}=x\) 

Since, x= \(cos^{-1}\frac{-1}{5}\) 

⇒ \(cot x= \frac{-1}{5}=\frac{\text{adjacent side}}{\text{opposite side}}\) 

By pythagorus theroem , 

(Hypotenuse )² = (opposite side )² + (adjacent side )²

Therefore, Hypotenuse =\(\sqrt{26}\) 

^⇒ sin x =\(\frac{\text{opposite side}}{\text{hypotenuse side}}\) =\(\frac{5}{\sqrt{26}}\)

Correct Answer is (C) \(\frac{3\pi}{2}\)

Let , x = \(sin^{-1}(\frac{-1}{2})+2cos^{-1}(\frac{-\sqrt{3}}{2})\) 

⇒ \(sin^{-1}(\frac{1}{2})+2[\pi -cos^{-1}(\frac{-\sqrt{3}}{2})]\) 

( \(\because\) sin-1(-\(\theta\)) = -sin^-1(\(\theta\)) and cos^-1(-\(\theta\))=π - cos^-1(\(\theta\))]

⇒ x= \(-(\frac{\pi}{6})\)+2\([\pi-\frac{\pi}{6}]\)

⇒ x= \(-(\frac{\pi}{6})\)+\(2[\frac{5\pi}{6}]\)

⇒ x = \(-\frac{\pi}{6}\)\(+\frac{5\pi}{3}\) 

⇒ x = \(\frac{3\pi}{2}\)

tan^-1[2sin(2cos^-1(√3/2))]

= tan^-1[2sin(2(π/6))] = tan^-1[2sin(π/3)] = tan^-1[2 x √3/2]

= tan^-1√3 = π/3

Volume of the frustum = \(\frac{1}3π(r'^2 + r"^2 + r'r")h\) = 12308.8

⇒ \(\frac{1}3π(20^2 + 12^2 + (20)(12))h\) = 12308.8

⇒ h = 15 cm 

Let l be the slant height of the bucket 

⇒ Slant height, l = \(\sqrt{(r' - r")^2 + h^2}\)

⇒ l = \(\sqrt{(20 - 12)^2 + 15^2}\)

⇒ l = 17 cm 

∴ length of the bucket, l = 17 cm 

Curved surface area = π(r’ + r’’)l + πr’’² 

= π(20 + 12)(17) + π (12)² 

= 2160.32 cm²

Karl Pearson’s method gives more reliable value of coefficient of skewness.

Coefficient of skewness is a relative measure, because it is a ratio of skewness and dispersion and hence free from unit of measurement.

The main objectives of studying skewness are as follows :

• To get complete information of the data with respect to the direction, shape and form of the frequency curve of the frequency distribution.
• To obtain more information about the population by studying the symmetry of the frequency curve.
• To study whether the frequency of observation at equidistant on both sides of mode is evenly distributed or not.

Skewness: Measures of central tendency and dispersion are useful for comparing two or more data. But they cannot give Information about whether the observations of the data are symmetrically distributed or not. This Information can be obtained from the frequency curve of thc frequency distribution. If frequency curve Is symmetric on both sides from Its peak point. It Is said that observations of the data are distributed symmetrically. But, if it is not so, we call it skewness in frequency distribution. Thus, skewness gives an idea about nature of the data with respect to shape of frequency curve. In short, skewness implies the lack of symmetry in frequency distribution of the data.

• The skewness of a frequency distribution may be positive or negative which can be determined by the shape of frequency curve.
• The measure of skewness can be obtained by using the measures of average – mean, median, mode and the measures of positional average – quartiles (Q₁, Q₃), median (M). Symbolically, it is denoted by S_k.
• Absolute measure of skewness is S_k = x – M₀ or S_k = 3(x – M)
  OR S_k = Q₃ + Q₁ – 2M.
• Absolute measure of skewness, cannot be used for comparing skewness of two or more frequency distributions.

Coefficient of skewness: It is a relative measure of skewness which is free from unit of measurement. It is denoted by symbol j.

• Coefficient of skewness can be used for comparing the skewness of two or more frequency distributions.
• It is obtained by dividing the absolute measure of skewness by an appropriate measure of dispersion, i.e., j = \(\frac{S_k}{s} \,or\, j = \frac{s_k}{Q_3−Q_1}\)

Under the following circumstances in which Karl Pearson’s formula j = 3\((\frac{\bar x−M}{s})\) is used to find coefficient of skewness :

• When the frequency distribution is bimodal.
• When the frequency distribution is of unequal class length.
• When the frequency distribution is of mixed type of discrete and continuous distribution.

Bowley’s method : Base : In a skewed frequency distribution, both quartiles Q₁ and Q₃ are not equidistant from the median M.

