The question asks for the common name of the Forty-Second Amendment to the Indian Constitution. This requires knowledge of Indian constitutional history and significant amendments.
A) Mini-Constitution — The Forty-Second Amendment Act, 1976, is widely known as the "Mini-Constitution" because it introduced a large number of changes to various parts of the Indian Constitution. These changes were so extensive and significant that they virtually rewrote large portions of the original document, leading to this popular nomenclature.
To find the average price per book, we need to calculate the total cost of all books and divide it by the total number of books. This involves calculating the cost from each shop and then summing them up.
Let's perform the calculations:
The average price Sachin paid per book is ₹1,014.65.
B) 1,014.65 is the correct average price per book calculated by dividing the total cost by the total number of books.
The question asks about the function of afferent neurons in the peripheral nervous system. To answer this, we need to recall the basic classification and functions of neurons, specifically afferent (sensory) and efferent (motor) neurons, and their roles within the nervous system.
D) Transmitting sensory information from receptors to the central nervous system. This statement accurately describes the primary function of afferent neurons. They are responsible for conveying sensory input from various parts of the body to the brain and spinal cord for processing.
The sum of the squares of two positive numbers a and b (where a > b) is 7 times their product. Find the difference between their squares.
\(\sqrt{15}\,ab\)
\(5\sqrt{3}\,ab\)
\(3\sqrt{5}\,ab\)
\(3\sqrt{5ab}\)
The problem provides a relationship between two positive numbers, \(a\) and \(b\), where \(a > b\). We are given that the sum of their squares is 7 times their product. We need to find the difference between their squares, expressed in terms of their product.
Formulate the given relationship:
The sum of the squares of two positive numbers \(a\) and \(b\) is \(a^2 + b^2\). Their product is \(ab\). The problem states that \(a^2 + b^2 = 7ab\).
Identify what needs to be found:
We need to find the difference between their squares, which is \(a^2 - b^2\). Since \(a > b\), \(a^2 - b^2\) will be positive.
Use algebraic identities:
We know the identity: \((a^2 - b^2)^2 = (a^2 + b^2)^2 - 4a^2b^2\).
Alternatively, we can use \((a-b)^2 = a^2 + b^2 - 2ab\) and \((a+b)^2 = a^2 + b^2 + 2ab\).
Since \(a^2 - b^2 = (a-b)(a+b)\), we can find \((a-b)\) and \((a+b)\) first.
From Equation 1, substitute \(a^2 + b^2 = 7ab\) into the identities:
For \((a-b)^2\):
\[ (a-b)^2 = a^2 + b^2 - 2ab \] \[ (a-b)^2 = 7ab - 2ab \] \[ (a-b)^2 = 5ab \] \[ a-b = \sqrt{5ab} \quad \text{(Equation 2, since } a>b \text{, } a-b \text{ is positive)} \]For \((a+b)^2\):
\[ (a+b)^2 = a^2 + b^2 + 2ab \] \[ (a+b)^2 = 7ab + 2ab \] \[ (a+b)^2 = 9ab \] \[ a+b = \sqrt{9ab} \] \[ a+b = 3\sqrt{ab} \quad \text{(Equation 3, since } a,b \text{ are positive, } a+b \text{ is positive)} \]Calculate the difference of squares:
Now, multiply Equation 2 and Equation 3:
C) \(3\sqrt{5}\,ab\)
As derived in the step-by-step solution, the difference between the squares of the two numbers is \(3\sqrt{5}ab\).
This problem involves setting up and solving a system of linear equations. We will define variables for the cost of a pen and a notebook, translate the given information into two equations, solve these equations simultaneously to find the individual costs, and then calculate the final required cost.
Define variables:
Let \(P\) be the cost of one pen (in ₹).
Let \(N\) be the cost of one notebook (in ₹).
Formulate the first equation:
The cost of 4 pens and 3 notebooks is ₹277.
\[ 4P + 3N = 277 \quad \text{(Equation 1)} \]
Formulate the second equation:
The cost of 16 notebooks exceeds the cost of 12 pens by ₹144.
This means: Cost of 16 notebooks - Cost of 12 pens = ₹144.
\[ 16N - 12P = 144 \quad \text{(Equation 2)} \]
Solve the system of equations:
We have:
(1) \(4P + 3N = 277\)
(2) \(-12P + 16N = 144\)
To eliminate \(P\), multiply Equation 1 by 3:
\[ 3 \times (4P + 3N) = 3 \times 277 \]\[ 12P + 9N = 831 \quad \text{(Equation 3)} \]
Now, add Equation 2 and Equation 3:
\[ (-12P + 16N) + (12P + 9N) = 144 + 831 \]\[ 25N = 975 \]\[ N = \frac{975}{25} \]\[ N = 39 \]
So, the cost of one notebook is ₹39.
Find the cost of one pen:
Substitute \(N = 39\) into Equation 1:
\[ 4P + 3(39) = 277 \]\[ 4P + 117 = 277 \]\[ 4P = 277 - 117 \]\[ 4P = 160 \]\[ P = \frac{160}{4} \]\[ P = 40 \]
So, the cost of one pen is ₹40.
Calculate the required cost:
We need to find the cost of 19 pens and 15 notebooks.
Cost \( = 19P + 15N \)
Cost \( = 19(40) + 15(39) \)
Cost \( = 760 + 585 \)
Cost \( = 1345 \)
The cost of 19 pens and 15 notebooks is ₹1,345.
D) ₹1,345 is the correct cost calculated from the derived values of pen and notebook costs.
The selling price of 34 books is equal to the cost price of 17 books. Find the loss or gain percentage.
