To expand the given expression \((x + 2y)(2x - y)\), we need to multiply each term in the first parenthesis by each term in the second parenthesis. This is often referred to as the FOIL method (First, Outer, Inner, Last) for binomials.
B) \(2x^{2} + 3xy - 2y^{2}\) is the correct expansion of the given expression, as derived in the step-by-step solution.
To find the mean (average) and mode of a given set of data, we need to apply their respective definitions. The mean is the sum of all values divided by the number of values. The mode is the value that appears most frequently in the data set.
C) 4 and 1. This option correctly states that the mean is 4 and the mode is 1, as calculated in the step-by-step explanation.
Four pairs of words are given. Three pairs are related and one is the odd one out. Identify the odd pair.
Hydrogen : Sulphuric Acid
Chlorine : Salt
Oxygen : Water
Nitrogen : Hydrochloric Acid
The question asks to identify the odd pair out of four given pairs of words. We need to find a common relationship among three pairs and then identify the one that does not fit this relationship.
D) Nitrogen : Hydrochloric Acid — In this pair, Nitrogen is not a constituent element of Hydrochloric Acid (\(\text{HCl}\)). Hydrochloric Acid is formed from Hydrogen and Chlorine. This pair does not follow the pattern established by the other three.
The question asks to identify which body part or system is primarily affected by osteoporosis. This requires knowledge of common medical conditions and their associated organs.
Correct Option: A) Bones
Osteoporosis literally means "porous bone." It is a disease that weakens bones, making them fragile and more likely to break. It develops when bone mineral density and bone mass decrease, or when the quality or structure of bone changes.
The question asks to identify the individual who has NOT received the Bharat Ratna, India's highest civilian award, from the given options. This requires knowledge of prominent Bharat Ratna recipients, especially those from the field of arts and music, as all options are renowned musicians.
Correct Option: B) A. R. Rahman has not received the Bharat Ratna. He is a highly decorated musician but this specific award has not been conferred upon him.
The question asks for the location of the Victoria Memorial Hall in India. This is a general knowledge question about famous landmarks in India.
Correct Option: C) Kolkata is the correct answer. The Victoria Memorial Hall is a well-known landmark located in Kolkata, West Bengal.
To find the value of the given expression, we will use fundamental trigonometric identities. Specifically, we will use the identity \( \sec^2 \theta - \tan^2 \theta = 1 \) and the reciprocal identity \( \tan \theta \cot \theta = 1 \).
The given expression is:
\[ (\sec ^{2}\theta + 2\tan \theta \cot \theta - \tan ^{2}\theta ) \]Rearrange the terms to group the \( \sec^2 \theta \) and \( \tan^2 \theta \) terms together:
\[ (\sec ^{2}\theta - \tan ^{2}\theta + 2\tan \theta \cot \theta ) \]Recall the fundamental trigonometric identity:
\[ \sec ^{2}\theta - \tan ^{2}\theta = 1 \]Substitute this identity into the expression:
\[ (1 + 2\tan \theta \cot \theta ) \]Recall the reciprocal identity for tangent and cotangent:
\[ \tan \theta \cot \theta = \tan \theta \cdot \frac{1}{\tan \theta} = 1 \]Substitute this identity into the expression:
\[ (1 + 2 \cdot 1) \]Simplify the expression:
\[ 1 + 2 = 3 \]Thus, the value of the given expression is 3.
D) 3 — By applying the trigonometric identities \( \sec^2 \theta - \tan^2 \theta = 1 \) and \( \tan \theta \cot \theta = 1 \), the expression simplifies to \( 1 + 2(1) = 3 \).
The question asks to identify the historical figure known as the 'Bismarck of India'. This requires knowledge of Indian history and the roles played by prominent leaders during the independence and post-independence era, particularly regarding the unification of the country.
Sardar Vallabhbhai Patel played a pivotal role in integrating over 500 princely states into the Indian Union after independence. His diplomatic skills, political acumen, and firm resolve were instrumental in achieving this unification, often referred to as the 'iron unification'. This achievement is comparable to Otto von Bismarck's role in unifying Germany in the 19th century, earning Patel the epithet 'Bismarck of India'.
B) Vallabhbhai Patel — Sardar Vallabhbhai Patel is widely recognized as the 'Bismarck of India' due to his crucial role in the integration of numerous princely states into the Indian Union following India's independence in 1947. His efforts were central to consolidating the territorial integrity of modern India.
This problem involves the concept of work and time, where the total work done is constant. We can use the formula: \(\text{Men} \times \text{Days} = \text{Total Work}\). By calculating the total work units, we can find out how many days it will take for the increased number of women to complete the same work, and then determine how many days earlier the work will be completed.
Given that 9 women can complete a work in 50 days.
Total Work \( = \text{Number of Women} \times \text{Number of Days}\)
\[ \text{Total Work} = 9 \text{ women} \times 50 \text{ days} = 450 \text{ woman-days} \]Initially, there are 9 women. If 6 more women join them, the new number of women will be:
\[ \text{New Number of Women} = 9 + 6 = 15 \text{ women} \]Now, 15 women need to complete the same total work (450 woman-days).
Let \(D\) be the number of days required by 15 women.
\[ 15 \text{ women} \times D \text{ days} = 450 \text{ woman-days} \]\[ D = \frac{450}{15} = 30 \text{ days} \]The original time to complete the work was 50 days. The new time is 30 days.
Days earlier \( = \text{Original Days} - \text{New Days}\)
\[ \text{Days earlier} = 50 - 30 = 20 \text{ days} \]Correct Option: B) 20
If 4x + 5y = 14 and x − 5y = 16 then the value of x and y are
10 and −\( \frac{6}{5} \)
6 and 2
10 and \( \frac{6}{5} \)
6 and −2
The problem requires us to find the values of \(x\) and \(y\) from a given system of two linear equations. We can solve this system using either the substitution method or the elimination method. The elimination method is particularly efficient here because the coefficients of \(y\) in both equations are additive inverses (\(5y\) and \(-5y\)).
Write down the given equations:
\[ 4x + 5y = 14 \quad \text{(Equation 1)} \]\[ x - 5y = 16 \quad \text{(Equation 2)} \]Add Equation 1 and Equation 2 to eliminate \(y\):
\[ (4x + 5y) + (x - 5y) = 14 + 16 \]\[ 4x + x + 5y - 5y = 30 \]\[ 5x = 30 \]Solve for \(x\):
\[ x = \frac{30}{5} \]\[ x = 6 \]Substitute the value of \(x = 6\) into either Equation 1 or Equation 2 to find \(y\). Let's use Equation 2:
\[ x - 5y = 16 \]\[ 6 - 5y = 16 \]Solve for \(y\):
\[ -5y = 16 - 6 \]\[ -5y = 10 \]\[ y = \frac{10}{-5} \]\[ y = -2 \]Thus, the values are \(x = 6\) and \(y = -2\).
Correct Option: D) 6 and \(-2\)