Measure of skewness: Using the distances of quartiles from median, i.e., (Q₃ – M) and (M – Q₁), the measure of skewness S_k is obtained as follows:
S_k = (Q₃ – M) – (M – Q₁)
= Q₃ – M – M + Q₁
S_k = Q₃ + Q₁ – 2M

Relative measure of skewness : Dividing the absolute measure of skewness by the sum of distances (Q₃ – M) and (M – Q₁), the relative measure of skewness-the coefficient of skewness (j) is obtained, i.e.,
J = \(\frac{Sk}{(Q_3−M)+(M−Q_1)}\)

= \(\frac{Q_3+Q_1−2M}{Q_3−Q_1}\)

-» If distances (Q₃ – M) and (M – Q₁) are positive, then their difference may be less than or equal to their sum. Hence, | j | ≤ 1, i.e., – 1 ≤ j ≤ 1.
-» This method is used when we have to compute the skewness by using positional average.
-> This is the only method to calculate skewness when the frequency distribution has open-end classes.

1. Because he was not good at advising.

2. Irrelevant

3. His conclusions lacked sense.

4. Because of Chitra Varma’s plans, actions, decisions, and conclusions

5. To seek help from the court’s jester Tenali Ramakrishna in this matter

The numbers of classes are obtained using Sturges’s Rule: k = 1 + 3.22 log N; N-number of items.

The number of classes are obtained using Sturges’s Rule: k = 1 + 3.22 Log N.

Because lateral meristem is absent in monocot plants.

Meiosis is the mode of cell division in which gametes are formed. Meiosis occurs in the germinal cells of the reproductive organs. Human beings have 46 chromosomes. Germinal cells with 46 chromosomes divide continuously two times. These divisions in meiosis are known as meiosis I and meiosis II. Two daughter cells “with half the number of chromosomes (23 chromosomes) are formed in meiosis I. Each daughter cell again divides in meiosis II. There is no change in the chromosome number in this division. Hence meiosis II is similar to. mitosis. As a result of meiosis, four daughter cells, each with 23 chromosomes, are formed from a germinal cell.

Mitosis leads to reproduction in unicellular organisms.

Growth of the body is brought about by cell division and cell growth

Meristematic cells

Because it is due to the presence of the cell wall in plant cell

Let the number of passengers in AC sleeper class, first class and sleeper class be x, 2x and 3x respectively. 

Let the fares of AC sleeper class, first class and sleeper class be 5y, 4y and 2y respectively. 

Then, the total incomes from the three classes = 5xy + 8xy + 6xy = 19xy 

Given, 19xy = Rs 54000 ⇒ xy = Rs \(\frac{54000}{19}\) 

∴ Income from AC sleeper = Rs \(\frac{54000}{19}\) x 5 = Rs 14210 (approx).

(d) 5 : 4

a : b : c = 7 : 3 : 5 

⇒ a = 7k, b = 3k, c = 5k 

∴ (a + b + c) : (2a + b – c) 

= (7k + 3k + 5k) : (14k + 3k – 5k) 

= 15 k : 12k = 5 : 4

After the first operation, 

Paint = 18 litres, Oil = 2 litres 

After the second operation, 

Paint = 18 - \(\frac{18}{20}\times2\) = 16.2 litres,

Oil = 2 – \(\frac{2}{20}\times2+2\) = 3.8 litres

After the third operation,

Paint = 16.2 - \(\frac{16.2}{20}\times2\) = 14.58 litres

Oil = 3.8 - \(\frac{3.8}{20}\times2+2\) = 5.42 litres

∴ Required ratio = \(\frac{14.58}{5.42} = \frac{729}{271}\)

The total charge of an isolated system remains always constant. 

S.I. unit of charge coulomb (C).

The electric charge is always an integral multiple of ‘e’ (charge on an electron). 

(i) It burns without smoke and leave no residue therefore causes no atmospheric pollution. (ii)Its heating capacity is high, i.e. it has high calorific value. Main constituents of bio gas: The composition of bio gas varies depending upon the nature of organic matter feeding in’the digester and advanced waste treatment technology. The typical composition of bio gas is

q = ne ; n is an integer (+ or -) and e is the charge on an electron.

(A) Computer hardware

(C) 1,024 PetaBytes

Correct Answer is  Computer hardware

(B) 1,024 TeraBytes

Correct Answer is  1,024 PetaBytes

Correct Answer is  1,024 TeraBytes

Correct Answer is 1,024 GigaBytes

Correct Answer is  1,024 KiloBytes

1. True

2. False

(C) 1,024 KiloBytes

(A) 1,024 GigaBytes

(D) 1,024 MegaBytes

1. False

2. True

Correct Answer is  1,024 MegaBytes

• Micro economics (Price theory) analyses the patterns of supply and demand so as to determine the price and output in the individual markets.
• The theories of product pricing and factor pricing help an entrepreneur in various ways.
• The price mechanism helps the businessmen to determine the prices of their goods and services.
• It also helps them to take important decisions related to cost of production.
• Moreover, by using this theory, the businessmen can efficiently allocate their resources as well as forecast the factors of demand for their product.

Thus, price theory is very useful to businessmen.

(d) All of the above

(i – c), 

(ii – d), 

(iii – a), 

(iv – b). 

(a) C 

(b) B 

(c) As density of the liquid increases, buoyancy also increases.

Viscosity decreases with increase in temperature.