Mayank, Deepak and Pawan, each of them working alone can complete a work in 5, 10 and 15 days respectively. If all three of them work together to complete a work and earn Rs. 11,000, What will be Deepak's share of the earnings?
When individuals work together to complete a task and earn money, their share of the earnings is proportional to the amount of work they do, which is inversely proportional to the time they take to complete the work individually. We will calculate the ratio of their efficiencies (work done per day) and then distribute the total earnings based on this ratio.
The work rate is the reciprocal of the time taken to complete the work.
The ratio of their efficiencies is \[\frac{1}{5} : \frac{1}{10} : \frac{1}{15}\].
To simplify this ratio, find the Least Common Multiple (LCM) of the denominators (5, 10, 15). The LCM of 5, 10, and 15 is 30.
Multiply each fraction by the LCM:
\[ \left( \frac{1}{5} \times 30 \right) : \left( \frac{1}{10} \times 30 \right) : \left( \frac{1}{15} \times 30 \right) \]\[ 6 : 3 : 2 \]The total earnings are Rs. 11,000. The sum of the ratio parts is \( 6 + 3 + 2 = 11 \).
Deepak's share corresponds to the ratio part of 3.
\[ \text{Deepak's share} = \frac{\text{Deepak's ratio part}}{\text{Sum of ratio parts}} \times \text{Total earnings} \]\[ \text{Deepak's share} = \frac{3}{11} \times 11,000 \]\[ \text{Deepak's share} = 3 \times 1,000 \]\[ \text{Deepak's share} = \text{Rs. } 3,000 \]Correct Option: D) Rs. 3,000 is the correct share for Deepak, calculated based on his work efficiency relative to Mayank and Pawan.
Below are given statements followed by two conclusions I and II. You have to take the given statements to be true even if they seem to be at variance from commonly known facts.
Statements:
All birds are trees.
All trees are papers.
Some papers are books.
Conclusions:
I. All birds are papers.
II. Some birds are books.
Decide which of the below options logically follows the given conclusions.
Both conclusions I and II follow
Neither conclusion I nor II follows
Only conclusion I follows
Only conclusion II follows
This is a Syllogism question. We will use Venn diagrams to represent the given statements and then check the validity of the conclusions based on these diagrams.
Correct Option: C) Only conclusion I follows
If \( 10x^{2} - 6xy + y^{2} - 4x + 4 = 0 \), then find the value of (x + 2y).
The given equation is a quadratic equation in two variables. We need to manipulate it to form perfect squares, which will allow us to find the values of \(x\) and \(y\).
The given equation is:
\[ 10x^{2} - 6xy + y^{2} - 4x + 4 = 0 \]Rearrange the terms to group them for completing the square. We can rewrite \(10x^2\) as \(x^2 + 9x^2\):
\[ x^{2} - 4x + 4 + 9x^{2} - 6xy + y^{2} = 0 \]Notice that the first three terms form a perfect square: \(x^2 - 4x + 4 = (x-2)^2\). The last three terms also form a perfect square: \(9x^2 - 6xy + y^2 = (3x-y)^2\).
Substitute these back into the equation:
\[ (x-2)^{2} + (3x-y)^{2} = 0 \]The sum of two squares can be zero if and only if each square term is zero. This is because squares of real numbers are always non-negative.
Therefore, we must have:
\[ (x-2)^{2} = 0 \]and
\[ (3x-y)^{2} = 0 \]From the first equation, \( (x-2)^2 = 0 \), we get:
\[ x - 2 = 0 \] \[ x = 2 \]From the second equation, \( (3x-y)^2 = 0 \), we get:
\[ 3x - y = 0 \]Substitute the value of \(x=2\) into this equation:
\[ 3(2) - y = 0 \] \[ 6 - y = 0 \] \[ y = 6 \]Now, we need to find the value of \( (x+2y) \). Substitute the values of \(x=2\) and \(y=6\):
\[ x + 2y = 2 + 2(6) \] \[ x + 2y = 2 + 12 \] \[ x + 2y = 14 \]C) 14 is the correct value for \( (x+2y) \).
The question asks for the city where the Nobel Museum is located. This is a factual question related to general knowledge about famous institutions and their geographical locations.
Correct Option: B) Stockholm is the capital city of Sweden and the primary location for most Nobel Prize ceremonies and related institutions, including the Nobel Museum. The museum is situated in the Old Town (Gamla Stan) of Stockholm.
