To find the average speed for a two-way journey where the distance is the same for both parts of the journey but the speeds are different, we use the formula for average speed. The average speed is defined as the total distance traveled divided by the total time taken. For two equal distances covered at different speeds, the average speed can be calculated using the harmonic mean of the two speeds.
Identify the given speeds: Let the speed from A to B be \(v_1 = 20 \text{ km/h}\). Let the speed from B to A be \(v_2 = 30 \text{ km/h}\).
Understand the concept of average speed: Average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]Assume a distance: Since the distance from A to B is the same as the distance from B to A, let's assume the distance between A and B is \(D\). A convenient way to choose \(D\) is the Least Common Multiple (LCM) of the speeds to simplify calculations. LCM of 20 and 30 is 60. So, let \(D = 60 \text{ km}\).
Calculate time for the journey from A to B: Time \[t_1 = \frac{\text{Distance}}{\text{Speed}_1} = \frac{D}{v_1}\] \[ t_1 = \frac{60 \text{ km}}{20 \text{ km/h}} = 3 \text{ hours} \]
Calculate time for the journey from B to A: Time \[t_2 = \frac{\text{Distance}}{\text{Speed}_2} = \frac{D}{v_2}\] \[ t_2 = \frac{60 \text{ km}}{30 \text{ km/h}} = 2 \text{ hours} \]
Calculate total distance: Total distance = Distance from A to B + Distance from B to A \(= D + D = 2D\) \[ \text{Total Distance} = 60 \text{ km} + 60 \text{ km} = 120 \text{ km} \]
Calculate total time: Total time = Time from A to B + Time from B to A \(= t_1 + t_2\) \[ \text{Total Time} = 3 \text{ hours} + 2 \text{ hours} = 5 \text{ hours} \]
Calculate average speed: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{120 \text{ km}}{5 \text{ hours}} = 24 \text{ km/h} \] Alternatively, for equal distances, the average speed can be calculated using the formula: \[ \text{Average Speed} = \frac{2 v_1 v_2}{v_1 + v_2} \] Substituting the given values: \[ \text{Average Speed} = \frac{2 \times 20 \times 30}{20 + 30} = \frac{1200}{50} = 24 \text{ km/h} \]
A) 24 km/h is the correct average speed calculated using both methods.
This problem involves determining relative directions, which is a common type of question in competitive exams. We can solve this by visualizing the positions of the towns on a 2D plane using standard cardinal directions (North, South, East, West).
D) North-West. Based on our step-by-step analysis, Town F is located to the North-West of Town D.
The question asks about the form in which autotrophs store their food. Autotrophs are organisms that produce their own food, primarily through photosynthesis. Understanding the primary storage carbohydrate in plants (the most common autotrophs) is key to answering this question.
Correct Option: C) starch — Autotrophs, especially plants, produce glucose through photosynthesis and store the excess glucose in the form of starch. Starch is a polysaccharide composed of many glucose units and is an efficient way to store energy for later use.
Refer to the following letter-symbol series and answer the question that follows:
N S S F S & H H Y T G B R ^ K D R S V % ^ V # D % C D @
How many such symbols are there that are immediately preceded by a letter and also immediately followed by a letter?
The problem asks us to identify symbols in a given series that are immediately preceded by a letter and immediately followed by a letter. We need to scan the series and look for the pattern: Letter - Symbol - Letter.
A) 4 is the correct answer because there are four symbols in the series that are immediately preceded by a letter and immediately followed by a letter.
The roots of the quadratic equation x2 − 10x + 21 = 0 are:
To find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\), we can use factorization, completing the square, or the quadratic formula. For the given equation, factorization is a straightforward method.
A) 7, 3. The roots of the equation \(x^2 - 10x + 21 = 0\) are \(x=3\) and \(x=7\).
The value of 2\(\sqrt{2}\)sin45°sin30° + 2\(\sqrt{3}\)cos60° tan60° − sin90° is __________.
To solve this problem, we need to evaluate the given trigonometric expression by substituting the standard values of trigonometric ratios for the angles 30°, 45°, 60°, and 90°. Then, we will perform the arithmetic operations to find the final value.
B) The final calculated value of the expression is 3.
If ‘+’ means ‘×’, ‘×’ means ‘÷’, ‘÷’ means ‘−’, and ‘−’ means ‘+’, then what will be the value of the given expression?
This problem involves substituting given symbol transformations into a mathematical expression and then evaluating the expression using the order of operations (BODMAS/PEMDAS).
The original expression is: \(215 \times 5 \div 4 - 18 \times 3\)
Applying the transformations:
So, the new expression is: \(215 \div 5 - 4 + 18 \div 3\)
BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. Perform operations in this order from left to right.
First division: \(215 \div 5\)
\[ 215 \div 5 = 43 \]The expression becomes: \(43 - 4 + 18 \div 3\)
Second division: \(18 \div 3\)
\[ 18 \div 3 = 6 \]The expression becomes: \(43 - 4 + 6\)
First operation (subtraction): \(43 - 4\)
\[ 43 - 4 = 39 \]The expression becomes: \(39 + 6\)
Second operation (addition): \(39 + 6\)
\[ 39 + 6 = 45 \]D) 45 — After transforming the symbols and applying the order of operations, the value of the expression is 45.
The question asks to identify the incorrect statement about Plaster of Paris. We need to recall the chemical name, formula, and common uses of Plaster of Paris to evaluate each option.
Based on the analysis, statement A is the incorrect one.
A) It is also known as calcium sulphate dihydrate. This statement is incorrect. Plaster of Paris is calcium sulfate hemihydrate (\( \text{CaSO}_4 \cdot \frac{1}{2}\text{H}_2\text{O} \)), while calcium sulfate dihydrate (\( \text{CaSO}_4 \cdot 2\text{H}_2\text{O} \)) is gypsum.
The question asks to identify the date observed as Rashtriya Ekta Diwas or National Unity Day in India. This is a factual question related to important days and dates in India's history and national observances.
Correct Option: D) 31 October is observed as Rashtriya Ekta Diwas or National Unity Day every year in India, commemorating the birth anniversary of Sardar Vallabhbhai Patel.
Which Indian Chess player was in attendance along with other officials to receive the official hosting rights of FIDE 44th World Chess Olympiad 2022 on behalf of India?
Dibyendu Barua
Viswanathan Anand
Koneru Humpy
Abhijit Kunte
The question asks to identify the Indian Chess player who was present to receive the official hosting rights for the FIDE 44th World Chess Olympiad 2022 on behalf of India. This requires knowledge of recent events in Indian chess and the prominent figures associated with them.
B) Viswanathan Anand — Viswanathan Anand, a legendary Indian Grandmaster and five-time World Chess Champion, was indeed present along with other officials to receive the official hosting rights for the FIDE 44th World Chess Olympiad 2022. He played a crucial role in bringing the prestigious event to India and has since been elected as the Deputy President of FIDE.