Chetna went from her home to the city at the speed of 8 km/hr but came back home by the same route at the speed of 5 km/hr. It took her 19\( \frac{1}{2} \) hours to travel two-way. What is the distance of the city from Chetna's house?
This problem involves calculating distance, speed, and time. We are given the speeds for the outward and return journeys and the total time taken. Since the distance for both journeys is the same, we can set up an equation using the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) and solve for the unknown distance.
Let the distance of the city from Chetna's house be \(D\) km.
Speed while going to the city (outward journey), \(S_1 = 8\) km/hr.
Time taken for the outward journey, \[T_1 = \frac{D}{S_1} = \frac{D}{8}\] hours.
Speed while coming back home (return journey), \(S_2 = 5\) km/hr.
Time taken for the return journey, \[T_2 = \frac{D}{S_2} = \frac{D}{5}\] hours.
The total time taken for the two-way journey is given as \(19\frac{1}{2}\) hours. Convert this mixed fraction to an improper fraction:
\[ 19\frac{1}{2} = \frac{(19 \times 2) + 1}{2} = \frac{38 + 1}{2} = \frac{39}{2} \text{ hours} \]
The total time is the sum of the time taken for the outward and return journeys:
\[ T_1 + T_2 = \frac{39}{2} \]
\[ \frac{D}{8} + \frac{D}{5} = \frac{39}{2} \]
To add the fractions on the left side, find a common denominator, which is the Least Common Multiple (LCM) of 8 and 5. The LCM of 8 and 5 is 40.
\[ \frac{5D}{40} + \frac{8D}{40} = \frac{39}{2} \]
\[ \frac{5D + 8D}{40} = \frac{39}{2} \]
\[ \frac{13D}{40} = \frac{39}{2} \]
Now, solve for \(D\):
\[ 13D = \frac{39}{2} \times 40 \]
\[ 13D = 39 \times 20 \]
\[ 13D = 780 \]
\[ D = \frac{780}{13} \]
\[ D = 60 \text{ km} \]
B) 60 km — The calculated distance is 60 km, which matches this option.
This problem can be solved by translating the given word problem into a quadratic equation. We will then solve the quadratic equation to find the possible values of the number.
Define the variable: Let the unknown number be \(x\).
Formulate the equation: The problem states that "The square of a number is 3 more than twice the number."
So, the equation is:
\[ x^2 = 2x + 3 \]Rearrange the equation into standard quadratic form: To solve a quadratic equation, we usually set it equal to zero:
\[ x^2 - 2x - 3 = 0 \]Solve the quadratic equation: We can solve this quadratic equation by factoring, using the quadratic formula, or completing the square. Let's use factoring.
We need two numbers that multiply to \(-3\) and add up to \(-2\). These numbers are \(-3\) and \(1\).
So, we can factor the equation as:
\[ (x - 3)(x + 1) = 0 \]Find the possible values of \(x\): For the product of two factors to be zero, at least one of the factors must be zero.
Thus, the possible numbers are \(-1\) or \(3\).
D) -1 or 3. This matches our calculated possible values for \(x\).
If one side of an equilateral triangle is 6\(\sqrt{3}\) cm, what is its height?
To find the height of an equilateral triangle given its side length, we can use the formula for the height of an equilateral triangle, or we can use the Pythagorean theorem by dividing the equilateral triangle into two right-angled triangles.
A) 9 cm — Both methods confirm that the height of the equilateral triangle is 9 cm.
Given below is a statement followed by assumptions numbered I and II.
You have to consider the statement followed by the given assumptions and decide which of the assumptions are implicit in the statement. Statement:
Where there is a will, there is a way.
Assumptions:
I. People make excuses to postpone something.
II. More important is the intention to do something.
Both I and II are implicit
Neither I nor II is implicit
Only assumption I is implicit.
Only assumption II is implicit.
The problem asks us to identify which of the given assumptions is implicit in the statement: "Where there is a will, there is a way." An implicit assumption is something that is not directly stated but is taken for granted or implied by the statement. We need to analyze each assumption in relation to the core meaning of the statement.
D) Only assumption II is implicit. Assumption II directly captures the essence of the statement, which emphasizes the paramount importance of having the intention or determination (the 'will') to achieve something.
The problem presents a coding pattern where words are transformed into coded words. We need to identify the rule governing this transformation by analyzing the given examples and then apply the same rule to the target word.
Correct Option: D) QLUE
This problem involves calculating the remaining number of households without electricity after a certain period, given an initial number and an annual electrification rate. We need to calculate the number of households electrified each year and subtract it from the remaining unelectrified households.
Initial number of households without electricity:
\[ \text{Initial Households} = 1200 \]Electrification rate:
\[ \text{Rate} = 75\% \text{ per annum} \]Calculate households remaining after 1st year:
If 75% are electrified, then \(100\% - 75\% = 25\%\) of households remain without electricity.
\[ \text{Households remaining after 1st year} = \text{Initial Households} \times (1 - \text{Rate}) \] \[ = 1200 \times (1 - 0.75) \] \[ = 1200 \times 0.25 \] \[ = 300 \]So, after 1 year, 300 households are still without electricity.
Calculate households remaining after 2nd year:
The electrification rate of 75% applies to the remaining unelectrified households each year. So, for the second year, we apply the rate to the 300 households that were still without electricity at the end of the first year.
\[ \text{Households remaining after 2nd year} = \text{Households after 1st year} \times (1 - \text{Rate}) \] \[ = 300 \times (1 - 0.75) \] \[ = 300 \times 0.25 \] \[ = 75 \]Thus, after 2 years, 75 households will still be living without an electricity connection.
Correct Option: B) 75
The question describes a fundamental principle of fluid mechanics related to buoyancy. We need to identify the scientific rule or principle that explains why a body immersed in a liquid experiences an upward force equal to the weight of the displaced liquid.
Correct Option: A) Archimedes' principle states that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object. This perfectly matches the description given in the question.
The question asks for the capital city of the Indian state of Uttarakhand. This is a factual question related to Indian geography and general knowledge.
C) Dehradun is the capital city of Uttarakhand. It serves as the provisional capital of the state and is also its largest city.
This is a syllogism problem that can be solved using Venn diagrams or by applying rules of logical deduction. We will represent the statements graphically and then evaluate the conclusions.
D) Both 1 and 2 are logical.
The question asks for the term that defines the sum of an object's kinetic energy and potential energy. This is a fundamental concept in physics, specifically in the study of mechanics and energy conservation.
Correct Option: A) Mechanical energy is the sum of kinetic energy and potential energy. This concept is central to the law of conservation of mechanical energy, which states that in an isolated system where only conservative forces do work, the total mechanical energy remains constant.