The mean of a distribution is 20 and the standard deviation is 4. What is the value of the coefficient of variation?
The coefficient of variation (CV) is a statistical measure that expresses the standard deviation as a percentage of the mean. It is used to compare the relative variability between different data sets, even if their means are vastly different. The formula for the coefficient of variation is given by:
\[ \text{Coefficient of Variation (CV)} = \frac{\text{Standard Deviation}}{\text{Mean}} \times 100\% \]Correct Option: B) 20% is the correct value for the coefficient of variation, calculated using the given mean and standard deviation.
This is a blood relations problem. We need to deduce the relationships between the family members and their genders based on the given statements. We will construct a family tree or list the relationships to identify the male members.
Statement 1: Chithra is the sister of Rakesh.
This tells us Chithra is female (\(F\)) and Rakesh is male (\(M\)).
Statement 2: Dilip is the father of Arun and grandfather of Rakesh.
This implies Dilip is male (\(M\)). Arun is Dilip's child and Rakesh's parent. Since Rakesh is male (from statement 1), Arun must be Rakesh's father or mother. Given Dilip is the grandfather of Rakesh, Arun must be Dilip's son or daughter, and Arun is Rakesh's parent. If Arun is Rakesh's father, then Arun is male (\(M\)).
Statement 3: Badhri is the brother of Eniya's husband.
This means Badhri is male (\(M\)). Eniya's husband is also male. Eniya is female (\(F\)).
Statement 4: There are two fathers, three brothers and a mother in the group.
Let's consolidate the known relationships and genders:
From the statement "Dilip is the father of Arun and grandfather of Rakesh", we know Dilip and Arun are fathers. So, we have two fathers: Dilip and Arun. This matches the condition "two fathers".
Since Arun is Rakesh's father, and Chithra is Rakesh's sister, Arun is also Chithra's father. So, Arun is the father of Rakesh and Chithra.
Now let's consider the mother. Since Arun is the father of Rakesh and Chithra, there must be a mother for Rakesh and Chithra. This mother must be Arun's wife. Let's assume Eniya is Arun's wife, making her the mother of Rakesh and Chithra.
Now let's verify the "three brothers" condition. We have Rakesh (\(M\)). We also have Badhri (\(M\)) who is the brother of Eniya's husband. Since Arun is Eniya's husband, Badhri is Arun's brother. So, the three brothers are Rakesh, Arun, and Badhri. This doesn't quite fit. Let's re-evaluate.
The statement "Badhri is the brother of Eniya's husband" means Badhri is male. Eniya's husband is male. Let's assume Arun is Eniya's husband. Then Badhri is Arun's brother. So, the brothers are Badhri and Arun. Rakesh is also a male. This gives us three males: Badhri, Arun, and Rakesh. The condition states "three brothers". This implies that there are three individuals who are brothers to someone. Badhri is a brother (to Arun). Arun is a brother (to Badhri). Rakesh is a brother (to Chithra).
Let's list all family members and their genders:
Let's check the conditions:
All conditions are satisfied with this family structure.
Now, we need to count the male members. The male members are:
There are 4 male members in the family.
C) Four. Based on our deduction, there are four male members: Dilip, Arun, Rakesh, and Badhri.
SubQuestion No: 2
Which of the following is a group of brothers?
Arun, Badhri, Dilip
Arun, Badhri, Rakesh
Badhri, Rakesh, Chithra
Badhri, Dilip, Rakesh
The question asks to identify a group of brothers from the given options. This requires referring to a previously established relationship diagram or text (which is implied to be available from the context of "SubQuestion No: 2"). We need to check each option against the known family relationships.
Correct Option: B) Arun, Badhri, Rakesh. This group consists entirely of brothers based on typical family relationship problem setups.
The question asks to identify the odd one out from the given options. This type of question usually requires finding a common characteristic among some options and identifying the one that does not share that characteristic.
C) Horse is the correct answer because, unlike the Lion, Tiger, and Giraffe which are wild animals, the Horse is a domesticated animal.
Which of the following objects directly orbits the Sun?
The question asks to identify which celestial body directly orbits the Sun. We need to understand the definitions and orbital characteristics of planets, satellites, stars, and galaxies.
Based on these definitions, planets are the objects that directly orbit the Sun.
Correct Option: A) A Planet is a celestial body that directly orbits a star, such as our Sun, and is gravitationally bound to it.