50% loss
\( \frac{100}{17} \) % gain
50% gain
\( \frac{100}{17} \) % loss
To find the loss or gain percentage, we need to establish a relationship between the selling price (SP) and the cost price (CP) of a single item. The problem states that the selling price of 34 books is equal to the cost price of 17 books. We will use this relationship to determine if there's a loss or gain and then calculate the percentage.
Set up the equation based on the given information:
Let \( \text{CP} \) be the cost price of one book and \( \text{SP} \) be the selling price of one book.
According to the problem statement:
\[ 34 \times \text{SP} = 17 \times \text{CP} \]Express \( \text{SP} \) in terms of \( \text{CP} \):
Divide both sides by 34:
Determine if there is a loss or a gain:
Since \( \text{SP} = \frac{1}{2} \times \text{CP} \), it means the selling price is half of the cost price. Therefore, \( \text{SP} < \text{CP} \), which indicates a loss.
Calculate the loss amount:
Loss \( = \text{CP} - \text{SP} \)
Calculate the loss percentage:
The loss percentage is calculated as:
Correct Option: A) 50% loss
To find the value of \(x\), we need to use the formula for average speed, which is the total distance divided by the total time taken. We will calculate the time taken for each segment of the journey and then sum them up. The total distance is the sum of the distances of all segments.
The man covers three distances: \(D_1 = 1980 \text{ km}\), \(D_2 = 1306 \text{ km}\), and \(D_3 = 540 \text{ km}\).
\[ \text{Total Distance} = D_1 + D_2 + D_3 = 1980 + 1306 + 540 = 3826 \text{ km} \]The formula for time is \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\).
For the first segment:
\[ T_1 = \frac{D_1}{S_1} = \frac{1980}{12} = 165 \text{ hours} \]For the second segment:
\[ T_2 = \frac{D_2}{S_2} = \frac{1306}{x} \text{ hours} \]For the third segment:
\[ T_3 = \frac{D_3}{S_3} = \frac{540}{36} = 15 \text{ hours} \]Given that the average speed for the whole journey is \(14 \text{ km/hr}\).
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \] \[ 14 = \frac{3826}{180 + \frac{1306}{x}} \]Rearrange the equation to solve for \(x\):
\[ 14 \left(180 + \frac{1306}{x}\right) = 3826 \]Divide both sides by 14:
\[ 180 + \frac{1306}{x} = \frac{3826}{14} \] \[ 180 + \frac{1306}{x} = 273.2857... \quad \text{(approximately)} \]Let's recheck the division: \(3826 \div 14\). \(3826 = 14 \times 273.28...\). It seems there might be a slight calculation error or the numbers are not perfectly divisible. Let's re-evaluate the total distance or average speed from the options.
Let's recheck the total distance: \(1980 + 1306 + 540 = 3826\). This is correct.
Let's recheck the average speed: \(14 \text{ km/hr}\). This is given.
Let's recheck the division \(3826 / 14\). \(3826 \div 14 = 273.2857...\). This is correct.
So, \(180 + \frac{1306}{x} = 273.2857...\)
\[ \frac{1306}{x} = 273.2857... - 180 \] \[ \frac{1306}{x} = 93.2857... \] \[ x = \frac{1306}{93.2857...} \] \[ x \approx 14 \]Let's work backward with \(x=14\) to verify.
If \(x=14\), then \(T_2 = \frac{1306}{14} = 93.2857...\) hours.
Total time \(T = 165 + 93.2857... + 15 = 273.2857...\) hours.
Average Speed \( = \frac{3826}{273.2857...} = 14 \text{ km/hr}\).
The calculation is consistent. The value of \(x\) is \(14\).
D) 14
The question asks about a specific achievement of Anahat Singh in women's squash in 2025. This requires knowledge of recent sports events and their outcomes, particularly those involving prominent Indian athletes. The key is to identify the correct tournament and location where she secured the title.
Anahat Singh won the women's title at the Indian Open 2025 PSA Copper event held in Mumbai. This was a significant win for the young Indian squash player.
A) Indian Open 2025 PSA Copper event in Mumbai is the correct answer. Anahat Singh indeed won the women's squash title at this specific event, which was held in Mumbai.
This problem involves profit distribution based on investment ratios after a portion of the profit is allocated to charity. We need to calculate the remaining profit, determine the investment ratio of the partners, and then find Ashok's share based on this ratio.
Pankaj's investment \( = \text{₹}41,000 \)
Meera's investment \( = \text{₹}39,000 \)
Ashok's investment \( = \text{₹}84,000 \)
Total investment \( = 41,000 + 39,000 + 84,000 = \text{₹}164,000 \)
Total profit \( = \text{₹}49,200 \)
Percentage given to charity \( = 29\% \)
\[ \text{Amount for charity} = 29\% \text{ of } 49,200 = \frac{29}{100} \times 49,200 = \text{₹}14,268 \]Pankaj : Meera : Ashok \( = 41,000 : 39,000 : 84,000 \)
Divide by 1,000 to simplify the ratio:
Pankaj : Meera : Ashok \( = 41 : 39 : 84 \)
Sum of the ratio parts \( = 41 + 39 + 84 = 164 \)
Ashok's share \[= \left( \frac{\text{Ashok's ratio part}}{\text{Sum of ratio parts}} \right) \times \text{Remaining profit}\]
\[ \text{Ashok's share} = \left( \frac{84}{164} \right) \times 34,932 \]\[ \text{Ashok's share} = 0.512195... \times 34,932 \]\[ \text{Ashok's share} = \text{₹}17,892 \]D) The share of Ashok is ₹17,892, which matches our calculation.
The question asks for the venue of the World Health Summit Regional Meeting 2025. This requires knowledge of current affairs and upcoming international events related to health.
Correct Option: C) New Delhi